İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

99

Transcript of İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

Page 1: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

���������� ���� �������

����������� ��!#"%$&$&')(*� +,.-/,.-02143657148:9.;�<>=@?A;6<65B-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F,.-HGI-02143657148:9.;�<>=CJ#K.8MLNLOK.;P-Q-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- R

ST�VUV'XWZY[� Y[�#$&'X\�]_^`'X�%'a��\�] �cbGI-/,.-ed�fhgiK[<6Kj0lkIm6n:oM1�8:giKp-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-q,srGI-HGI-ed�fhgiK[<6Kutwvjm6n:oM1�8:giKp-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-q,�FGI-yx>-ez{K.;>LO8ML|gi=�;6148M5�-D-D-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}GI,GI-~r6-ez{K.;&�7=4�O�7=4;61�8�5 -D-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}GI,GI-H�I-02LNgi57�68:nM8M<657< -D-D-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}G.GGI-y�>-Q�#=4K.�N=4vj5u<>=C��5�k�-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}G�r

�T�V���&����!���]_��]4�2Y[$7��� \��)Y[] S��x>-/,.-��K.�NvB36n�5�148M97;�-D-D-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}G.�x>-HGI-J#K.nM8:;6K.vu8MK.L�<6=Cz{�6=4�&�[LO�6=h��-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}G7F

x>-HGI-/,.-���8:�)=��O=�;6148M57148:9.;�-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- x&Gx>-yx>-Q¡¢m6�OKskI8Mvj57148:9.;¤£E57nM=��O¥c8M;¦-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- x7rx>-~r6-ed�fhgiK[<6Ku<6=E�§573 -D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- x.Rx>-H�I-ed�fhgiK[<6Ku<6=Cz{K.nMK[1s571�8M9.; -D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- x.R

,

Page 2: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¨����2©4Y[ª«�%�)��] ¨#�r6-/,.-��8/�)=4�N=4;61�8�57L¬�§8:;68:gi57L -D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r6,

r6-/,.-/,.-�02143�571�8M9.;¤<6=C?A;6<�5 -C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r6,r6-/,.-HGI-�02143�571�8M9.;¤<6=DJ%K78MLOLNK.; -D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r.rr6-/,.-yx>-�d�f�gOKI<>Kj<6=C®.571�K.�68E-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r&�r6-/,.-~r6-�d�f�gOKI<>Kj<6=C£E5736LNL|¯|°[=48M<>=4n±-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­rc�r6-/,.-H�I-�d�f�gOKI<>Kj<6=C°IK7�6�O=D²³=4n�5�´O57148:9.;¤°I361�=4LO8/��5¶µ·°6?�²�¸¹-D-D-C-D-C-­rc�

r6-HGI-ed�fhgiK[<6Kj0ºLNm�=�1�gO�i57n�<6=Cz{K.n:KI1457148:9.; -D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r&Rr6-HGI-/,.-�02143�571�8M9.;¤<6=DJ%K78MLOLNK.; -D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-­r&R

r6-yx>-ed�fhgiK[<6Kj0ºLNm�=�1�gO�i57n��%573 -D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}�I,r6-yx>-/,.-�02143�571�8M9.;¤<6=D»{36�N�.=4�NL¼-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}�I,r6-yx>-HGI-½twvjm>nM=4vu=4;&gi57148:9.;�<6=4n�vuf�gOKI<>Ku�%573e-D-C-D-C-D-E-C-D-D-C-D-C-}�.G

¾T��¿ÀY[]s"#�)�s\�!#�*]_Á¶"#ª«�c��'·$.�*] ¾�Â�I-/,.-02143657148:9.;�<>=@?A;6<65B-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}�7F

�I-/,.-/,.-�²¬=�LO36n/gÃ57<>K.L¬Ä¢36vuf4�N8M14K7L³57m6nM8:1s57;><6K_��8:�)=��O=�;6148M57L³�§8M;>8:gÃ5�LQ-C-}�7F�I-HGI-02143657148:9.;�<>=CJ#K.8MLNLOK.; -C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- �.x

�I-HGI-/,.-�²¬=�LO36n/gÃ57<>K.L¬Ä¢36vuf4�N8M14K7L³57m6nM8:1s57;><6K_��8:�)=��O=�;6148M57L³�§8M;>8:gÃ5�LQ-C- �.x�I-HGI-HGI-�²¬=�LO36n/gÃ57<>K.L�;c36vuf4�N8M1�K.L¤57m6nM8:1s57;><6Ke=4nCd�f�gOKI<>KÅ<6=«z{K.n:K�¯

1s571�8M9.;¤02LOm*=41hgi�O57n%-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}�.��I-HGI-yx>-�z{K.vum�57�i5�148M97;Æ=�;&gi�O=_n:K.LDd�f�gOKI<>K.LCz{K.nMK[1s571�8M9.;Ç02LNm�=�1�gi�O57n

��<6=D��8/�)=4�O=�;6148M57L¬�§8M;>8:gÃ5�LÈm�57�O5_nM5B=41�3�57148:9.;¤<6=DJ%K.8:LOLOK7;¦-C-}�.��I-yx>-02143657148:9.;�<>=C»�36�O�.=��OL -C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C-}�7R

É �*Á�����ª«\�]_!�Y¶ÊDY[$7������Y[]@ËÅ��\>����'X$.YI] Â��¡@-/,.-³Ä¬K.�Ovj57L`<6=�̺=41hgiK.�N=4L�-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F.x¡@-HGI-³Ä¬K.�Ovj57L`<6=E36;�5jd¶5�gO�O8:ÍÎ-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F7r¡@-yx>-³Ä¬K.�Ovj57L¬z{K.vum�5�gO8M�6n:=4L­-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F7r

G

Page 3: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¡@-~r6-³Ä¬K.�Ovj57L`<6=E36;�5jd¶5�gO�O8:ÍD°I36�*K.�O<68:;�57<�5Æ-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F&�¡@-H�I-¬d�fhgiK[<6Ku<6=D²¬36;>�.=h¯wÏE3IgOgÃ5Ð-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F.�¡@-y�>-³��5u0lkIm�5�;6LO8:9.;¶�¢8MLO1��O=hgÃ5�<6=D��K.36�O8:=4��-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- F&�¡@-H�I-¬°IK7nM361�8M9.;À<>=�36;�°I8ML|gi=4vj5@�#�O8:<68�57�7K.;�57n*14K.;�nM5Bm6�O8:vj=���Ñ�nM5@<68:LNgi8:;&gÃ5

<6=E1�=4�OKÅ-C-D-D-D-D-D-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- R.Ò¡C-Ó�I-/,.-u¡�n:�.K.�O8/givuK6Ô#-C-D-C-D-D-C-D-E-C-D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- R7r

¡@-yF>-¬d�fhgiK[<6Ku<6=D0ºn:8Mvu8M;�5�148M97;Õ<6=C£E5736LOL -D-C-D-D-C-D-C-D-E-C-D-D-C-D-C- R7r

Page 4: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

Ö ������ ���e��× �T����ؤÙÛÚ

��nM=��&57�¢5Õ=4L|gÃ5Àm�57�|gi=D�)36=BvB3>�¤<68/�)oM1�8MnZÜ>m�=��OKÀvjÝ7L�<>8:�)oM1�8Mn�=�LA=4;61�K.;&gi�O57�An�5�L¢m�5�n/¯57�>�i57L¬14K.�N�O=�1�gÃ5�L¢m�5��i5_=hkIm6�O=�Li57��=4n�m6�OK��)36;6<6Kj57�.�O57<6=41�8Mvu8M=�;cgOKu5Õ��t|?E°V�¤5_giK[<�57LnM57L`m�=��OLNK.;�57L³Þc36=E=4L|gi3>�[8M=��OK.;�14K.;>vj8:�.KB<636�O57;&gi=D=4L|giK.L³148:;614K_nM57�O�.K7L³57ß6K.L�-

J236=4Lº�68:=4;TÜ7m6�O8:vj=��OKAÞc368:=4�OKE5��.�i57<>=414=��º5E��t|?E°�=�;&gi�O=³vB3>1Ã��57L2K7gO�i57L21�K.Li57L2m*K.���5���=��Ovu=¢m*=4�Nvj8/gi8M<>KEgO=4�Nvj8:;�57�º=4L|gi=¢gO�i57�65 ´NK6-c¡àvu8��X57vj8:nM8M5>Ô�vu8�vÕ5�vÕÝD£�8MLO=�n�5>Ü&vu8m�5�m�Ýu£E8:nM<�57�N<6K6ÜIvj8:LÈ�6=4�NvÕ5�;6K.L¬Ä�57<>8�5>ÜI�`�6=4n:vÕ5>Ü>twvjvj57;c36=4n·Ü>ÌA8:K.nM=hgÃ5C��Ì�8M1hgiK.�N8�5m*K.�ºgiK[<�57LáLO36L�=4;6LN=4ß�57;>Ís57L�Ü6LO36L�14K.;>LO=â´NK7L4Ü>m*K.�á��57�*=4�áLOK.m*K.�|gÃ57<6KCvj8�vj57n�1s5��i571hgi=4�LNK.�6�O=�gOKI<6KÇ<63>�i57;&gi=Vn:K.L�ã6n:gO8MvuK.L@vu=4LN=4L�Üám*K.�BgOKI<6KÇ=�n`1s57�N8Mß6K��e57m*K��.KÇÞc36=¤vj=�6�N8M;6<657�OK.;V1�3�57;6<6KuvjÝ7L³nMK�;6=�14=4LN8:gi57��5>Ü�m*K.�`;6Kum�=��Ovu8:gi8:�{Þc36=Dvj=E�N8M;6<68:=4�O5B=4;¤nMK.LvuK.vu=4;&giK.L`vjÝ7L³<68/Ñ�148:nM=4L�-�Ä¢K_gi=�;6�.K6Ü�;68�gO=4;6<6�N=CnM5�vÕ57;>=4�i5_<6=C57�.�O57<6=41�=4�NnM=4L`gOKI<6Kn:K�Þ[3>=C��5�;¤�>=41Ã�6Kjm�K.�`vu8Z-

z{K.vuK¹K7n:�[8M<�5��Ovu=�<6=�vu8ML_vu=â´NK.�N=4Lj57vu8M�.K.Lu°I57�i578`�e¡¢;cgOK.;68:K«5ÆÞ[3>8M=4;>=4LÕn:=4L57�7�i57<6=�Í41�K«m*K.�j�657��=��Ovu=��>�O8M;><�57<6KÆLO3ä5�vj8:LNgÃ5�<å<>36�i57;&gO=¶=4L|gi=¶gO8M=�vjm*K6Ülm*K.�OÞc36==�nMnMK7LBgÃ57vB�>8Mf4;�)K.�NvÕ5�;åm657�NgO=�<6=V=�LNgi=¤gi�O57��5 ´NK6Ü2m*K.�_��57�*=4�_1�K.vjm657�NgO8M<6K�1�K.;6vu8/¯�.Kvu8MLu�)�O36L|gi�O57148:K.;6=4L�1�3�57;6<>KånM57LÕ1�K.Li5�L¤;>Kvj=æLO57nMoM57;��68:=4;TÜ�m�K7�À=4LN14361Ã��5��Ovu=1�3�57;6<6K¤;6=�14=4LN8:gi57��5VÞ[3>=un:KV�68M1�8M=4�O57;TÜ�m�K.�ELNK.m�K�gÃ57��vu8lvj57n§1457�i571hgi=��E�æLO=4�7368M�Nvj=<6=�vjK7LNgi�O57;6<6K_LN3¶57m*K��.K_��m*K.�ÈgÃ57;&gi57L³14K.LO57L³Þ[3>=C1�K.vjm657�NgO8MvuK.L4-

ç 368M=��OKä5��.�i57<>=414=���=4LNm�=�148�5�nMvu=4;&gi=«57nB����-�²³8M1s5��O<6Kä»�=414=��O�N8MnB»{57�O1�=4;�57L�m*K.���5���=��Ovu=jgi=�;68M<>K¶gÃ57;&gi5¶m�571�8M=4;>148�5IÜ#m*K.�NÞ[3>=ÀLOfÀÞc36=ÀgO�i57�65 ´O57�@1�K.;6vu8M�.K¤;6Kæ=�LB�XÝ�¯1�8MnZÜ�m*K.�¬��57�*=4�¢LO8M<>KÕ36;æ=èkI14=4n:=4;&gi=_5�LO=4LNK.�4Ü*m636=�L�giK[<�57L¢LO36L¢=4;6LN=4ß�5�;6Ís57LA�VgOKI<6Kj=4n57m*K��.KÀÞc36=Cvj=C�6�O8:;6<69j=4L�vjÝ7L¢<6=@n:KÕÞc36=@3>;¶gi=�LO8ML|gÃ5Õm>36=4<6=@=4LOm*=4�O57�4-PJ%=��OKÀLNK.�6�O=

Page 5: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

gOKI<6K>Ü6�.�i5�148�5�L¬m*K.�³��57�*=4�Nvj=E�6�N8M;6<657<6K_LO3¶57vj8:LNgi57<���14K.;&�7=4�NgO8M�NLO=@=�;¶36;>Kj<6=Dvu8MLvu=â´NK.�N=4L³57vu8M�7K.L4Ü6d¶é�ê�zÈ颡�°�£D²³¡¬z{tw¡�°*-

�%57vB�68:f4;u57�.�O57<6=4Í�14KC57n�����-[£E=��i57�N<6KE�`8M;6K[14K��B5DnM5���8:17-&J%5�gi�N8M1�8�5Ed¶57;>�O8MÞc36=�Í7Üm*K.�OÞc36=@LO8§=�nMn:K.L¢;>KÀvj=C�c36�68M=��i57;�<657<6KÀnM5jK.m*K.�NgO36;68:<�57<æ<6=@8M;6�.�N=4LO57��5À=�LNgi5��*5�1h¯36n/gÃ57<PÜ�;6K=�LNgi57�OoM5«=4LN14�O8:�68M=�;6<6K=�LNgÃ5�L�nMoM;>=s57L�-È£��i571�8�57Ljm�K7�Õ��57�*=4�Nvj=¶m�=��Ovu8:gi8:<6K<6=�vjK7LNgi�O57�EÞc36=jn:KVã6;68M1�K�Þc36=uLO=j;6=41�=4LN8:gÃ5Vm�57�O5VgO=4�Nvj8:;�57��=4L|gÃ5¤1s57�N�O=4�O5¤=�L@=�n2=4L|¯�)36=��OÍ�Kj�V=4n�<6=�LO=�KÕ<6=DnMK.�7�i57�NnMK6ÜILO8M;¤8Mvum*K.�Ngi57�`LO=��¢m6�NKI1�=4<6=�;&gi=@<>=C36;���571Ã�68:nMn:=4�i5�giK<6=Õ;6K¶gÃ5�;Q�636=�;åm>�O=4L|gi8:�.8MK6-%£��i571�8�57LCJ§5�gw�«m*K.�CgOKI<>Kæ=4n{57m�K��7K�Þ[3>=ÀLO8:=4vum6�O=jvj=�6�N8M;6<657LNgO=7Ü>�7�i571�8�57LÈm*K.�ÈLN=4�`36;65B<6=�n�57L{m�K[1s57L{m�=��OLOK7;�57L`Þc36=ELO=�m6�N=4K[1436m65Bm�K.�{nMK.L57n:36vu;6K.LÈ<6=E=�LNgÃ5u��57143>n:gÃ5�<V5736;6Þc36=D;6K_nMK_�O=�14K.;>K.Í41457;T-

¡�giK[<6K.L³nMK7L¬m>�OK7�)=�LOK.�N=4L¬Þ[3>=D�)K.�NvÕ5��OK.;�m�5��Ngi=D<6=Evu8�57m6�O=�;6<68:Ís5 ´N=�Ô���8:17-�J%5�gi�N8/¯1�8�5>ÜI���4-6£E=��i57�N<6KB�`8:;6KI1�K6Ü[���4-6®7K&57Þc368:;TÜ6����-6��368:L�Ìá57n:=4�OK>ÜI�¢�4-�d�57�O8:K_z{=4LO57�4Ü>���4-²³8M1457�O<6Kum*K.�¢vu=4;>148MK7;�57�¢57nM�.3>;6K.L4-�¡¢nl°I=�14�N=�gÃ5��O8MKj¡`1s57<>f4vu8M14KÕ��8M17-�z{=�nM=4�N8M;6K>Ü*m*K.�gOKI<6K_=�n�57m�K��7Ku<>36�i57;&gO=CnM5B1s57�N�O=4�O5_�Õ<636�O57;&gi=D=4n�m6�NKI1�=4LOKu<6=�gi8/gi36nM57148:9.;T-

¡��7�i57<6=�Í41�K_gi57vB�68:f4;V57nP���4-6���O57;61�8MLO1�Ku°I8M<><�57�NgO��5_£�ã6Í4vj57;TÜIm�K.�{��57�*=4�`571�=4mI¯gi57<6KÀLN=4��LO8:;6K[<�57n#<6=�=4L|gÃ5Õgi=�LO8:L4ÜT5�LOol1�K.vjKÕm�K.�¢n�57LA5714=��NgÃ5�<�57LAK.�>LO=4�|��57148:K.;6=4L�Þc36=vu=�m�=��Ovu8:gO8M=4�NK.;Àvu=â´NK.�O57�È=�n�14K.;&gi=�;68M<>Kj<6=E=�LNgi=Egi�O57��5 ´NK>-

J%K7�Õ36n/gi8MvuK6Ülm*=4�NK«;6K«m�K.�u=4LNKQvu=4;6K.Lu8Mvum*K.�Ngi57;&gi=7ܺ5��.�i57<>=4Í41�K57n³tw;6L|gi8:gO3>giK<6=C�§o:LO8M145B��d¶5�gi=�vÕÝ�gi8M1457L`=4;�=4LNm�=�148�5�n%5_n�5_��8M1�-*z{n�573><68�5_»È57�N�O8:�&5>Ü6m*K.�³��57�*=4�Nvj=m6�NK.m*K.�O1�8MK.;�5�<6K�36;¤=4LOm657148:Kum657�i5��O=457nM8:Ís57�{vj8PgO�i57�65 ´NK6-

¡�giK[<6K.L`vB3>1Ã��57L4Ü�m*=4�NKjv@361Ã��57L³�.�O57148M57L4-

Page 6: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ë�ì�í�î¬ïðì*ñ

02;¹=�LNgi5¤gi=�LO8:LDLN=_��5714=_36;65¤=èk>m*K.LN8M148:9.;Ç<6=un�57L��.=�;6=4�O57nM8:<�57<6=�L�<6=_<6K.LE<6=�n2nMK.Lvuf�gOKI<6K7Lá;c36vuf4�N8M1�K.L�vjÝ7Lº3>gO8MnM8:Ís57<>K.Llm�57�O5C�N=4LOK7n:�.=���=�143�5�148M97;6=4L�<68:�)=��O=4;>148�5�nM=4L2m657�|¯1�8�57n:=4L4Ô[=4nP<6=@òEó ôsõhöiõh÷PøhóXù�ú³ûºóa÷�óaüýù�úAµ·d¶�A�2¸á�ÕnMK.L¬þ¹ÿhü�������ú��ºú���õÃøhüXöOù�MõÃúAµ·d¶0á¸h-6��K.Lm6�N8M;61�8Mm68:K.LA��Ý�LO8M1�K.LE=hkIm636=�LNgOK.L4Ü%LO=_8Mn:36LNgO�i57;Ç14K7;¹=w´N=4vum6nMK.LE<6=_=�143�571�8MK.;>=4LD8Mvum�K.�â¯gi57;&gi=4Lj=4;�nM5Ç�)o:LO8:1s5>Ô2nM5Æ=�143�571�8M9.;Å<6=¶1457nMK7�4Ü2n�5Ç=41�3�571�8M9.;Å<6=¶K.;><�5� ��.ܺnM5Æ<6=»{3>�O�.=��OL4Ü6�Àn�5B<6=CJ%K.8:LOLNK.;T-�02LNgi5uã6n/gi8:vÕ5DLO=D�O=4LN36=4n/�.=D14K.;�57vB�657L³57m6�NK�kI8:vÕ571�8MK.;>=4Lµ·d¶�A�Å�¶d¶0ḳ�¤LN=B�O=457nM8:Ís5_36;�5Õ1�K.vjm657�i571�8M9.;¤<6=Cn�5��.=4n:KI1�8M<�5�<¶<6=B1�K.;&�.=��O�.=�;6148M5<6=D57v@��K.L`vuf�gOKI<>K.L4-

Page 7: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� �����*Ø����TÙ �� ��ؤ��Ù �����å�e�����

02nT�6K7vB�6�O=�<>=4LO<>=CLN36L³8M;68:148MK7L4Ü[��5Bgi�i5�gÃ57<6K�<>=D=�;614K7;cgO�i57�³36;65_=èkIm6nM8:1s571�8M9.;V5@gOKI<6K�)=�;69.vu=4;6KjÞ[3>=@LO=Cm6�O=�LO=�;cgi5À5jLN3�57n:�O=4<>=4<6K.��-�J2�O8Mvu=4�NK6Ü�<68:=4�OK7;¶=hkIm6n:8M1457148:K.;6=4L¬=4v�¯m6o:�O8:1s57L�Ü�m*=4�NKÀ14K.;�=�nlm�57LNK�<6=4n#gi8:=4vum�Kjf4LNgi57LA�)36=��OK.;�8:;6LO3>Ñ6148M=�;&gi=4L�ÜTm*K.�AnMKjÞc36=�=4n�6K7vB�6�O=uLO=j�I8:K¶=�;Qn�5V;6=41�=4LN8M<�57<«<6=À�636LN1s57�CK7gi�NK.LDvjfhgiK[<6K.LDÞ[3>=ÀnM=um�=��Ovu8:gO8M=4�O57;K.�Igi=4;>=4�³=hkIm6n:8M1s5�148MK7;6=4L`vjÝ7L³571�=4�Ngi57<�57L�-��=C=4L|gÃ5jvj57;6=��i5uLO36�N�.8M9un�5��)K.�Ov@36n�571�8M9.;Vvj5�gi=�vÕÝ�gi8M145>Ü>Þc36=C=4;ænM5uvj5s�.K.�No�5_<6=DnMK.Lm6�NK.�6n:=4vj57LCÞc36=�<>=4LO1��O8:��=�ܧ8M;&�7K.nM361��i5¶1457vB�68:K.LD14K.;�N=4LOm*=41hgiKÇ5æ36;65¶KævjÝ7LC��57�N8/¯57�>nM=4L{8M;6<6=�m�=�;6<68:=4;&gi=�L4Ü�nMK�1�3�57nT;>K.L`14K.;><63614=D5B=�143�571�8MK.;>=4L³<68:�)=��O=�;6148M57nM=�L4ÜI�.5�LO=s5�;K.�N<68M;657�O8M57LÕKm�5��O148M57nM=�L4-{Ä¢K.LNK7gi�NK.LÀgi�O57��5 ´O57�N=4vuK.LÀ1�K.; =4143657148:K.;6=4LV<68/�)=4�N=4;61�8�57n:=4Lm�5��O148M57nM=�L¢<>=@LO=��.36;6<>K�K.�O<6=�;TÜ*n�57L¬143�5�nM=4L�Ü�=�;�<6K.L¢<68Mvu=4;6LN8MK.;>=4L¬LO=B=�LO14�N8M�*=4;æ=�;�n�5�)K.�NvÕ5

�����! ��" � ��# ���! �$"%�'& �)( ���! �'& � ��* �% �$" ��+ �% �'& �-, �/.1032 µ|,.-/, ¸<6K7;6<6= � Ü4#�Ü5( Ü5+.Ü[�1.jm>36=4<6=�;�LO=����)36;61�8MK.;>=4L{<>=An�57Lº��57�O8M57�6n:=4Lº8M;6<>=4m*=4;6<68:=4;&gi=�L " = & Ü�D<6=�n�5`�75��O8�5��6nM=º<6=�m�=�;6<68M=�;&gi= 768"�9:&<; -���8:1Ã��5¢=�143�5�148M97;BLO=�14n�5�LO8:Ñ61s5¢<>={n�5³LO8:�.368:=4;&gi=vj57;6=��i5>Ô�LN8=# �?>/@ � ( AB2j5jnM5u=�143�571�8M9.;�LO=CnM=EnMnM57vÕ5B=�nMo:m>gi8:1s5>Ü6LN8=# �C>)@ � (EDF2jLO=n:=�nMn�5�vÕ5D�68Mm*=4�N��9.n:8M145C��LO8�# �G>H@ � (I0B2�LO=DnM=�n:n�57vj5Cm657�i57�*9.n:8M1s5I-

Page 8: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

02;61�K.;&gi�i5��D3>;�5¤LNK.nM3>148M97;¹5V36;�5V=41�3�571�8M9.;ÆK¤5¤36;ÆLO8:LNgO=4vj5¤<6=_=�143�571�8MK.;>=4LD<68:�)=��|¯=�;6148M57n�m657�O1�8�57n:=4L4Ü�;>Kæ=4LB3>;�5�gÃ5��O=s5�LO=4;>148Mn:n�5>ÜT�¹vjÝ7LC1�3�57;6<>Kæn�57LD=41�3�571�8MK.;6=�LBLOK7;;6K�¯wnM8:;6=s5�nM=4L�-*0ºL�148M=��NgOK6Ü�Þc36=�=�;Ç57nM�736;�57L¢LO8:gO3�571�8MK.;6=�L4Ü�=4L�m*K.LO8:�6nM=D=4;61�K.;&gi�O57��LOK.n:3I¯1�8MK.;6=�LÈ=hk>571hgÃ57L³m*K.�{vuf�gOKI<>K.L`57;�5�nMo:gO8M1�K.L��7=4;6=��i57vu=4;&gi=�LNK7Ñ�L|gi8M1457<6K.L�Ü>m*=4�NKu=�;�vB3I¯1Ã�6K7L¬K�gi�OK7L`1s57LNK.L4Ü�=�L¬m>�i571hgi8M1457vu=4;&gi=A8:vjm*K.LN8M�6n:=¢�N=4LOK7n:�.=��OnM57L³57;�57n:o:gO8M1s5�vj=�;cgO=7-KJ�LO8nM57L¢1�K.;6<68:148:K.;6=4L¢<6=@�)�NK.;&gi=4�O5ÕLOK7;�14K.vum6nM8:1s57<657L4Ü�nM5jLO8/gi3�5�148M97;¶m636=�<6=���571�=4�NLO=_57ã6;vjÝ7L`14K7vjm6n:=â´O5>-02Lám�K.�º=�LNgiK>Ü[Þc36=¢=4n��6K7vB�6�O=¬LO=³�[8MKD=4;jn�5�;6=41�=4LN8M<�57<j<6=¬<6=4LO57�O�NK.nMnM57�2�6=4�N�i57vu8M=�;&gÃ57LvjÝ7Ll=hÑ�1s571�=4L�Ü&Þ[3>=`nM={m*=4�Nvj8/gi8:=4�i5�;@=4;>14K.;&gi�O57�2LOK7nM361�8MK.;6=�L25A<68:1Ã��57L2=41�3�571�8MK.;6=�L4-.¡¢LOoLN36�O�.8:=4�NK.;j��57�N8MK.LávjfhgiK[<6K.L�;[3>vjf��O8M1�K.L4-c�#�O=�LÈ<6=An:K.L�vjÝ7Lá14K.;>KI1�8M<6K.Lá�uvÕÝ7L{57vum6nM8 ¯57vu=4;&gO=¬3Igi8Mn:8MÍ457<6K.L%LOK.;TÔcvuf�gOKI<6K7L2<6=³<68:�)=��O=4;>148�5�LºÑ6;68:gi57L4Ü.<>=¢=�nM=4vu=4;&gOKEÑ6;68:gOKE�BnMK.L=�LOm*=41�gO�i57n:=4L�-02nÈm6�NK.m*9.LO8/giKæ<6=�=�LNgO=VgO�i57�65 ´NK6ܧ=�L�=hkIm*K.;6=4�_�6�N=��7=4vu=4;&gi=VnMK7LB=4n:=4vu=4;&giK.L��6Ý7LO8:14K.L<6=Ç<6K7L�vjfhgiK[<6K.L�;c36vuf4�O8:14K.L�ÔÈ=4n@þÆÿhü������)�[õ¶ò�ó ôsõhöÃõh÷PøhóXù�ú¤ûºóa÷�óaüZù�úu�änMK.L¤þ¹ÿhü��ML����úN�áú���õÃøhüXöNù�MõÃúhÜT57m>nM8M1457<6K.L¢14K.vuKÀ�&5�LN=�vj=�;6148:K.;69�57;&gO=4�O8:K.�Ovu=4;&gO=_5À=4143657148:K.;6=4L<68/�)=4�N=4;61�8�57n:=4L�m�57�N148�5�nM=4L�-�ê¬gi8:nM8:Ís57;6<>K¬nM5¢=�143�571�8M9.;C<6=`J#K.8MLNLOK.;Bm657�i5¬��5714=��§36;65�14K.v�¯m�5��i571�8M9.;�<6=EnM5��7=4n:KI1�8M<�57<�1�K.;&�.=��O�.=�;6148M5j=4;&gO�O=DnMK.LÈ<>K.L³vjfhgiK[<6K.L4-

OEP�OEP QFRTSVUNRCWYXNZ []\ ^_ZV[VU��5Õ=�143�571�8M9.;«,.-ÓGu=4LAnM5j�)K.�Ovj5j�7=4;6=��i57n#<6=BnM5Õ=41�3�57148:9.;æÞc36=B<>=4LO1��O8:��=�5ÀnM57L¬K.;6<�57L=�;�<68:vj=�;6LO8:9.; ,a`B,.Ôcb�n�5�LO=��.36;6<65«<6=4�N8:��57<�5Ç<6=�n¬<>=4LOm>n�57Í457vj8:=4;&giKÆ<6=¤n�5ÇK.;6<�5 &�N=4LOm*=41hgiK�5Dn�5E14K[K.�O<>=4;�57<65 " =4L�8M�.3657n�57n�8M;&�.=��OLNKB<6=�nM5E�7=4n:KI1�8M<�57<u<6=¢m6�OK.m657�&571�8M9.;<6=@n�5ÕK.;><�5�57n§1�3�57<6�O57<6K�m*K.�AnM5ÕLO=��.36;6<�5�<6=��O8/�75�<�5Õ<6=�nl<6=4LNm6n�5�Ís57vu8M=�;cgOKÀ<6=B=�LNgi51�K.;��N=4LNm�=�1�giKj57nPgO8M=�vjm*K6-db

� � &��" � > e � � � &�$f � 032 µ|,.-HG.¸

F

Page 9: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

°>5���=�vjK.LjÞc36=æn�5Æ�)K.�Ovj5¹�.=�;6=4�O57n¬<6=æn�5¹LOK.n:36148:9.;�<6=¶K.;6<�57Lj�I8M5 ´N=4�O57LÕm657�i5«=�LNgi5=�143�571�8M9.;j=4L &�6g"�9hfi; 0j, 6g"ck e fi; m*=4�OKCn�5��)K.�Ovj5�=4LNm�=�148:Ñ61s5@<6=�m�=�;6<6=¢<6=¢nM57Lá14K.;6<>8/¯1�8MK.;6=�LC8:;68M1�8�57n:=4L��Ç<>=u�)�NK.;&gi=4�O5>-#02LNgÃ5¤=4LCn�5�=41�3�57148:9.;«1�K.;«nM5�Þ[3>=ÕnM8:<6=s5��O=4vuK.L�=�;=�LNgÃ5�gi=�LO8:L4-J#=4�OK�nM5C=�143�571�8M9.;�<>=AK.;6<�5B;>KBLOK.n:KBLO3>�O�.=A=�;�n�5Dvj=�1sÝ7;68:1s5@LN8M;6KCgÃ57vB�>8Mf4;Õ=4;�K7gO�i57LLN8:gi3657148:K.;6=4L�Üági57n¬=4LÀ=�n�1s5�LOKQ<6=¶=4nM=�1�gO�OK.vj57�.;6=hgi8:LOvuK6Ü2LO8¢36Li57vuK.Ljn�57Lj=4143657148:K.;6=4L=�nM=41hgi�NK.vÕ5��.;6f�gO8M1457L³Ü[K7�>gi=�;6<6�O=�vjK7L{36;�5C=4143657148:9.;Õ<6=¢n�5Dvj8:LOvj5��)K7�Ovj5DÞc36=¢n�5CÞc36=<6=�LO1��O8M�*=æ5Æn�57L_K.;><�57LÕvu=414Ý7;68:1s57L�ÜáLNK.nMKÆÞc36=¶=�n¢<>=4LOm>n�57Í457vj8:=4;&giK & LO=�LO36L|gi8/gi3>�7=m*K.�ÈnM5@8M;&gi=�;6LO8:<�57<�<6=�nT1s5�vjm*K�=4n:f41�gO�O8:14K@�ÕvÕ57�7;6f�gO8M14K>->02;V=4L|gi=�1s5�LOK_n�5C�7=4n:KI1�8M<�57<<6=um6�NK.m�57�&5�148M97;Æ<>=jnM57LEK.;6<�57LD=4n:=41hgi�OK7vÕ57�7;6f�gO8M1s5�L�Þ[3>=u�[8�5�´O57;¹=�;«=�nº��571�oMK¤=4LDn�5�7=4nMK[148:<�57<V<6=En�5Bn:36Í ( -�l/,!mOEP:n1P QFRTSVUNRCWYXNZ []\porqEW�s�stquZvPê¬;�5C<6=¢n�57Lá3>gO8MnM8:<�57<6=�Lº<6=�nM5D=�143�571�8M9.;j<6=AJ#K.8MLNLOK.;ÕLO=A5�m6�O=�148�5EgÃ57;&gOKB=4;Õ0ºn:=41hgi�OK[<I¯8:;�Ý7vu8M1s5IÜP14K7vjK¤=4;e£��i5s�[8:gi57148:9.;TÜTm>36=4LCf4LNgi5�<6=�LO1��O8M�*=un�5��75��O8�5�148M97;Æ<>=jm*K7gi=�;6148M57n=�nMf41hgi�N8M14KuKu�.�i5s�[8:gi57148:K.;�57nT=�;æ=4n�8:;cgO=4�N8MK.�`<6=C<68ML|gi�N8M�6361�8MK.;>=4L³=4n:f41�gO�O8:1s57L³KÕ<>=Cvj57Li5µ)=417-/,.-yx&¸h- w �yx 0 >{z�| µ|,.-yx&¸f�LNgÃ5Ç=4Lun�5�n:n�57vj57<�5}�¬ø!~>ù[øhó���÷r�[õ�����óMúÃú���÷�Ül<>K.;6<6= z 0 ���� m�57�i5Ç=4n`1457LOKÆ<6=¤n�5=�nM=41hgi�NKI<68:;�Ý7vu148M5D� z 0 @��� m�57�i5�=�n�1s57LNKj<6=C£��i5s�[8:gi57148:9.;T-��5�LOK.n:36148:9.;�m�57�O5unM5B=41�3�57148:9.;Ç,.-yxum�5��i5 | =�;�1�8M=4�|gÃ5B�N=4�.8:K.;�Ì�=�LNgiÝu<�5�<�5�m�K.� lyGMm

x 6��; 0 @�t�h� �'� |%68�M��;� ��>�� � � * e � µ|,.-~rc¸

R

Page 10: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� �����*Ø����TÙ �� ���º���¤�����e����Ú � �����*Ø���Ú

02;�=�LNgO=æ1s57m>o:gi3>nMKÇ<6=4LN14�N8M�68:�O=4vuK.Lj5Æ�7�i57;6<>=4LÕ�O57LO�7K.Lj=�nAd¤f�giK[<6K¹<6=æ��8/�)=4�N=4;61�8�57L�§8:;68:gi57L4Ül57LOo³14K7vjKÇ��57�O8:K.L_14K.;61�=4m>gOK.LÕÞc36=¤f4L|gi=�8M;&�.K.n:3614�O5>ÜlgÃ5�nM=4Lu14K7vjK6Ô{þÆÿhü�������=�Y�%���Zøhóaü����§þÆÿhü������N�y���%���Zøhóaü����I�C��÷>úhóMúhüâõh÷PøhóXùM� �ºúhüýùK¡4óg� ó��&ùK�����C��÷�¢7õhö�£[õh÷PøhóXùV¤�¥�õ¦�MLöiõ!�uù��[õ¨§Tù©�Yª�-02;¶òEó ôsõhöiõh÷PøhóXù�ú¬ûºóa÷�óaüýù�úhÜ[n�5�LÈ=41�3�57148:K.;6=�LÈ<68:�)=��O=�;6148M57nM=�L{LO=��N=4LN36=4n/�.=4;¤=4;�1�8M=��NgÃ5B�N=h¯�.8:K.;¬« 5714K�gÃ57<�5�m�K.�D36;�5¤�)�OK.;&gi=��i5>Ü#<6K.;><6=­« m*=4�|gi=�;6=41�=V57nº=4LOm657148:Kæ=4;Q=�n�1�3�57nLN=@<6=�LO14�N8M�*=4;�nM57L¢=�143�5�148MK7;6=4LA<>8:�)=4�N=4;61�8�57n:=4L�Ü�µ)=4L�<>=4148:�4Ü*LO8#n�5u=4143657148:9.;æ<6=�m�=�;6<6=B<6=<6K7Lu��57�N8�57�6n:=4L�Üá=�;&giK.;61�=4LÀ=�LNgi57�O=�vjK.Lu=4;�36;�=4LNm�571�8MK«<>=¶<6K.Lu<68:vj=�;6LO8:K.;6=4L�ܺm�57�i5;6K7LOK7gO�OK.L³<68:1Ã�6K_=4LOm657148:KuLN=4�O51« � ¸è-02LNgO=�vjfhgiK[<6KÀ1�K.;6LO8:LNgO=7Ü�gi57n§1�K.vuKÀLO3�;6K7vB�6�O=@nMKÀ<68:14=�ÜP=4;Ç5�m6�OKskI8Mvj57��m*K.�¢vj=�<68MK<6=�n�gO=4K.�N=4vj5@<6=A�%5s�In:K.�ÈnM57L{<6=��O8/�75�<�57L{8:;&�.K.n:3614�O57<�57LÈ=�;�n�5D=4143657148:9.;�<68:�)=��O=�;6148M57nP5nM5CÞc36=�LO=�Þ[3>8M=4�N=A<�57�{LOK.n:36148:9.;T-[��=�=4L|gÃ5Bvj57;6=��i5@K.�Igi=4;><6�O=�vjK.L{36;�5B=hkIm6�N=4LN8M9.;�=�;gOf4�Ovu8M;>K.LE<6=Õ<68:1Ã��57L@57m6�OKskI8Mvj571�8MK.;6=�L4Ü��¹57LOoá;6K.LDLO=45ævÕÝ�LCLN=4;61�8MnMn:K�=�;614K7;cgO�i57�Cn�5LNK.nM361�8M9.;Õ=4;V1s57<65_m>36;&giK_<6=�n�5C�O=4�78M9.;�«�-6��K_5�;cgO=4�N8MK.�ÈLN=�<6=4LN14�N8M�*=�<6=�36;�5B�)K.�NvÕ5vjÝ7L³57vum6nM8M5@5_14K.;&gO8M;c3�571�8M9.;T-°I=45V®uÜT3>;�5À�)36;6148:9.;ægÃ5�nlÞc36=_f4LNgi5V�¶LO36LE<6=��O8/�75�<�57L�LO=457;Æ36;68:��57n:3�57<�5�L4Ü�Ñ6;68:gi57L��

,�Ò

Page 11: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

1�K.;&gi8M;c3�5�LB14K.;�N=4LNm�=�1�giKÆ5æ1�3�57nMÞc368:=4�D�75��O8�5��6nM= " Ül=4;&gOK.;614=�L_m�K.�C=4n��%=�K.�O=�vÕ5æ<6=�%5s�In:K.�Ègi=�;6=4vuK.L³n�5�L`LO8M�7368M=�;cgO=4L³57m>�OKskI8Mvj57148:K.;6=4L�-® 6g" ��¯ ; 0B® 68"°; ��¯�® � 68"°; � ± ¯ � ® � � 68"°; � ² ¯�³:® � � � 68"°; �r´µ´¶´ µZGI-/, ¸� ® 68"­> ¯ ; 03® 6g"°;=> ¯�® � 6g"°; � ± ¯ � ® � � 68"°;·> ² ¯ ³ ® � � � 6g"°; �F´¶´µ´ µZGI-HG.¸57�>K.�i5�LO3>vj=�vjK.L¬G[-:,��¤GI-HGIÜ6K7�>gi=�;68M=�;6<6Ku14K.vuK��O=�LO36n/gÃ57<6K® 6g" ��¯ ; ��® 68"�> ¯ ; 0j±¸® 6g"°; �-¯ � ® � � 6g"°; �)¹ 6 ¯$º ; µZGI-yx&¸=�;CnM5È=4143657148:9.;CGI-HxIÜ�¹ 6 ¯ º ; �O=4m>�O=4LN=4;&gÃ5¬nMK.LTgOf4�Nvj8:;6K.LTÞc36=º14K7;cgO8M=�;6=4;D143�5��NgÃ5�L#m*K7gO=4;I¯1�8�57LÈ��LN36m*=4�O8:K.�O=�L³<6=N¯�-�z{K.;6LO8:<6=4�O57;6<6KuÞc36=E<68M1Ã�>K.L`gif4�Nvj8:;6K.LÈLNK.;¤8M;6LN8M�.;68/Ñ�1s5�;cgO=4L1�K.vjm657�i57<>K.L�1�K.;ænMK7L¢gif��Ovu8M;6K7L¢Þc36=@gO8M=�;6=4;�m*K7gi=�;6148M57L�<>=»¯�8M;>�)=��O8MK7�O=4L�Ü�m�K[<6=4vuK.Ln:nM=4�.57�È5

® � � 6g"°; 0�¼ * � ®* " ��½¿¾!À<¾�Á ¯ �T ® 68" �-¯ ;=> ±® 68"°; �)® 6g"Ã> ¯ ;�Ä µZGI-~rc¸1�K.;�3>;¤=��O�NK.�³<6=EK.�O<6=�;ů � -°I8á57�>K.�i5V��571�=4vuK.LCnM5�<68:�)=��O=4;>148�5V=4;&gO�O=ÕnM57LE=41�3�57148:K.;6=�L�GI-/,u�¹G[-ÓGIÜ�<6=�LOm6�N=41�8�57;6<>Kn:K.LÈgif��Ovu8M;6K.L{<6=EK.�O<6=�;ů ³ ;6K.L³<�57�OÝ�14K.vuK_�O=4LN36n:gi57<6K_36;�5_=hkIm6�N=4LN8M9.;¤m�57�O5Æ® �86g"°;® � 6g"°; 0�¼ *Ç®* " ½ ¾!À<¾ Á ±È¯  ® 68" �-¯ ;·> ® 68"­> ¯ ;�Ä µZGI-H�.¸1�K.;�3>;¤=��O�NK.�³<6=EK.�O<6=�;ů � -J#KI<6=�vjK7L�K.�6LO=��N��57��=�; ;c36=�LNgO�i5QÑ��.36�O5åGI-:,7Ü{Þc36=Çn�5«=�143�5�148M97;àG[-Ó�Q57m>�OKskI8Mvj5«n�5m*=4;6<>8M=4;&gO=¬<>=¬nM5AgÃ57;6�7=4;&gi=³=4;ÕJ«m*K.�2n�5Am*=4;6<>8M=4;&gO=¢<6=³nM5E1�36�N��5E¡A»�Ü7=4LNgi5C=�143�571�8M9.;=�L�n:n�57vj57<�5äù!�*ö:�Y�.óg�uù[øhó���÷½õh÷É�7ó ôsõhöiõh÷PøhóXù�ú«øÃõh÷�üXöOù�MõÃúh-³J#=4�NKÅgÃ5�vB�68Mf�; m�K[<6=4vuK.L

,.,

Page 12: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

x

F(x)

0x−h x x+h

B

P

A

F(x−h)

F(x) F(x+h)

�§8:�.36�O5ÀGI-/,.Ô%Ê·Ë<Ì:ÍÏÎÈÐ�ÑNÒMÓ�Ð�ÔyÕÅÖK×TصÒcË�×�Õ$ÖaÐ�×�ÕÚÙ�×{ÖK×IصÒÆÙ�ÒYÕÈÛK×�ÕÚÙ�ר×�Õ�Ü'Ý�Ë�ÍyÌ!Þ�صÒcË�×�Õ$ÖaÐ�×�ÕÚÙ�×ÖK×ßصÒNÓ!à©Ìyá¦Ò�ÊCâCÝÇصÒIË�×�Õ$ÖaÐ�×�ÕÚÙ�×cÖK×ßصÒNÓ!à©Ìyá:Ò�Ê?ÜHÍ�صÒTË�×�Õ$ÖaÐ�×�ÕÚÙ�×cÖK×GصÒEÓ!àMÒYÌyá¦Ò{Ü�â57m>�OKskI8Mvj57�En�5�m*=4;><68M=�;cgO=À<6=ÀnM5¤gi57;6�.=�;cgO=À=4;eJ#Ü#�.5æLO=45¶m�K7�@n�5¤m�=�;6<68:=4;&gi=À<>=Àn�51�36�N��5uJỢÜ�<�57<�5�m*K.�`n�5B=�143�571�8M9.;«µh�7ó ôsõhöiõh÷PøhóXù�úcã>ù[øhóXù��[õ!�/ù7÷�üâõO¸® � 68"°; Á ¯ ® 6g" ��¯ ;=> ® 6g"°; µZGI-y�&¸KVm*K.��n�5�m*=4;6<68:=4;&gi=_<6=_nM5V1�36�N��5V¡AJ#Ü�<�57<�5�m*K.�En�5À=41�3�571�8M9.;�µh�7ó ôsõhöiõh÷PøhóXù�úuã>ù[øhóXùù7üXöiä�úO¸h- ® � 6g"°; Á ¯  ® 68"°;=> ® 68"­> ¯ ;�Ä µZGI-H�.¸°I3>m�K.;>�&57vuK.LÕ5��6K.�i5ÆÞc36=¶nM5Æ�)36;61�8M9.;-®Î<>=4m*=4;6<6=¶<6=¶nM57Lu��57�N8�57�>nM=4L " � & -á02;I¯gOK.;614=�LÈn�5C�O=4�78M9.;Æ«q=4LNgi57�iÝB1�K.;&gi=�;68M<�5C=4;�=4n�m6n�5�;6K "'& -6°I3>��<68/�[8M<68:�O=4vuK.Lá=4n�m6n�57;>K6Ü<6=Ègi57n6vj57;6=��i5AÞc36=³=4n6=w´N=`�6K.�N8MÍ4K7;cgi57n>Þc36=�<�57�OÝD<68/�[8M<68:<6KA=4;_LN=4�.vu=4;&giK7Lá<>=ÈgÃ57vj57ß6Kå " 0�¯TÜ�<6=@=4LNgi5Õ�)K.�Ovj5_n�5j1�KIK7�O<6=�;�57<�5 " =�LNgÃ5��iÝÕ�N=4m6�N=4LO=�;&gÃ57<�5�m*K.� "�æ 0Éç�¯æ14K7;

, G

Page 13: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ç7032 9yk 9yk ± 9 ´µ´¶´/Üs��=4n&=w´N=º�7=4�|gi8M1457n&nMK`<>8:�[8M<6=��O=�vjK.L�=�;CLN=4�.vu=4;&gOK.L%<>=ºgi57vj57ß6K å & 0z Ü>57LOoPn�5@1�KIK.�N<6=4;657<�5 & =4LNgi57�iÝ��N=4m6�N=4LO=�;&gÃ57<�5_m*K.� &©è 0-é z 14K.;Æéu0B2 9yk 9yk ± 9 ´¶´µ´/-02n>��57n:K.�l<6=¿® =�;j1457<�5�m636;&gOKuê 68"�æ�9:&©è!; 0Bê 6 çë¯ 9 é z<; <6=³n�5A�N=4<_=4L|gÃÝE�O=4m>�O=4LN=4;&gÃ5�<6Km*K.�¨® ìH0í® 6 ç�¯ 9 é z$; �¶m�57�O5VLN8Mvum6nM8/Ñ�1s5��¢nM5Õ;6K7gi57148:9.;TÜP=�;Æ57<6=4nM57;&gi=Bn:K�<6=4;>K7gÃ57�N=h¯vuK.L`14K7vjK1® ædî è -��=D571�36=4�N<6Ku14K.;¤n�5B=�143�571�8M9.;«µZG[-yrc¸

¼ ��� ®�$" �7½ ì 0�¼ ��� ®�$" � ½ ædî è Á ®»ï 6 ç'�r ; ¯ 9 é z%ðc> ±®»ïaç�¯ 9 é z%ð ��®»ï 6 ç > ; ¯ 9 é z%ð¯ � µZGI-yF&¸=�;¤gOf4�Ovu8M;>K.LÈ<6=En�5�;c36=h�75_;6K7gi57148:9.;¼ ��� ®�$" � ½ ædî è Á ® æòñ � î èC> ±® ædî è �)® ædó � î è¯ � µZGI-yR&¸1�K.;�3>;¤=��O�NK.�³<6=EK.�O<6=�;ů � -��¢=D8:�.3�57n��)K.�Ovj5Bm�57�O5¼ ��� ®�$f �¿½ ædî è Á ® ædî èhñ � > ±® ædî è �)® ædî è¦ó �z � µZGI-/,�Ò&¸1�K.;Q36;=��O�OK7�@<6=ÀK7�O<6=�; zÚ� -%��=V57143>=4�O<>Kæ14K.;nM5�;6K�gÃ571�8M9.;¹<6=�57m6�NKsk>8:vÕ5�148M97;¹<6=<68/�)=4�N=4;61�8�57L`��5�148�5�5�<6=4nM57;&gi=Em�57�O5_nM5B=hkIm6�N=4LO8:9.;-ôyõô ¾ =4;�J�=4L|gÃÝ_<�57<�5�m*K.�

� ®�$" Á ® ædî èiñ � > ® ædî è¯ µZGI-/,., ¸1�K.;æ36;æ=��O�NK.�¢<>=@K.�N<6=4;�<6=u¹ 6 ¯ ; Ü�LO8Mvu8MnM57�Ovu=4;&gO=Am�57�O5vôyõô¦ö Ü�m*=4�NKÀ14K7;æ36;æ=��O�OK7�¢<6=K.�N<6=4;j<6={¹ 6�z$; -[é¢=4vuK.LáK.�>gO=4;68:<6K6Ücn�57Lº<6=��O8/�75�<�57Lá=4;ÕgOf4�Ovu8M;>K.Lº<6=�<68/�)=4�N=4;61�8�57L�Ücgi57n1�K.vjK�LN=Cvu=4;>148MK7;69_57�O�N8M��5I-J%57�i5Æn�5Ç=hkIm6nM8:1s571�8M9.;e<6=�n�57L_LN=4141�8MK.;>=4LÕ<>=4n¬vjfhgiK[<6K«0§kIm6n:oM148/giKÇ=¤twvjm6n:oM1�8:giK>ܧ;6K.L573[k>8:nM8M57�O=�vjK.L�14K.;VnM5_=�143�5�148M97;¤<>=@zÈ5�nMK.�

,�x

Page 14: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�ø÷�$ù 0Bú ����÷�$û � µZGI-/, G.¸1�K.;ÅúV14K.;>LNgÃ5�;cgO=7-?A�>gO=4;68:=4;6<6K_36;65u=�143�571�8M9.;�57<68:vj=�;6LO8:K.;�57n*<6=En�5�=�143�571�8M9.;�GI-/, GIÜ><6=En�5B�)K.�Ovj5

��ü�$f 0 � � ü�$" � µZGI-/,�x&¸1�K.;¹nM5V5��[36<65¤<6=un�5�L�LO36�6L|gi8/gi361�8MK.;6=�L ü 0 ýý � Ü " 0�þ ÿà� f 0 ú �ÿ��.Ü�<>K.;6<6= ÷ =4L36;65@gi=4vum*=4�i5�gi36�O5B5@148:=4�|gÃ5@<68:LNgi57;6148M5 û 5B36;ÀgO8M=�vjm*K ù � ÷7� =�L`n�5DgO=4vum�=��i5�gO36�i58:;68M1�8�57n�5Bgi8M=�vjm*K�14=��OK6-

n1P�OEP � ����qÆ[Vq Q�� ���YR?W���qê¬;BvjfhgiK[<6K�m657�i5�=�;614K7;cgO�i57�§n�5¢LNK.nM3>148M97;@<6=È=�143�5�148MK7;6=4Ll1�K.;B<68/�)=4�N=4;61�8�57L#Ñ�;68:gi57L4Ü�=4L3>gO8Mn:8MÍs5�;6<6KD=4n�þ¹ÿhü������u�=�Y�%���Zøhóaü���Ü[=�nP1�3�57n*8MnM3>LNgi�O57�O=�vjK7L�1�K.;�5s�[36<�5B<6=AnM5C=�143�571�8M9.;<6=E1457nMK.�È57<68Mvu=4;>LO8MK7;�57nZ-ê¬;�5_57m6�OKskI8Mvj571�8M9.;�=�;�<68/�)=4�N=4;61�8�57LÈÑ�;>8:gÃ5�L`m�57�i5�nM5B=4143657148:9.;

�ø÷�$f 0 � � ÷�$" � ´ µZGI-/,src¸°I=�K.�>gi8:=4;6=�57m6n:8M1s5�;6<6KDn�5C=�143�5�148M97;¤GI-:,7,A57n*n�57<>K@<6=��O=41Ã�>K_�jn�5D=41�3�57148:9.;VGI-yR�µ)m�57�i5nM5@�75��O8�5��6nM= f ¸`57nPn�57<6K�8:Í4Þc368:=4�O<>Ku<>=@G[-:,sr>Ü

ü�ædî èhñ � >Hü�ædî èz 0 ü�æòñ � î èC> ± ü�ædî è � ü�ædó � î è¯ � µZGI-/, �.¸<6K7;6<6= ü =4L�n�5jLNK.nM3>148M97;¶=hk>571hgÃ5À<6=@n�57L¢=4143657148:K.;6=4L�<6=B<68/�)=4�O=�;6148M57LA5�m6�OKskI8Mvj57<�57L�Ü1�K.;

,sr

Page 15: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

x

t

Valores Iniciales

Con

dici

ones

de

Fron

tera

uu u

u i, j+1

k

h

i−1, j i, j i+1, j

valor desconocido

valores conocidos

�§8:�.36�O5EGI-ÓG[Ô���× Ë<Ì:×��:×�ÕÚÙ�ÒMÓ�Ð�ÔyÕNÛaÌ����$ÓhÒ�ÖK× Ø�Í��Gá:ÒYØ�ÍyÌ:×��ßÓ¦ÍyÕ$͸Ó�Ð�ÖKÍ���� ædó � î è Ý�� ædî è Ý�� æòñ � î è�� ÖK×�Øá:ÒYØ�ÍyÌTÖK×��¦Ó¦ÍyÕ$͸Ó�Ð�ÖKÍ�� ædî èiñ ���"�æ 0Fçë¯ & f�è 0�é z 6 ç 9 é 032 9 9 ± 9 ´¶´µ´ ;°I8TgOK.vj57vjK7L � 0 � ö� � ¾� � 0 !" � Ü�m�K[<6=4vuK.L`=�LO14�N8M�68:�`n�5B=�143�571�8M9.;¶GI-/, �B14K.vuK

ü�ædî èiñ � 0 �Mü'æµó � î è � 6 > ± �;Ïü�ædî è � �Mü'æ�ñ � î è µZGI-/,��&¸��5«=�143�571�8M9.;äGI-:,s�>Ü�;>K.LÕLN=4�iÝQãIgi8Mn¬m�57�O5«=4;>14K.;&gi�O57�ÀnM5Ægi=�vjm*=4�|gi36�O5«<6=�LO14K7;6KI1�8M<�5ü�ædî èiñ � Ü>=�;�=4n�m636;&giK 6 ç 9 éI�- ; <6=�n�5D�O=�<TÜ6=4;ÀgOf4�Nvj8:;6K.L{<>=An�57Lági=�vjm*=4�O5�gi36�O57LÈ1�K.;6K�¯1�8M<�57L³5�n:KunM57�O�7K�<6=En�5¨é7¯wf�LO8Mvj5@Ñ�nM5B<6=4n�gi8:=4vum�K@=4;¤n�5��O=�<TÜ6Ñ��6-�G[-ÓGI-02;&giK.;>14=4L%�.=4vuK.L%Þ[3>=Èm�K[<6=�vjK.L#1s5�nM143>n�57��1s5�<�5¢36;65�<6={nM57L�gi=4vum*=4�i5�gi36�O57L%<>=4LO1�K.;6K�¯1�8M<�57L�=4;Ç=4n§gO8M=�vjm*K f 0 z ܺµ)=4L�<6=�148:�4ÜT=�;ÆnM5Õm6�N8Mvu=4��Ñ�nM5Õ14K7; éV0 ¸èÜP1�K.;Æ5��[36<65<6=EnM57L`14K.;><68M1�8MK.;6=�L³8M;68:148�5�nM=4LAµ§Ñ�n�5�14=��OKu14K.;­é»0B2.¸È�À<6=��)�OK.;&gi=��i5�µX1�K.;6LO8:<6=4�O57<�57LnM5C14K7nM36vu;�5@ÒD�Õã6n/gi8:vÕ5>Ü&1�K.;]ç7032@�Æç70$#¹ÜILO8�=4nP=w´N=A=4LNm�571�8�57n*LO=�<68/�[8M<68:=4�i5E=4;%#

, �

Page 16: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 0.2 0.4 0.6 0.8 1x

0

0.1

0.2

0.3

0.4

0.5

u(x)

t = 0.005

t = 0.02

t = 0.04

t = 0.06

t = 0.08

t = 0.1

�§8:�.36�O5�GI-yx>Ô·à'& Î<ݶÙ)(¨Ó¦ÍyÕÅ×�ØtÑ+*!Ù�͸ÖKÍ-,�ÎaË<Ø/.�Ó�Ð8Ù�Í�Ë'ÒYÌiÒÃá¦ÒYÌ�Ð�Í�� Ù�Ð�×�Ñ¿Ë�Í�� �10 ÍyÕ$ÖaÐ�Ó�Ð�ÍyÕ$×���Ð�ÕKÐ32Ó�еÒYØ�×��!Þ��54)6 æ87�9 6 æ Ë'ÒYÌiÒ 6 æ;: < �>= � �54)6 æ?7�9A@CB 6 æ Ë'ÒYÌiÒ < �>=ED 6 æ ÝCÓ¦ÍyÕF 9HGJI ÝLK 9MI�NOIQP Ý'RTSVU�×�2XW �ZY 9[< �]\^=^�m�5��Ngi8:148:K.;6=4Li¸#5¢nMKAnM57�O�.K¬<6= f 0B2>-���=ÈnM5�vj8:LOvj5¬vj57;6=��i5�m*K[<6=4vuK.L§=�;614K7;cgO�i57�§1s57<�536;65_<>=DnM57LÈgi=�vjm*=4�O5�gi3>�i57L³=4;¤=4n�gi8M=�vjm*K f 0B± z Ü61�K.;¶5s�[36<�5_<6=En�5�LÈ�&5_14K.;6K[148:<�57LgO=4vum�=��i5�gO36�i57L³5�n:KunM57�O�7KB<6= f 0 z Ü�<6=E¥%_ ,.Ü/-/-:-/ÜHÄ`¯Ã,.-02;��.=�;6=4�O57nZÜIm�K[<6=4vuK.LÈ=�;614K7;cgO�i57�{nMK.L{��57n:K.�O=�L{<6=�nM57L�gO=4vum�=��i5�gO36�i5�LÈ<6=4LN14K.;>KI1�8M<�57L=�;æ148:=4�NgOKugO8M=�vjm*K;`�gsÜ�=�;¶�)36;61�8M9.;�<6=@n:K.L¬��57n:K.�O=�L³<6=CgO=4vum�=��i5�gO36�i57L¬14K7;6KI1�8M<�5�L¢=�;36;Vgi8:=4vum�KB57;&gi=4�N8MK.�³5 f `IÜy=�L¢<>=4148:�4Ü f ` ó � -�¡½=4L|gÃ5B�)K.�Ovj5B<6=E=4;>14K.;&gi�O57�³n�5�LOK7nM361�8M9.;<6=`3>;�5A=41�3�57148:9.;�<68/�)=4�O=�;6148M57n>=�;_1s57<�5Am>36;&giKE<6=`n�5¢�O=4<_LN=`nM={1�K.;6K[14=`1�K.vjK@þÆÿhü�������=�Y�%���Zøhóaü���-

,��

Page 17: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 0.2 0.4 0.6 0.8 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

u(x)

�§8:�.36�O5_GI-yr>Ôba·Ì����$ÓhÒNÖK×c�54)6 Nede7 Ó¦ÍyÕ]×�Øgf *!Ù�͸ÖKÍ�,�ÎaË<Ø/.�Ó�Ð8Ù�ÍTË'ÒYÌiÒ{Ø�Í��¿Ù�Ð�×�Ñ¿Ë�Í�� d19MI�NOIhIi@&�Ø/.�Õ$×hÒ?ÛaÌyà×��iÒ�( � dj9kI�N>IhIi@ml &�Ø/.�Õ$×hÒ{ÖK×�Ø ÛÒMÖÒ�( �L0 ÍyÕNصÒn� Ó¦ÍyÕ$ÖaÐ�Ó�Ð�ÍyÕ$×��=Ð�ÕKÐ�Ó�еÒYØ�×��b�o4)6 æp719 6 æË'ÒYÌiÒ�6 æo:q< �>= � �54)6 æp7j9@rB 6 æ Ë'ÒYÌiÒ < �>=TD 6 æ Ý<Ó¦ÍyÕ F 9@sIhI ÝtK 9[< � < U�ÝuRTS =^�>= ×�2< = �vY 9[< �>=h=^�¡}m�=�Li57�B<>=ÀÞc36=À=4L@vÕÝ7LCLO=�;6148:nMnMK¤3>gO8MnM8:Ís57�E=�n³þÆÿhü��������=�Y�%���Zøhóaü���Ü#f4L|gi=ÕgO8M=�;6=À36;�5LN=4�O8M5�<6=4L|�.=4;&gi5 ´O5>ܧm>36=4LD=4LD8M;6=�LNgi57�6nM=�µaÑ��6-#GI-~rc¸A5Vm�57�NgO8M��<6=uÞc36= � giK.vj5À��57nMK.�N=4Lvj5s�.K.�N=4L³Þc36= �� Ü>=4L¬<6=41�8M�4Ü>=4n�vuf�giK[<6K�LNK.nMK_=4L`��Ý7n:8M<6K�m�5��i512»A !" �-w �� µ)Ñ��6-�GI-Hx.¸h-02LDm�K.��=�LOK�Þ[3>=Àz{�O57;6¥¶�ÆĬ8M14K7nMLOK7;/lyx¸mºm>�OK.m*K.;6=�;«36;ÆvjfhgiK[<6KVÞc36=u�)3>=4�i5���Ý7nM8:<6Km�5��i5_143�57n:Þc368M=��È�75�nMK.�ÈÑ�;>8:giK_<6= � µ)=4L¬<6=41�8M�4Ü�Þc36=D=4n�vuf�gOKI<6K_LN=s5_=4LNgi57�6n:=�¸hÜ�=�LNgi=D=4L=�n§þ¹ÿhü������u�!���%���Zøhóaü���-

, �

Page 18: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

n1P:n1P � ����qÆ[Vq xgy %���YR?Wm��qz{�O57;6¥���Ä¢8M1�K.nMLNK.;¤14K.;6LN8M<6=��i57�NK.;æÞc36=Cn�5j=�143�5�148M97;¶<68/�)=4�O=�;6148M57n�m�57�N148�5�nIô ýôyö 0 ô � ýô ¾ �LN=¬LO5�gi8:LN�X571�=³=4;u=4n�m636;&gOKCvu=4<68:K ç�¯ 9a6 é�� �� ;iz ���N=4=�vjm6nM57Ís5�;6<6KE=4n�m6�NK.vu=4<68:K�<6=³n�57L57m>�OKskI8Mvj57148:K.;6=4LP<6=�<>8:�)=4�N=4;61�8�57LTÑ6;68:gi57L�=4;D=4naé¢� 6 é$�1 ; ;>8:�.=�nM=�L#<>=4n.gi8:=4vum�K`m657�i5³n�5m�5��Ngi=�<6=4�N=41Ã��5�<6={n�5¢=�143�5�148M97;Bm�57�N148�5�nZ- 02;&giK.;61�=4Lºn�5¢57m6�NK�kI8:vÕ571�8M9.;D<6=ÈnM5¢=�143�571�8M9.;

¼ �ø÷�$f ½ ædî èhñ{z� 0 ¼ � � ÷�$" � ½ æµî èhñ{z� µZGI-/, �.¸=�LNgÃÝ_<�57<65_m*K.��Ü

ü�ædî èhñ � >vü'æµî èz 0 ± | ü�æòñ � î èiñ � > ± ü�ædî èhñ � � ü'æµó � î èhñ �¯ � � ü�æòñ � î èG> ± ü�ædî è � ü�ædó � î è¯ � }LN36LNgO8:gi3I�.=4;><6K � 0 !" � K.�Igi=4;>=4vuK.L³n�5BLO8:�.368:;cgO=�=41�3�57148:9.;

>¿�¸ü�ædó � î èiñ � � 6 ± � ± �È;�ü�ædî èhñ � > �¸ü�æòñ � î èiñ � 0 �¸ü�ædó � î è � 6 ± > ± �È;�ü'æµî è � ü'æµó � î è µZGI-/,�F&¸J#KI<6=�vjK7LuK7�6LO=��N��57�_Þc36=�=�;ÅnM5�=4143657148:9.;�G[-:,�FIÜl=4n³n�5�<6K�8MÍ�Þ[3>8M=4�N<6KÆ14K.;&gi8:=4;6=�gO�O=4LgOf4�Ovu8M;>K.L³<6= ü <>=4LO1�K.;6K[148:<6K.L�=�; f 0 6 é �3 ;iz Ü��V=4n#n�57<>Kj<6=��O=�1Ã�6KÀgi�N=4L¢gOf4�Nvj8:;6K.L<6= ü 14K.;6K[148:<6K.L�Ü>Ñ��6-*GI-H�°I8�m�57�O5æ1s57<�5¶gO8M=�vjm*K f�è ܧ=�nÈ;cã6vj=��OK�<6=Õm636;&giK.L�8:;&gi=4�N;6K.LB<>=�n�5¶vj57nMnM5¤=�L�8M�.3657n5ÇÄCܧ=4;&giK.;>14=4Lum�57�O5VéH0 2�=�çu0p 9 ± 9s~<9 ´¶´µ´ 9 #Æܧn�5æ=�143�5�148M97;�G[-:,�Fæ;>K.L_<�57�i5ÇÄ=�143�571�8MK.;>=4L`LO8:vB36n/gÃ57;6=457Lám�57�i5Bn:K.LÈÄ �75�nMK.�N=4LÈ<6=�LO1�K.;6K[148M<>K.L`5BnMK@n�57�N�.K@<6=�nTm6�N8Mvu=4�gO8M=4vum*K6Ü�=4;B�)36;>148M97;@<6=Èn:K.L��75�nMK.�N=4L#8M;68:148�5�nM=4L��D<6={�)�NK.;&gi=��i5¢�.5�14K.;>KI1�8M<6K.L�-7�¢=È8M�.3657nvj57;6=��i5>ÜIm�57�i5¨éu0��<6=�nM5B=4143657148:9.;¤GI-/,�FBLO=�K.�Igi=4;><6�iÝ7;¤Äà��57n:K.�O=�LÈ<6=4LN14K.;>KI1�8M<6K.Lm�5��i5 ü 5�nMK¤n�57�N�.K�<>=Àn�5VLO=��.36;6<�5¶Ñ6n�5�<>=4nºgi8M=�vjm*K6Ü�=�;«gif��Ovu8M;6K.LD<6=jnMK.LE��57nMK.�N=4L<6= ü �.5¢1457nM1�36n�57<>K.L�5¬n:K³n�57�N�.K`<6=án�5`m6�O8:vj=��i5{Ñ�nM5`<6=4n&gi8:=4vum�K>Ü��D57LNo&LO361�=4LO8/��57vj=�;&gi=7-02;&giK.;>14=4L�Ü�=�n#d�f�gOKI<>Kjtwvum6nMo:148/giKB1�K.;6LO8:LNgO=D=�;��N=4LNK.n:�7=4�¬36;�LN8ML|gi=4vj5�<6=D=4143657148:K.;6=4Lm�5��i5æ1s5�<�5¶gi8:=4vum�K f�è 14K.; éH0 2 9 9 ± 9 ´µ´¶´/Ü%m�5��i5æ=4;>14K.;&gi�O57��nMK7L@��57nMK.�N=4LB<>= ü =�;

,�F

Page 19: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

x

t

j+1

j

i −1 i i+1

Valores desconocidos de "u"

Valores conocidos de "u"

�§8:�.36�O5ÇGI-Ó�[Ô��KÌ:×��Æá¦ÒYØ�ÍyÌ:×�� ÖK×T�/Ó¦ÍyÕ$ÍÓ�Ð�ÖKÍ��»×�Õ}صÒn�cËtÍ���Ð�Ó�Ð�ÍyÕ$×��T6 ædó � Ý16 æ � 6 æ�ñ � ×�Õ�×�ØÙ�Ð�×�Ñ¿ËtÍ d�è Ý � Ù�Ì:×��TÖK×��:Ó¦ÍyÕ$ÍÓ�Ð�ÖKÍ��I×�ÕV×�Ø�Ù�Ð�×�Ñ¿ËtÍ dgèhñ � Ë'ÒYÌiÒ{صÒn� ѨÐ���ÑNÒn� Ë�Í���Ð�Ó�Ð�ÍyÕ$×�� �

,�R

Page 20: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

u(x)

t = 0.01

t = 0.03

t = 0.07

t = 0.13

t = 0.17

�§8:�.36�O5�G[-H�>Ô{a=Ì����$ÓhÒÃÖK×��54)6 7 Ë'ÒYÌiÒ]á¦ÒYÌ�Ð�Í��EÙ�Ð�×�Ñ¿Ë�Í��EàKÙ�Ð�Ø�Ð/�YÒYÕ$ÖKÍÃ×�Øbf *!Ù�͸ÖKÍ�� Ñ¿Ë<Ø/.�Ó�Ð8Ù�Í �0 ÍyÕ$ÖaÐ�Ó�Ð�ÍyÕ$×��ßÐ�ÕKÐ�Ó�еÒYØ�×��c�54)6 æ 719 6 æ Ë'ÒYÌiÒ�6 æ : < �>= � �54)6 æ 719�@�B 6 æ Ë'ÒYÌiÒ < �>=�D 6 æÓ¦ÍyÕ F�� U <h< ÝLK � < � < U�ÝLR�SVU�×�2XW �ZY � U �1457<�5�m636;&giKu<6=En�5��N=4<T-0§kI8ML|gi=4;¤��57�O8:K.L`vjfhgiK[<6K.L³m�57�O5u�N=4LNK.n:�7=4�¬=4n�LO8:LNgO=4vj5u<6=D=41�3�571�8MK.;6=�L¢Þc36=DLO=D�.=�;6=4�O5=�;À1s57<65Dgi8:=4vum�K f�è <6=A5�1436=��O<6KB1�K.;ÀnM5D=�143�571�8M9.;�GI-:,sF>Üc36;6K@<>=�f4L|giK.L{=4L{=�n�þÆÿhü�������[õC�?� óg��óa÷*ù[øhó���÷Ã�[õ-�³ù~&úÃúȵ·57m*f4;6<>8M14={¡�¸#�C=4L2=�n>Þc36=`LO=`3Igi8Mn:8MÍ457�iݳ=4;�=�LNgi=ÈgO�i57�65 ´NK6-J%57�i5åLi57�*=4��LO8D=4nDvjfhgiK[<6KÅÞc36=«3>gi8:nM8MÍ457vuK.LÀm�57�O5Å=�;614K.;&gO�i57��n�5ÅLNK.nM3>148M97; <6=¹n�5=�143�571�8M9.;¤<68/�)=4�O=�;6148M57n�=4L¬=4n#57m>�OK.m68M57<6K_gi=�;6=4vuK.L³Þc36=C�7=4�O8/Ñ�1s5��¢Þc36=D<68M1Ã�>KÕvuf�giK[<6K1�36vjm>nM=¬14K.;ÀnM57LáLO8M�7368M=�;cgO=4Lá14K.;><68M1�8MK.;6=�L4Ô7�C��÷>úhóMúhüâõh÷PøhóXùM�C�C��÷�¢7õhö £[õh÷PøhóXù+�N�ºúhüZùK¡4óg��Ló��&ùK�

G7Ò

Page 21: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

n1P��­P � quZVstWYsj��\�ZVR?WYU°I=45%� 6ë÷{; 0í2ÀÞc36=_�O=4m>�O=4LN=4;&gi=un�5j=4143657148:9.;Ç<68:�)=��O=4;>148�5�n%m�5��O148M57n§<>=4m*=4;6<68:=4;&gi=_<6=" � f ÜT1�K.;ÇLOK.n:36148:9.;�=hk>571hgÃ5 ÷ ÜT�}® 6�ü°; 0 2À�N=4m6�N=4LN=4;&gÃ5¤n�5À=41�3�571�8M9.;�<6=u57m>�OKskI8/¯vj57148:9.;�<6=E<68/�)=4�O=�;6148M57LÈÑ�;68/gÃ57L`<>=4m*=4;6<68:=4;&gi=D<6= " � f Ü�14K.;VLOK7nM361�8M9.;V=hk>571hgÃ5 ü -��=hÑ�;68:=4;6<6Ku=4n�=��O�NK.�`<6=�gi�N36;61s5�vj8:=4;&giK ù%ædî èM6 e ; <6=En�5BLO8:�.368:=4;&gi=A�)K.�NvÕ5IÔ

ù%ædî èM6 e ; 0B® ædî èM6 e ;=> � ædî èM6 e ; µZGI-/,�R&¸<6K7;6<6= e =4L#36;�5¬�)3>;6148:9.;@1�K.;&gi8M;c3�5¬=4; " � f Ü n�5³143�57n&gO8M=4;>=�36;C;cã6vj=��OK¢LO3>Ñ�1�8M=�;cgO={<6=<6=��O8/�75�<�57L{1�K.;&gi8:;[3657LÈm�57�O5Cm*=4�Nvj8/gi8M�á5+� 6 e ; LO=��È=���57n:3�57<�5@=�;�nMK.Lám636;&gOK.L 6 çë¯ 9 é z<; -°I8 ù%ædî èM6 e ;�� 2>Ü�1�3�57;6<6K z�� 2IÜø¯ � 2>Ü*LO=@<>8M14=CÞc36=@nM5j=41�3�571�8M9.;�<6=@<68/�)=4�N=4;61�8�57LÑ�;>8:gÃ5�L`=4L³14K.;>LO8ML|gi;6=hgi=DK_14K.vum�57�O57�6nM=�14K.;¤n�5B=�143�571�8M9.;¤<68/�)=4�O=�;6148M57nTm�5��O148M57nZ-��5VvÕ5s�.K7�Oo�5�<6=unMK7LE573>giK7�O=4LDm*K.;6=4; e 0 ÷ Ü�<6=u=4L|gÃ5¤�)K7�Ovj5Àn�5V=4143657148:9.;QG[-:,�RVLO=�N=4<6361�=@5

ù%ædî è¸6�÷{; 03® æµî è6ë÷¨; µZGI-HG7Ò&¸=�;cgOK.;61�=4L4Ü&=�n>=��O�OK7�l<6={gO�O36;61457vu8M=4;&gOKA14K.8:;6148:<6={1�K.;�=4n>=4�O�NK.�§n:KI1457n[<6={gi�N36;61457vj8:=4;&giK, ædî èM6�÷{; -�¡�LOoIm636=�L4Ü.nM5¢=41�3�571�8M9.;�<6={<68:�)=��O=4;>148�5�L§=4L§14K.;6LN8ML|gi=4;&gO=`LO8>=�nI��57n:K.�%<>=4n>n:8Mvu8:gO=<6=�n�=4�N�OK.�`n:KI1457nT<6=�gi�N36;61457vj8:=4;&giK�=�L¢1�=4�NKj1�3�57;6<6K­¯ � 2>Ü z;� 2I-n1Pe�]P � quZ�� \����E\�Z�RCW�U°I=45 ÷ nM5¹LOK7nM361�8M9.;�=èk>571�gi5e<>=æn�5«=�143�5�148M97;ä<68:�)=��O=4;>148�5�n¢m�5��O148M57n¬�O=4LNm�=�1�gOKÅ5¹n�57L��57�N8�57�6n:=4L¢8M;6<6=�m�=�;6<68:=4;&gi=�L " � f Ü��¶LO=s5 ü n�5jLOK.n:36148:9.;æ=hk>571hgÃ5À<6=�nM5Õ=41�3�571�8M9.;�<6=<68/�)=4�N=4;61�8�57L@Ñ6;68:gi57L4-l°I=�<68M1�=ÀÞc36=Vn�5¶=�143�5�148M97;<6=�<68:�)=��O=�;6148M57L@Ñ�;68/gÃ57L@=4L_14K.;&�7=4�|¯

GI,

Page 22: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�.=�;&gi=E143�5�;6<6K ü�� ÷ =�;�36;¤m636;&gOKjÑ�´NKu5_nMK�n�5��O�.K�<6=D36;¤;68:�7=4n�<6= f Ñ�´NKu1�3�57;6<>Kå "�9 å f�� 202n�gi=4K7�O=4vj5@<6=A�#5 kÕ<>8M14=�Þ[3>=A��57L|gÃ5@1�K.;�Þc36=ALO=�1436vum6n�5Dn�5D=4LNgi57�68:nM8M<657<Õm�5��i5B57LN=h¯�.3>�i57�`Þc36=ELO=D143>vjm6n:=�n�5B1�K.;&�.=4�N�.=4;>148�5I-

n1P��1P QFso� U-�VW��YWY[VUN[J%57�i5B36;�m>�OK.�6n:=4vj5@nM8:;6=s5�n*1�K.;���57nMK.�N=4L�8M;68:148�5�nM=4L2�Õ<6=��)�OK7;cgO=4�O5>Ü���5�kj�À²³8M1Ã�&givC�.=4��N=4n�5�148MK7;�57;en�5Ç=4L|gÃ57�68:nM8:<�57<Q�å14K.;&�7=4�O�7=4;61�8�5¹�[o�5Ç=4n{�C� ¥�õ¦��öiõ!�uùÅ����~[óg¢�ù�Mõh÷�üwõ �[õ§Tù©�ÆlyxamZÜ�<6=hÑ�;68:=4;6<6K_=�LNgÃ5��68Mn:8M<�57<PÜI=�;¤gOf4�Ovu8M;>K.LÈ<6=En�5�LÈLOK.n:36148:K.;6=�L³5714K7gi57<�57L`<>=�n�57L=�143�571�8MK.;>=4Lá<6=¢<>8:�)=4�N=4;61�8�57L2Ñ�;>8:gÃ5�Lám657�i5�36;j;68:�7=4n�<6=³gi8M=�vjm*K f Ñ�´NKD143657;6<6K z;� 2>Ü=�L�<6=�148M��Ü�143�5�;6<6K»é � � Ü�=4L|giK�5�LO36vu8M=�;6<6KuÞ[3>= å " 0ɯ¶=�LNgÃÝu�O=�n�571�8MK.;�5�<6K_14K.; z<6=�gi57nTvj57;6=4�O5�Þ[3>=u¯ � 2�143�5�;6<6K z;� 2>-

¡�LN36vu8M=4;><6KBÞc36=�=4n��.=�1�gOK.�`LOK.n:36148:9.; ü�èiñ � 0�� ü � î èhñ � 9:ü � î èhñ � 9 ´µ´¶´ 9:üg ßó � î èhñ �X¡ <>=An�5=�143�571�8M9.;¤<6=D<68:�)=��O=�;6148M57L`Ñ�;68:gi57L`=4;�=4n�;68/�.=4n�<>=DgO8M=4vum*K 6 é{�3 ; =�LNgiÝu�N=4n�5�148MK7;�57<6K1�K.;«=�nº�7=41hgiK.�D<6=ÕLNK.nM3>148M97; ü�è =�;«=4nº;68/�.=�ná<>=ugO8M=4vum*K­é7¯ý=4LO8:vjKVm*K.�Evj=�<68MK¤<6=un�5=�143�571�8M9.;

¢ èiñ � 0¤£ ¢ è �¦¥ è µZGI-HGI, ¸<6K7;6<6=§¥ è =�LC=�nl�.=�1�giK7�C1�K.nM36vu;�5À<6=unMK7LA�75�nMK.�N=4L�<6=_�)�OK.;&gO=4�i5�1�K.;6K[148M<>K.L4ÜT�E£În�5vj5�gi�N8MÍá<6= 6 # > ;g¨Æ6 # > ; <6=`14K[=�Ñ6148M=�;&gi=4Lº1�K.;6K[148M<>K.L4-70º;&gOK.;614=�L4Ü&vuK.LNgO�i57�N=4vuK.LÞc36=D36;�5u14K.;6LN=41�36=4;61�8�5Õ<>=C=�LNgÃ5u<6=�Ñ6;68M1�8M9.;¤<6=D=4LNgi57�68:nM8M<657<�=4L¢Þ[3>=C36;65j;6K.�NvÕ5�<6=nM5BvÕ5�gO�O8:ÍC£±1�K.vum�5�gi8:�6nM=�14K.;V36;�5_;6K.�NvÕ5B<>=D3>;¤�7=41�gOK.� ¢ Li5�gO8MLN�X5�14=�µ� £ �d� w

G.G

Page 23: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

1�3�57;6<6K�n�5¢LNK.nM3>148M97;C<6={n�5³=4143657148:9.;B<6=�<68:�)=��O=4;>148�5�L§;6K¢<68:�7=4�O�7=È143�5�;6<6K f 573>vj=�;cgi5>ÜK �d� £ �d� w C��¹ 6�z$; µZGI-HG.G.¸1�3�57;6<6Kun�5BLNK.nM3>148M97;�<6=CnM5B=41�3�57148:9.;V<6=E<68:�)=��O=�;6148M57L`<68:�7=4�N�.=E143�57;><6Kug¢573>vj=�;cgi5>-02nTvuf�gOKI<6K�vj5�gO�O8M1�8�57n*<6=E57;6Ý7nM8:LO8ML�vB36=�LNgi�O5�Þ[3>=DnM57L`=41�3�571�8MK.;6=�L³LOK.;V=4L|gÃ57�>nM=4L³LN8T=4nvu9I<>36nMKuvÕÝ�L¢�.�O57;6<6=C<6=Bn:K.L¬��57n:K.�O=�L�m6�NK.m68:K.L¢<>=Bn�5uvÕ5�gO�O8:Í�£æÜ*=4L¢<6=41�8M��Ü�=�n§�i5�<68MK=�LOm*=41�gO�i57n |%6 £ ; <6=-£±LO5�gi8:LN�X571�=

|%6 £ ; wÉ 91�3�57;6<6K¬n�5ÈLNK.nM3>148M97;D<>=ánM5È=41�3�57148:9.;D<6=º<68:�)=��O=4;>148�5�LOn.;6K³<68/�.=��O�.=º143657;6<6K f 573>vj=�;cgi5>ÜK

|%6 £ ; w C��¹ 6 z$;1�3�57;6<6Kun�5BLNK.nM3>148M97;�<6=CnM5B=41�3�57148:9.;V<6=E<68:�)=��O=�;6148M57LOn�<68/�.=��O�.=�1�3�57;6<6K f 5�36vj=�;&gÃ5>-¡¢36;6Þc36={=�LNgi5¢14K7;6<68M1�8M9.;C57LN=4�.36�O5¢Þc36=án�5³LOK.n:36148:9.;E=4L|gÃÝA571�K7gÃ5�<�5>Ü�f4L|giK¢;6K��&5��i57;&gi8:Ís5nM5æ14K.;&�.=��O�.=�;6148M5¹5ævu=4;>K.L�Þc36=VnMK.L@��57n:K.�O=�LBm6�NK.m68MK7L�<6=%£ =�LNgOf4;å�O=4L|gi�N8M;6�.8:<6K.L�5LO5�gi8:LN�X571�=4� |%6 £ ; w �µ� £ �d� w D1�3�57;6<>K�# � � ´°I=45ª© æ 36;��57n:K.�@m6�NK.m68:K�<6=�n�5¶vj5�gO�O8MÍ�£ <>= 6p«¬¨k«t; �®­¯ æ =�nÈ14K7�O�O=�LOm*K.;6<68:=4;&gi=�7=41�gOK.�³m6�OK7m68MK6-60º;&gOK.;614=�L

£ ¯ æ 0$© æ ¯ æ?A�>gO=4;68:=4;6<6K_nM5_;>K.�Ovj5B<6=En�5�=41�3�571�8M9.;¤57;&gi=��O8:K.�Ègi=�;6=4vuK.L³Þc36=7Ü�d� £ ¯ æ �µ� 0 �d� © æ ¯ æ �d� 0 � © æ �d�d� ¯ æ �d� ´J%57�i5�36;�5_vj5�gi�N8MÍA1�K.vjm65�gi8:�6nM=�Ü[��;>K.�Ovj57L`<6=��.=�1�gOK.�O=�L�µ·5�m�f�;6<68M1�=�¡�¸ÈnMn:=4�&5�vjK.L{5

G7x

Page 24: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� © æ �µ�d� ¯ æ �µ� 0 �d� £ ¯ æ �d� w �d� £ �d�µ�d� ¯ æ �µ� ´J#K.�³nMKBgÃ5�;cgOK6Ü � © æ � w �µ� £ �d� 9 ç70 9 ´µ´¶´ 9°« µZGI-HG7x&¸02;&giK.;>14=4L�Ü�<6=En�5�=41�3�57148:9.;�GI-HG7x�LO=�gi8:=4;6=EÞc36=

|%6 £ ; w �µ� £ �µ� ´ µZGI-HG�rc¸02L³m�K7�`=4LOKuÞc36=7Ü �µ� £ �d� w =�L¬1�K.;6<68:148:9.;V;6=41�=4Li5��O8�5B��LN3>Ñ�1�8M=4;&gO=Cm�5��i5�Þc36=CLN=E1436vum6n�5BnM5�=4LNgi57�68:nM8M<657<T-

n1P�±­P ² \IqT�=\³y U [V\µ´vU[��A5�<6K�36;æm6�NK.�6n:=4vj5ÕnM8:;6=s57n�<6=C�75�nMK.�N=4L�8:;68M1�8�57n:=4L¬57m6�NK.m68M57<6Kj��36;�5�5�m6�OKskI8Mvj57148:9.;n:8M;6=457n§<6=_<68/�)=4�N=4;61�8�57LAÑ�;>8:gÃ5�L�n�5�143�5�n2Li5�gO8MLN�X5��&5Àn�5�14K7;6<68M1�8M9.;�<>=�¶�·g¸b¹JºX¹�»h¼u¸�½iº?¾TÜ¿ ¹�»h¾'¥�ºXÀpºXÁ1¾ÂÁ¦=�LÀn�5Æ14K.;6<>8M148:9.;å;6=41�=4Li5��O8�5Æ��LN3>Ñ�148:=4;&gi=¶m�57�i5¹Þ[3>=¶LO=¶1436vum6n�5Æn�5¶�·g¸ÄÃg¼ÆÅ�Çȼu¸b½iºp¾ l~x¸mZ-02LNgO=�1s57m>o:gi3>nMK@LN=��O=�LO36vu=A=4;VnM5CÑ��.36�O5�GI-H�IÜ~<6K.;6<6=E5�n�gi=�;6=4�{36;�5B=41�3�571�8M9.;�<68/�)=4�N=4;I¯1�8�57n[m�57�N148�5�nZÜ LO=È�65714=`36;65A<68MLN14�O8:=�gO8MÍs5�148M97;@m�57�O5�nMn:=4�&5��§5�3>;�LO8ML|gi=�vÕ5A<6={=4143657148:K.;6=4L57n:�.=4�>�iÝ78:1s57L�Ü4<68:1Ã�6K¢LN8ML|gi=4vj5`<6=4�*=ºLO=��%1�K.;6LO8:LNgO=4;&gi=á14K.;Dn�5`=�143�571�8M9.;E<68/�)=4�N=4;61�8�57n.m657�|¯1�8�57n·Ü>�¤57<6=�vÕÝ7L³<6=DLO=��¢=�LNgi57�6nM=D14K7;¶n�5�LNK.nM361�8M9.;¤;c36vuf4�O8:1s5>Ü�nM5�143�57n�1�K.;&�.=4�N�.=B5un�5LNK.nM361�8M9.;V5�;�57nMo/gi8:1s5>-

G�r

Page 25: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ECUACIONDIFERENCIAL

PARCIAL

F(U)

U

ANALITICA

SOLUCIONSOLUCIONNUMERICA

SISTEMA DE

ALGEBRAICAS

ECUACIONES

DISCRETIZACION

ERROR DE TRUNCAMIENTO

0CONSISTENCIA

ESTABILIDAD

x, t

CONVERGENCIAx, t 0

�§8:�.36�O5_GI-Ó�[Ô

G.�

Page 26: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� �����*Ø����TÙ ÉÊ Ø¤Ùð� ÙÛÚ Ë Ú�� ���¹Ø¤�¶���T��Ú

��K.L¤vjfhgiK[<6K.LV=4LNm�=�1�gi�O57nM=�L�8:;&�.K.n:3614�O57; nM5å�>ã6LOÞc36=�<�5�<6=ÆLOK.n:36148:K.;6=�L¤<>=«36;�5=�143�571�8M9.;�<>8:�)=4�N=4;61�8�57n�K�<6=�36;VLO8:LNgO=4vj5B<6=�=41�3�57148:K.;6=�L`<68:�)=��O=4;>148�5�nM=4L�Ü>=�;VgOf4�Nvj8:;6K.L<6=��)36;>148MK7;6=4LVLO365��7=4L�-�02LNgOK.L¤vjfhgiK[<6K.L�LO=Ç=�vjm*=4Í457�OK.; 53>gi8:nM8:Ís57�Õ57vjm>nM8�5�vj=�;cgO=<6=�LO<6=j��571�=j36;>K.L@x7Ò�57ß>K.L4Ü��ÇLOK7;«571hgi3�5�nMvu=4;&gi=�Ü�36;�5¶5�n:gi=��O;�5�gi8:��5�57nºvuf�gOKI<6KV<6=<68/�)=4�N=4;61�8�57L�Ñ�;68/gÃ57L�Üs�E<6=4nc=�nM=4vu=4;&gOK¬Ñ�;>8:giK>- ��K7L#vuf�gOKI<>K.L�=4LOm*=41hgi�O57nM=�L%��5�;@m6�NK.��57<6KLN=4�B=�;Q=hk[gi�N=4vuK�ã>gi8:nM=�LC=�;<68M;�Ý�vj8:1s5V<6=ÍÌ�3>8M<6K.L�Ü%<>K.;6<6=�5��6K.�i5�LO=À36LO57;149[<68:�.K.L=�LOm*=41�gO�i57n:=4L³�68:<6�OK[<68M;6Ý7vj8:14K.L�m�57�O5_=�n�=4L|gi36<68:K_<6=En�5Bgi36�N�636n:=4;61�8�5>ÜI8M;6=�LNgÃ5��68Mn:8M<�57<>=4L�68:<6�OK[<68:;�Ý7vu8M1s5�L4Ü�m6�N=4<68:14148:9.;<6=�nÈ14n:8Mvj5>Ü��Q=4;e=�n{vuK �[8:vj8:=4;&giKæ<>=Àn�57L@14K.�N�O8M=�;&gi=4Lvj57�Oo/gi8:vÕ57L�-�J%=��OK�LN3�36LOKA;>KALO=È�65AnM8Mvu8:gi57<6K³5�nM5¢<68:;�Ý7vu8M145¬<6={nMK.L1Ì�368:<6K.L4-702;�57ß6K.L�N=4148:=4;&gi=�L4ÜILO=��657;Õ8:vjm6n:=4vu=4;&gÃ5�<6KC1�9I<>8M�.K.Lºm�5��i5C�N=s57n:8MÍ457�ºLO8Mv@36n�571�8MK.;>=4Lº;c36vjf��O8:1s57L=�;À14K.;><636141�8M9.;�<>=A1s57n:K.�4Ücvu=414Ý7;68M145C1�3�Ý7;&gi8:1s5>Üc�Õ=�;�=4n�gi=��O�O=�;6K�<6=AnM5@57L|gi�NK7�)oMLN8M1s5��nM5B�O=4nM5�gi8/�[8M<�57<Õ�.=4;6=��i57n·Ü�=4;¤m�57�|gi8M1�36n�5��{=4;¤=4n�=�LNgO36<68MKu<6=EnMK7L`�6K��.K.L³;6=��.�OK.L�-

02;¹=�LNgO=j1457m6o:gO36nMK>Ü���57vuK.LE5¤<657��36;�5V=hkIm*K.LO8:148M97;Æ�7=4;6=��i57n2<6=un�5Õ�)K7�OvB36nM57148:9.;<6=³=�LNgiK7Lávuf�gOKI<>K.L4-&ĬKEm6�O=hgi=4;><6=4vuK.Lº<�57�º36;�5�m6�N=4LN=4;&gÃ571�8M9.;_�)K.�NvÕ57n[���O8:�.36�NK.Li5A<6=n:K.L�vu8MLNvjK.L�Ü�m�=��OK�LOo%36;�5Õ�[8MLN8M9.;¶m�57;6K.�OÝ7vu8M1s5u<6=�LO3æ�)K7�OvB36nM57148:9.;TÜ�8Mn:36LNgO�i57;><6KÕLN336LNKj1�K.;¶=w´N=4vum6nMK7L`LO8Mvum6n:=4L`m6�N8Mvu=4�OK>Ü>��<6=�LOm6ã6=�L�=�;¶nMK7L`LO8M�7368M=�;cgO=4L³1s5�m6o:gO36nMK.L�Ü614K7;=w´N=4vum6nMK.L`vjÝ7L`1�K.vjm>nM=â´NK7L4-

G7�

Page 27: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�­P�OEP ÎÃqT�ry S��YUNRCWYXNZ��K.LuvjfhgiK[<6K.Lu=4LOm*=41hgi�O57nM=�LOLO=�36Li5�;�m�57�O5¹=4;>14K.;&gi�O57�uLOK.n:36148:K.;6=4Lj5¹m>�OK.�6n:=4vj57L

Þc36=�gi8:=4;6=�;�nM5@�)K.�Ovj5

��ü768"�9hfi;�$f 0 Ï 68"�9hfi;�ü=6g"�9hfi; "%��Ð µ·x>-/, ¸Ñ 68"°;�ü=6g"�9hfi; 0 2 "%�:�'Ð µ·x>-HG.¸ü=6g"�9hf 0F2 ; 0 . 68"°; "%��Ð µ·x>-yx&¸

LN8M=4;><6K Ð =4nE<6K.vu8M;68:K=4LNm�57148M57nE<6=4nEm6�NK.�6n:=4vj5>Ü{� �'Ð LO3 �)�OK.;&gi=��i5>Ü[Ï 68"�9hfi; =4L36;¶K.m�=��i57<>K.�¢<>8:�)=4�N=4;61�8�57n·Ü6� Ñ �O=�m6�O=�LO=4;&gi5Àn�57L¬14K.;><68M1�8MK.;6=�L�<6=D�)�OK.;&gO=4�i5I-P��5j8:<6=s5��Ý�LO8M145�<6=`nMK.L2vuf�gOKI<>K.L2=4LNm�=�1�gi�O57nM=�Lº=4Lºn�5A�6ã>LOÞc36=4<65C<6=³36;�5�LNK.nM361�8M9.;_5�m6�OKskI8Mvj57<�557nPm6�OK.�>nM=4vj5�x>-/,�¯âxI-HxB<6=EnM5��)K7�Ovj5

üg �6g"�9hfi; 0  Ò ` À � � ` 6gfi;°Ó ` 6g"°; µ·x>-~rc¸<6K7;6<6= Ó ` 68"°; LNK.;Ç36;�14K.; ´N36;&giKV<�57<6K�<>=B�)36;6148:K.;6=�LAK.�NgOK.�.K.;657nM=�L�14K7;6KI1�8M<�5�L4-P¡¢LOoÞc36=¬57Þc36o·Ü.n�57L28:;61497�.;68:gi57LlLNK.;jn:K.Ll1�KI=hÑ�148:=4;&gi=�L � `I-cê¢;�5A�.=�Í¢K7�>gi=�;68M<6K7L2=4LNgOK.Lá14K[=�Ñ>¯1�8M=4;&gO=4L4Ü�n�5`LNK.nM361�8M9.;E�&5¬=4L|gÃ57�No�5³14K.vum6nM=hgÃ5>Ü��C;>K¢LN=�gi=�;6<6�No�5³;68M;6�736;�5³<68MLN14�O=hgi8:Ís571�8M9.;=�LOm�571�8�57n·Ü7=�L�<6=�148:�4Ü.nM5ALOK.n:36148:9.;�LO=³gi=�;6<6�OoM5A=4;¤ü�������õ!�%���M��óa÷�ó��@õÃú��>ù[øhóXù� -.02LNgi5D1457�|¯571hgi=��OoMÍhgi8:1s5Õ1�K.;&gi�i5�LNgÃ5�1�K.;Ç=4nlvuf�gOKI<6K�<>=�<68:�)=��O=4;>148�5�LAÑ�;68:gi57L4Ü�<6K.;6<>=�n�5ÀLOK7nM361�8M9.;LN=D<65uLN9.nMK�=�;�n:K.L`m636;&giK7L¢<6=EnM5BvÕ57n:n�5>-

z{9.vuKB=4n:=4�.8:�{=�nT14K.; ´N36;&giK�K.�|giK.�.K7;�57n*3>gi8:nM8MÍ457<6KC=4;Vn�5@=èkIm�57;6LN8M9.;Vx>-~r6ÜI<6=4m*=4;6<>=vB3>1Ã�6KQ<6=ænM57Lu14K.;6<>8M148:K.;6=�LÕ<6=¤�)�OK.;&gi=��i5>-�J§5��i5¹�)36;>148MK7;6=4LÀ<>=��)�OK7;cgO=4�O5«m�=��O8:9I<68 ¯1457L4Ü�n�5V=4n:=41�148M97;¹;�5�gi36�O57nº<6=jnM5¤�657LO=u<6=j�)3>;6148:K.;6=4LD=4L Ó ! 6g"°; 0ÕÔ�Öt× 6 ç z�"°; -%J%57�i51�K.;6<68:148MK7;6=4L%<6={�)�OK7;cgO=4�O5A;6K�m�=��O8:9I<68:1s57L�Ü ��5s�B36;�5¢�.�i57;B��57�N8M=�<�57<B<6={m�K.LN8M�6n:=4L%=4nM=�1h¯1�8MK.;6=�L4Ü>5736;6Þc36=E=4;VnM5@m6�iÝ�1�gi8:1s5>ÜIn�57L{vÕÝ7L{36Li57<657L`LOK.;VnM57L{�)36;61�8MK.;>=4L4Ü>LO=4;>K6Ü614K7LO=4;>KKEnMK7Llm�K7nM8M;>K.vj8:K.L�<6=¢z{�6=4�&�[LO�>=��*-6z{3657;6<6KD��5s��LO8:vj=hgi�No�5�m�5��2K�8Mvum�57�4Ü�LO=³36LO57;_n�57L

G.�

Page 28: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�)36;>148MK7;6=4L³14K7LO=4;>KjK_LO=�;6KÕ�N=4LNm�=�1�gi8/��57vj=�;&gi=7-�J#=4�NKj=�;�LN8:gi3657148:K.;6=4L`vj57L`�.=�;6=4�O57nM=�L4Ün:K.Llm*K.nM8:;6K.vu8MK.L�<6=¬z{�6=��&�ILN�6=��uLNK.;j3>;�5�vj=w´NK.�§K.m*148:9.;T-70º;uLO3_=�;6148:14nMK7m�f�<68M1�K�nM8M�>�OKØ z{�6=4�&�[LO�6=h�_57;6<B��K736�O8:=4�§°Im�=�1�gO�i57n6d�=�gO�6KI<>L�Ù{»{K��[<�=4LO1��O8:�68M9A3>;B<68M1Ã�>KAm6�OK��7=4�O�>8�57nLNK.�6�O=EnM5B=4nM=�141�8M9.;V<6=En�5���5�LO=E<6=��)36;61�8MK.;6=�L4Ô

µ)8a¸Àz{3657;6<6KÅ=4LNgOfÇ=4;à<>36<�5>Ü`36LN=ÇnMK.L�m*K.nM8:;6K.vu8MK.Lj<6=¹z{�6=4�&�[LO�>=��*ÜA5vu=4;>K.L¢Þc36=Cn�5uLOK.n:36148:9.;�LN=s5Õ=�LOm�5�148�5�nMvu=4;&gi=Dm�=��O8:9I<68:1s5>Ü>=4;æ1�3>�.KÀ1457LOK=�L³vj=w´NK.�`36;�5�LO=��O8:=D<>=C��K.3>�O8M=��ÈK.�N<68M;�5��O8�5I-µ)8M8�¸�¡�vu=4;>K.L¢Þc36=�Ü�=�LNgOfC36L|gi=�<æ�O=457nMvu=4;&gi=ELN=4�.36�NKÀÞc36=DK7gi�NKj1�K.;�´N36;&giK<>=E�)3>;6148:K.;6=4L³��5�LO=E=4L³vu=â´NK.��Ü636LO=Em*K.nM8:;6K.vu8MK.Lá<6=Cz{�6=��c�[LN�6=��*-µ)8M8:8a¸C¡ vj=�;6K.LuÞ[3>=¶=4L|gifæ3>LNgi=�<��O=457nMvu=4;&gO=7ܺm�=��OK¹�O=s5�nMvu=4;&gi=¤LO=��.36�OKÞc36=CK7gi�NKj1�K.;�´N36;&gOKÀ<6=C�)3>;6148:K.;6=4L¬��57LN=@=�L�vu=â´NK.��Ü*3>LO=@m*K.n:8M;6K.vu8MK7L{<6=z{�>=4�&�[LO�6=h��-

��K.L#m�K.n:8M;6K7vj8:K.LP<6=Èz{�6=4�&�[LO�>=��*Ü.;6K¢LOK.nM57vj=�;&gi=ºLO={57vjK7nM<�57;�v@3>�E�68M=�;B5¬14K.;6<>8/¯1�8MK.;6=�L§<6=á�)�OK.;&gO=4�i5�57�O�68/gi�O57�O8:K.L4Ü�Þc36=È=�;�<68:�)=��O=�;6148M57L#Ñ6;68:gi57L§LN=4�OoM5¢<68/Ñ�148:nc8Mvum6nM=�vj=�;I¯gi57�4Ü�LO8:;6K¶Þc36=jgÃ57vB�>8Mf4;PÜ#m*K.�DLO=��@36;�5�LO=4�N8M=j<6=À��K.36�N8M=�� Ø <68ML|�)�i57Í457<�5�ÙiÜ�LO=Õm>36=4<6=�;36LO57��n:K.L�m*K[<6=4�NK.LOK.L�57nM�7K.�O8/givuK.L`<6=BnM57L¢gO�i57;6L|�)K.�Ovj57<�5�L¢<6=B��K736�O8:=4�¬�iÝ7m68:<�57L�-Pz{K7;K7gO�i57L���57LO=�L�;6K�=hkI8:LNgi=�=�LNgi5Àm�K.LN8M�68:nM8:<�57<TÜ�m63>=4L�;6K�1�36=4;&gÃ5�;¹1�K.;�<68M1Ã�657L�gi�O57;6L|�)K.�|¯vj57<�57L�-

02n�149.vuKE<6=�gO=4�Nvj8:;�57�2nMK7L214K[=�Ñ�1�8M=4;&gO=4Lá<6=³n�5A=hkIm�5�;6LO8:9.;ÕxI-yrD<�5�nM36�.57�º5�nMK.LlgO�O=4Lvuf�gOKI<6K7L�=4LOm*=41hgi�O57nM=�L#vjÝ7L�3>gO8Mn:8MÍs5�<6K.L4Ôs£D5�nM=4�N¥c8M;TÜsz{K.nMK[1s571�8M9.;PÜh���%573TÜ�Þc36=º�.=��O=�vjK.LvjÝ7L³57<6=�n�57;&gO=7Ü6m*=4�NKum>�O8Mvu=4�NK���57�N=4vuK.L`36;�5��6�N=��.=D=hkIm*K.LO8:148:9.;V<6=EnMK.LÈm*K.n:8M;6K.vu8MK7L<6=Cz{�6=��&�ILN�6=��¤��LN36L³m6�OK.m>8M=4<657<6=4L³vj57L`�O=�nM=h�75�;cgO=4L¬m�57�O5_=�LNgO=EgO�i57�65 ´NK6-

�­P:n1P orq��!WYZVq�y W!qus []\ ��Ú]\��§Û�s1Ú]\��02n*m�K.n:8M;6K7vj8:K�<6=�z{�6=��&�ILN�6=�� ù ! 68"°; <>=��.�O57<6K z =4LNgiÝB<6=�Ñ6;68M<6KÀµXm�5��i5 "ÝÜ � > 9 ¡ ;m*K.�

G7F

Page 29: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ù ! 6g"°; 0ßÞnà�á 6�z Þnà�á ó � "°;!9 z 0B2 9 9 ± 9 ´¶´µ´ 9 µ·x>-H�.¸m*K.�`nMKBgi57;&giK6Ü > -w ù ! w .-��%8 ´O57;6<>K " 0âÞ�à�ágã>Ü6gi=�;6=4vuK.L

ù ! 0¤Þ�à�á z ã µ·x>-y�&¸m*K.�`nMK�Þc36=D=4L`�XÝ71�8MnT<>=4<6361�8M�`n:K.L`m6�O8:vj=��OK.L`m*K.n:8M;6K.vu8MK7L�<6=Cz{�6=��&�ILN�6=��

ùø� 0 9 ù � 0¤Þ�à�áÈãN0 "�9 ù � 0¤Þnà�á%±^ãu0j±»(�äæå � ã > c0j± " � > 9 ´¶´¶´d¶Ý�L¬�7=4;6=��i57n:vj=�;cgO=7Ü6m657�i5�n�5B�)9.�NvB36nM5@<6=Cd�K.8/�I�N=7Ü6gO=4;6=�vjK.L

Þnàæá z ãu0r«V+Lç 6 Þnà�áÈãc�)çiá�è)é³ã ; !Lê��=�;&giK.;61�=4L�Ü�5�m6nM8:1s57;6<>K�n�5@�)9.�NvB36n�5B�>8M;6K.vu8�5�nZÜcm�K[<6=�vjK.L`=èkIm6�O=�Li57� ù ! 1�K.vuK

ù ! 0 z ±Íë !sì �îíÒï À � 6Ï> ; ï 6 z»>Mð_> ;mñðòñd6�z > ± ð�;mñ 6 ± "°; ! ó � ï µ·x>-H�.¸<6K7;6<6=Í�ôó ¡ <>=4;6K7gi5unM5Bm�57�|gi=E=4;&gi=��i5_<6=-óA-��=EnM5B8M<6=�;cgO8M<�5�<VgO�O8:�.K.;6K.vuf�gO�O8:1s5

Þnà�á 6 z �F ; ã��HÞnà�á 6�z»> ; ãN0j±�ÞnàæáÂãcÞ�à�á 6 z ã ;LN=D<>=4<6361�=CnM5��O=4nM57148:9.;�<6=E�O=�1436�N�O=�;6148M5

ù ! ñ � > ± "�ù ! � ù ! ó � 0B2 9 z�õ 9 µ·x>-yF&¸nM5B143�57nP;6K.L³m�=��Ovu8:gO=7Ü>=�;�m657�NgO8M143>n�57��ÜI<6=�<6361�8M�³n�5B=èk>m>�O=4LN8M9.;¤<6=�n:K.L`m�K7nM8M;>8Mvu8MK.L ù ! Üz�õ ±IÜ><>=4nT1�K.;6K[148Mvu8M=�;&giKB<6= ù°� � ù � -���5B�.�iÝ�Ñ�1s5B<6=�n:K.LÈm6�N8Mvu=4�NK.LÈm*K.nM8:;6K.vu8MK.LºLO=vB3>=4LNgO�i5�=4;�n�5@Ñ��736�i5_x>-/,.-

G7R

Page 30: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1T_0(x),...,T_4(x)

�§8:�.36�i5�xI-:,.ÔL,b�yÙ�Ò��ÈÛ�à©ÌiÒ�Ñuà×��yÙ�ÌiÒ�Ø�Í��7Ë<Ì�Ð�Ñ ×�Ì:Í���Ó!àMÒÙ�Ì:Í?Ë�ÍyØ�Ð�Õ$ÍyѨÐ�Í��IÖK× 0Äö ×�÷ � � ö ×!á��5�=hkIm6�N=4LO8:9.;�xI-Ó��m636=�<6=�LN=4�uã>gi8:nÈ=4;å148:�O1�36;6LNgi57;61�8�57Lu=4LNm�=�148�5�nM=4L_m*=4�NK¹nM5æ�O=4m>�O=h¯LN=4;&gÃ571�8M9.;åx>-y�æ=4L_�.=�;6=4�O57nMvu=4;&gO=À36Li5�<�5�=4;å14K.vum63>gi57148:9.;57LNoÈ14K.vuK�=4;e=�LNgi3><68MK.LgO=49.�N8M14K7L4-¡¢�6K.�i5B=�;c36vj=��i57vuK.L`5�nM�.36;657LÈ1s5��i571hgi=4�NoML|gi8M1457LÈm�5��i5Bn�5B1�K.vum6�O=�;6LO8:9.;À��57m>nM8M1457148:9.;<6=¢nMK.Lºm*K.nM8:;6K.vu8MK.L§<6=�z{�6=��&�ILN�6=���5Cn�5ELOK7nM361�8M9.;u<6=�=�143�5�148MK7;6=4L�<68:�)=��O=4;>148�5�nM=³K.�O<>8/¯;�5��O8�5�LÈKjm657�O1�8�57n:=4L4-��K.LÈ��57n:K.�O=�L`<6= ù ! ��LN36L³m6�O8:vj=��i57L`<6=��O8/�75�<�57L ù �! =�; " 0 k C=�LNgiÝ7;�<657<�57L³m*K.�

ù ! 6 k ; 0 6�k ; ! 9 ù �! 6 k ; 0 6�k ; ! ñ � z � ´ µ·x>-yR&¸02n�14K.;>KI1�8Mvu8M=4;&gOK�<6=C=�LNgOK.L¢��57n:K.�O=�L`m636=4<>=@LO=��¢<>=C8:;cgO=4�Nf4L¬143�57;><6KÕLN=C=�LO14�N8M�*=4;�n�57L1�K.;6<68:148MK7;6=4L`<6=��)�NK.;&gi=4�O5>-�02L³8Mvum�K.�|gÃ57;&gO=A;6K7gi57�`Þc36=

x.Ò

Page 31: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ù ! 6Ï>�"°; 0 6Ï> ; ! ù ! 6g"°; µ·x>-/,�Ò&¸f�LNgÃ5À=�LAn�5um�57�N8M<�57<¶<6=4n%m�K.n:8M;6K7vj8:K�Þ[3>=B=4L�8:�.3�57n#5ÕLN3��.�i57<>K z -T02n#m�K7nM8M;>K.vj8:K ù !LN=C5�;[3>n�57;V=4;¤nMK7L`m636;&giK.L "�æ µXm63>;cgOK.L³<6=C£D5�36LOLi¸`<>=�Ñ�;68:<6K.L³m*K.�

"�æ 0âÞ�à�á{¼�ç%� ± ½ � z 9 ç70B2 9 ´µ´¶´ 9¦zu> µ·x>-/,., ¸�E5�nM1s5�;6Ís5³LO36L���57nMK7�O=4L�=èk[gi�O=�vjK7L k �=�;@n:K.L�m636;&giK7L�ø"�æ µXm63>;cgOK.L#<6=È£E5736LNL|¯w��K7��5�gOgOKc¸<6=hÑ�;68:<6K.L³m�K7�

ø"�æ 0¤Þ�à�á � çz 9 ç70B2 9 ´µ´¶´ 9yz ´ µ·x>-/, G.¸ê¬;�5À�O=�n�571�8M9.;æ<>=��O=41�36�O�N=4;61�8�5�<6=BnM5À<6=4�N8:��57<�5 ù ! m63>=4<6=�;ÆK7�>gi=�;6=4�NLO=��XÝ7148:nMvu=4;&gi=�-J2�O8:vj=��OK6ÜIn�5B<68/�)=4�N=4;61�8�571�8M9.;�<6= ù ! ;6K.L³<�5

ù �! 0 **Æã 6 Þ�à�á z ã ; *Æã* " 0 z á�è)é z ãá°èùé³ã 9<6K7;6<6=236LO57vjK7L�n�5{�N=4m6�N=4LO=�;&gÃ571�8M9.;Õµ·x>-y�&¸è-s02;&giK.;61�=4L�Ü�5�m6nM8:1s57;6<>K{�)9.�Ov@36n�57L*gi�N8M�.K.;>K.vjfhgi�N8/¯1457L4Ü�K.�Igi=4;>=4vuK.L³n�5B�O=�n�571�8M9.;

ù��! ñ �z �r > ù��! ó �z > 0j± ù ! µ·x>-/,�x&¸��Ý7n:8M<�5Dm�57�i5 z DB7-�ê¢;�5@�)97�OvB36nM5CLN8Mvu8Mn�5��ám657�i5BnM5@mI¯wf�LO8:vÕ5@<>=4�O8/��57<�5B=4L`K7�>gi=�;68M<�5<68/�)=4�N=4;61�8�57;6<>K�LO361�=4LO8/��57vj=�;&gi=C5�µXx>-/,�x&¸h-��K.L`m*K.nM8:;6K.vu8MK.Lá<6=Cz{�6=��c�[LN�6=���LOK.;¤K.�|giK.�.K7;�57nM=�LÈ=4;ú� > 9 ¡ 14K.;¤=4n�m*=4LNK

û 0 6 >�" � ; ó � ì � µ·x>-/,src¸��1�K.;¤=4n�m6�NKI<63>1�giK_=�LO1s5�n�57�`<6=hÑ�;68:<6Kjm*K.��Ü

6�ü�9 e ;ýü 0 � �ó � ü eÆû * " ´ µ·x>-/, �.¸x>,

Page 32: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

��=º=�LNgi=ºvuKI<>K`n�5È14K7;6<68M1�8M9.;�<>=áK7�NgiK7�.K.;�57n:8M<�5�<�m�57�O5`nMK.LTm*K.n:8M;6K.vu8MK7L�<6=áz{�>=4�&�[LO�6=h�=�L

6gù ! 9hù5þµ;eü 0 � �ó � ù ! ù5þ û * " 0 � ± ( ! å ! î þ ´ µ·x>-/,��&¸<6K7;6<6= å ! î þ =4L³n�5B<>=4n:gi5�<6=DÏE�NK.;6=�1Ã¥7=4�¬�V( ! =4L³<6=�Ñ6;68M<6K_m*K.�

( ! 0 ÿ ± å©ç z 032 9 å©ç ¥ õ .-���������� ��� ����������������������5A=hkIm6�N=4LN8M9.;_<6=`<6=��O8/�75�<�57L2=4;_nM5���57LN=¬<>=¬z{�6=4�&�[LO�6=h�u=4L2vjÝ7Ll14K7vjm6n:8M1457<�5�Þc36=`=�;nM5Õ<6=���K.36�N8M=��4-���=��6=41Ã�>K6Ü�n�5Õ=èkIm6�O=�LO8M97;Æ<>=Bn�5Õ<>=4�O8/��57<�5À<6= ù ! 6g"°; 8Mvum6nM8:1s5�giK[<6K.Ln:K.L{m*K.nM8:;6K.vu8MK.L2<>=�m�57�O8:<�57<ÀK7m636=4L|gÃ5B�j�.�i57<>KBvÕÝ�LÈ��5 ´NK�vu8M=�;cgO�i57L�n�5@<>=4�O8/��57<�5@<6=+ æ ! ¾ =�L³LO8Mvum6n:=4vu=4;&gi=¨ç z + æ ! ¾ -��=EnM5B�O=4nM57148:9.;V<6=E�O=�1436�N�O=4;>148�5_x>-/,�x>Ü>LO=DK.�>gi8:=4;6=

ù �! 0j± z �Ò ` À � ( ! ó � ó � ` ù ! ó � ó � ` 6g"°;!9 µ·x>-/, �.¸<6K7;6<6=�� 0 � 6 zu> ;�� ± ¡ -6J#K.�`nMKCgÃ57;&giK>Ü614K.;>LO8M<>=4�i5�;6<6K_n�5@m>�O8Mvu=4�O5@<6=��O8:��57<65B<6=�n�5�)36;>148M97; üg c68"°;

ü �   6g"°; 0  Ò ! À � ü ! ù �! 6g"°; 0  Ò ! À � ü � � ! ù ! 68"°; ´ µ·x>-/,�F&¸ê³gi8Mn:8MÍ457;6<6KÈnM5`=417-�x>-/,�x³=4;Cn�5`=41�3�57148:9.;C57;&gi=4�N8MK.��ÜsnMn:=4�&5�vjK.LT5¬nM5³LN8M�.3>8M=4;&gO=á=èkIm6�O=�LO8M97;

z ü ! 03( ! ó � ü � � ! ó � > ü � � ! ñ � z%õ "! µ·x>-/,�R&¸<6=EnM5B=4143657148:9.;¶5�;cgO=4�N8MK.�ÈLN=C<>=4<6361�=4;�nMK.L`1�KI=hÑ�148:=4;&gi=�L ü � � !

x&G

Page 33: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ü � � ! 0 ±( !  Ò#%$'&)( z* #%(+&),.-./0#�1%2 3 ü54È9 z 032 9 ´¶´¶´ 9 # > µ·x>-HG7Ò&¸

� ü � �   0B2>-60ºL|giKuLO=Dm636=�<6=E=4LO1��O8:�68M�`=�;¤�)K.�NvÕ5Bvj5�gO�O8M1�8�57n*14K7vjK ÷ � � 0 Ð ÷u9 µ·x>-HGI, ¸

<6K7;6<6= ÷ 0 6 ü'�©9 ´¶´µ´ 9 üg G; � Ü ÷ � � _ 6 ü � � � 9 ´µ´¶´ 9 ü � �   ; � � Ð =�L`36;�5�vÕ5�gi�O8:Í¢gO�O8�5�;6�.36nM57�LN36m�=��O8:K.�È��LO3>L¢=�;cgO�i57<657L³LO=E<6=4<>3614=�;¶<6=En�5�=�143�571�8M9.;¤x>-HG7Ò>-��5�=hkIm�57;>LO8M97;�<6=EnM5BLO=4�736;6<�5u<6=4�N8:��57<�5�<6=En�5@�)3>;6148:9.; üÄ  =4L

ü � �  0  Ò ! À � ü � � ! ù ! 6g"°; µ·x>-HG.G.¸1�K.;

ü � � ! 0 ( !  Ò#%$'&)( �* #%('&),6#�1%2 3 6 3 � >)z � ; ü54È9 z 032 9 ´¶´µ´ 9 # > ± µ·x>-HG7x&¸

� ü � �   ó � 0 ü � �   0B2I-�0§k>m>�O=4LO57;6<6KÀx>-ÓG�xB=4;V�)K.�Ovj5BvÕ5�gi�O8:148M57n ÷ � � 0 Ð � ÷ 9 µ·x>-HG�rc¸

<6K7;6<6= ÷ � � 0 6 ü � � � 9 ´¶´µ´ 9 ü � �   ; � -��57LA=èkIm6�O=�LO8MK7;6=4LD57;�5�nMo:gO8M1457L�xI-ÓG7ÒÕ�¶x>-HG.GÕLOK7;Æ<>=�8M;&gi=��Of4L�1457<�5Õ�7=4Í_Þc36=�nMK.L�14K[=�Ñ>¯1�8M=4;&gO=4Lº<6=³n�5�=hkIm�5�;6LO8:9.;j<>=¬nM57L2<6=��O8:��57<657L2LO=³8M;&�7K.nM361��i57;_=�;j14Ý7nM1�36nMK.Lº5�nM�.=��6�i578:14K.L�-J#K.��K7gO�i5um�57�NgO=7Ü*LO8#LOK.n:KÕLN=B;6=41�=4LN8:gÃ5�;ænMK.L³��57nMK7�O=4L¬;c36vuf4�O8:14K.L¢<6=@n:K.L¬14K[=�Ñ�1�8M=4;&gO=4L4ÜLN=�m636=4<>=4;Æ1s57n:1436nM57�¢<>=4nlm6�NKI<>361�gOK�<6=�vÕ5�gi�O8:14=�L����7=41hgiK.�N=4LDx>-HGI,B��x>-HG�r�K�<6=@n�57L

x.x

Page 34: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�)9.�NvB36nM57L�<>=��O=�1436�N�O=�;6148M5BÞ[3>=ALO=A<>=4<6361�=4;V<6=�xI-:,�xI->02;��.=�;6=4�O57nPnM5D�)9.�NvB36n�5E<6=A�N=h¯1�36�O�N=4;61�8�5jm657�i5unMK.L³14K[=�Ñ6148M=�;&gi=4L ü � 4 ! m�57�O5jn�5 3 ¯ýf4LO8:vÕ5_<6=��O8:��57<65u=�L�K.�>gO=4;68:<�5jm*K.�nM5B<68:�)=��O=4;>148�5�148M97;�LO361�=4LO8/��5>Ü

( ! ó � ü � 4 ! ó � 0 ü � 4 ! ñ � ��± z ü � 4�ó � ! 9 z�õ 9 µ·x>-HG.�.¸m�5��i5�n�5Bm6�N8Mvu=4�O5B<6=4�N8:��57<�5

ü � �   032 9 ü � �   ó � 03±æ# üg �9 µ·x>-HG7�&¸�7Ü�m�57�i5�nM5BLO=4�736;6<�5u<6=4�N8:��57<�5

ü � �   0 ü � �   ó � 0F2 9 ü � �  ßó � 0B± 6 # > ; ü � �  ßó � 0B±^# 6 # > ; üg  ´�µ·x>-HG.�.¸

�­P��­P 7 ��=q³��W�y UNRCWYXNZ 8 UT�!\��:9ÆW�Zz{3�5�;6<6KÆ<6=4LO57�O�NK.nMnM57vjK7L@36;�5æ�)36;>148M97;å�>8M=4;e1�K.vjm*K.�|gÃ57<�5eµ)14K.;&gO8M;c3�5æ�<6=��O8:��57�>nM==�L¤LN3 <6K.vu8M;68:K«<6=Ç<6=hÑ�;68M1�9.;�¸Õ=�; gOf4�Nvj8:;6K.LÀ<6=Ç36;65eLN=4�N8M=ægi�N36;61457<�5<6=Æ��K.36�O8 ¯=��jKQ<>=�z{�6=��c�[LN�6=��*Ü`nMK7Lu1�KI=hÑ�148:=4;&gi=�LÀ<6=�n�5Ç=hkIm�57;6LN8M9.;�<6=41��O=41�=4; vB3>�e�OÝ7m68:<�5�¯vu=4;&gi=�-6J%57�i5�8Mn:36LNgO�i57�á=4L|gi=��6=�1Ã�6K6Ü�<6=�Li57�N�OK.n:n�57vuK.Lá57n:�.36;�57L��)36;6148:K.;6=�L�, 68"°; Ü>1�K.vjK=�; µ·x>-~rc¸hÜ�=�;Qgif4�Nvj8:;6K.LD<6=Àm*K.nM8:;6K.vu8MK.L�<6=Vz{�6=��c�[LN�6=��*Ü2�ÆvjK.L|gi�O57vjK7LC1�9.vjK¶<6=h¯1457=4; nMK7LÕ14K[=�Ñ6148M=�;&gi=4L���149.vuKQLO=Æ57m>�OKskI8Mvj5¹n�5Q=èkIm�57;6LN8M9.; 5QnM5«�)36;6148:9.;F, 6g"°; -02LNgO=C1�K.vum�K.�|gÃ57vu8M=�;&giK�=4L¬8Mvum*K.�Ngi57;&gi=�=4;�=4n�vuf�gOKI<6K_=�LOm*=41�gO�i57n§£E57nM=��O¥c8M;TÜ>14K.;¤=4nÞc36=_LO=�m>�O=�gO=4;6<>=j��5�nMn�5��¢LNK.nM361�8MK.;>=4L�<6=�n2LO8ML|gi=�vÕ5ǵXx>-:,h¯âx>-yx&¸h-Pz{3�57;><6K�LN36LNgO8:gO368MvuK.LnM5¤=èk>m657;6LO8:9.; µXx>-yr&¸E=�;äµ·xI-:, ¸èÜ�LO=jgO8M=�;6=ÕÞc36=j=4n�57LOoºnMnM57vj57<6K<;³õÃúhó��~'�>= 68"�9hfy9 � ` ; Ü=h�I8:<6=4;&gO=4vu=4;&gi=D;6K_LO=C57;c36n�5

=j0 �'üg �68"�9hfi;�$f > Ï 68"�9hfi;�üg �6g"�9hfi;@?0B2 µ·x>-HG7F&¸

x7r

Page 35: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

z{3�5�<6�OK_x>-/,.Ôt,�ջصÒEÙ�Ò�÷�صÒ��¦×=Ñuà×��yÙ�ÌiÒYÕ1Ø�Í��Iá¦ÒYØ�ÍyÌ:×��?ÖK×·Ø�Í��?Ó¦Í×ý�$Ó¦×�ÕÚÙ�×��IÖK×ßصҨ×�ÎaË'ÒYÕÆ��Ð�ÔyÕ×�ÕÃË�ÍyØ�Ð�Õ$ÍyѨÐ�Í��NÖK× 0Äö ×�÷ � � ö ×!áMÝ'Ë'ÒYÌiһصÒBA�à©Õ$Ó�Ð�ÔyÕ<CEDGF�4 P 6 �IH P 6 ³ 7 Ý=ÖK×��:ÖK×KJ 9 @JN�LMLML/N FÓ¦ÍyÕ F 9MP�@ �Ä z{K[=�Ñ�1�8M=4;&gO= Ä z{K[=�Ñ�1�8M=4;&gO=Ò r6-/,�x7r&x7=è¯âÒ>, ,7, ,7-Ó�7�7R.R�¯Ã,s�, ,.-H�7Ò.Ò7Ò7=�`�Ò.Ò ,�G GI-HG7�&G7�7=è¯âÒ&�G �I-~r&Ò&�.��=è¯âÒ>, ,sx ,.-H�7�.R.R7=è¯Ã,��x �I-yÒ.Ò.Ò.Ò7=è¯âÒ>, ,4r GI-H�.G7F.x7=è¯âÒ.�r ,.-HG�r6,��7=è¯âÒ>, ,�� x>-HG7�&�7F7=è¯Ã,��� ,.-H�.�7�.�7=è¯Ã,�� ,s� ,.-yÒ>,�F>,s=è¯âÒ&�� x>-HG7R&G7x7=è¯Ã, G ,�� r6-/, G.G7F7=è¯Ã,��� GI-yx.R.�>,s=è¯Ã,�� ,sF F>-yF.x7r6,s=è¯âÒ.RF GI-y�&�7�&G�=è¯âÒ.x ,sR ,.-HG�r&�.R7=è¯Ã,��R GI-H�.�7Ò.F7=è¯Ã,�� G�Ò GI-yF7rcG7�7=è¯Ã,�Ò,�Ò r6-HG7Ò.x&G�=è¯âÒ7r G[, ,�x>-yx&�I,�R7=è¯Ã,��

x&�

Page 36: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1x

−3

−2

−1

0

1

2

f(x)

�§8:�.36�O5jx>-HGIÔ 0 ÍyÑ¿Ë'ÒYÌiÒMÓ�Ð�ÔyÕÅÖK×TصÒ@Aëà©Õ$Ó�Ð�ÔyÕNCEDGF�4 P 6 � 7 H P 6 ³ &�Ø/.�Õ$×hÒ-�:ÔyØ�Ð�ÖÒ�( � صÒ�A�à©Õ$Ó�Ð�ÔyÕÒ:Ë<Ì:ÍÏÎÈÐ�ÑNÒMÖÒcÓ¦ÍyÕ»à©ÕÚÒ¿×�Î�Ë'ÒYÕÆ��Ð�ÔyÕ ×�Õ Ù�×�Ì�ѨÐ�Õ$Í�� ÖK×°ËtÍyØ�Ð�Õ$ÍyѨÐ�Í��·ÖK× 0Äö ×�÷ � � ö ×!át&�Ø/.�Õ$×hÒ Ë�à©Õu2Ù�×hÒMÖÒ�(TÓ¦ÍyÕPO �RQ �TS ×=Ì:×�UKà©Ð�×�Ì:×�Õ Ì:×hÒYØ�Ñ ×�ÕÚÙ�×GÑuà � Ë�͸ӦÍ��TÙî*�Ì�ѨÐ�Õ$Í���Ë'ÒYÌiÒ¿Ì:× Ë<Ì:×��:×�Õ<Ù�ÒYÌCÒIصÒA�à©Õ$Ó�Ð�ÔyÕ]Ò:Ë<Ì:ÍiË<еÒMÖÒYÑ ×�ÕÚÙ�× � ÊCØ'Ð�ÌTÒà©Ñ ×�Õ<Ù�ÒYÕ$ÖKÍ ×�Ø á:ÒYØ�ÍyÌ¿ÖK× F Ý�صÒC�¦ÍyØ8àÓ�Ð�ÔyÕÆÕÚà©Ñ+*�Ì�Ð�ÓhÒ��:×á:Ò¨Ò:Ë<Ì:ÍiÎÐ�ÑNÒYÕ$ÖKÍNÓhÒMÖÒ á�×ý��Ñ��n�CÒcصÒC�:ÍyØ8àÓ�Ð�ÔyÕ]×�Î�ÒMÓ!Ù�Ò �

x.�

Page 37: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

02LNgi5D�)36;>148M97;Õ�O=�LO8:<63�57n�m63>=4<6=A=èkIm�57;6<68:�OLN=7Ü[14K.vuKC1�3�57n:Þ[3>8M=4�2�)36;>148M97;TÜ&=4;ÕgOf4�Nvj8:;6K.L<6=E36;¤14K7;�´N36;&giKuK.�NgOK.�.K.;657nZÜ

= 68"�9hfy9 � ��9 ´µ´¶´ 9 �  C; 0 VÒ ! À � � ! 6 � ��9 ´¶´µ´ 9 �  C;°Ó ! 6g"°; µ·x>-HG7R&¸<6K7;6<6=7Ü2<>=4�68:<6KÇ5�n�5æK.�|giK.�.K7;�57nM8:<�57<¹<6=Vn�5æ��57LN=V<6=À�)36;61�8MK.;6=�L Ó ! 68"°; ÜlnMK7L�14K[=�Ñ>¯1�8M=4;&gO=4L � ! LO=D<6=hgi=4�Nvj8:;�57;�m*K.�³=4n�m6�NKI<>361�gOKu8:;&gi=4�N;6K

� ! 0 6 = 9mÓ ! ; 0 � = Ó ! 68"°; * " ´ µ·x>-yx.Ò&¸02;«=�n2vjfhgiK[<6K¤=4LOm*=41hgi�O57n�£D57n:=4�N¥[8:;TÜTLN=jm68:<6=_Þc36=j=�nº�O=4LN8M<63>K�LN=s5�m*=4Þc36=�ß6K6Ü�=4;¹=4nLN=4;&gi8:<6K�<6=BÞc36=�n:K.L�m6�N8Mvu=4�NK.L 6 # �B ; gif4�Nvj8:;6K.L¬<6=�n�5uLO=��O8M=@=4LOm*=41hgi�O57n³µ·xI-ÓG7R.¸¬LO=57;c36n:=4;T-���57LDLO=��O8:=4LD<6=Õ��K.3>�O8M=��A�Ç<6=Õz{�6=4�&�[LO�6=h�«<>=u�)36;>148MK7;6=4LDLO3�5s�7=4LB<6=�14�N=414=�;=èk>m*K.;6=�;6148M57nMvu=4;&gO=A14K.; z Ü><6=�=�LNgi=�vuK[<6K6ÜcgiK[<6K.LÈn:K.L � ! m�57�i5 z D# m>�O=4LN36vu8M�6n:=h¯vu=4;&gi=¢LO=��iÝ7;�vB3I�jm*=4Þc36=�ß6K.LÈm657�i5+#±�.�i5�;6<6=4L�->¡¢LOo*Þ[3>=7ÜI�)K7�OÍs5��{Þc36=�n:K.L{m>�O8Mvu=4�NK.L� ! LN=s57;¹14=4�NK6Ü�vj8:;68Mvu8MÍ45Õ5W= 68"°; -%02;¹=4nºnMo:vj8/gi=@=�;«=�n2Þ[3>=�# � � = 6g"°; <6=���=��� 2>Ü2��m*K.�unMK¹gi57;&giK6Üán�5Æ57m6�NKsk>8:vÕ5�148M97;TÜ2<6=4�*=æ1�K.;&�.=4�N�.=4�ÕvB3>�å�iÝ7m>8M<6K«5«n�5LNK.nM361�8M9.;�=èk>571�gi5>-��%K[<6Ku=4L|giKu=4;&giK.;>14=4L¬LO=�gO�i57<63>14=C5>Ô�>ù7öNùÃ��óa÷�óg��óYX�ù7öCõ!�TöiõÃúhó��~���ú�õT�*ó��[õ ��~�õ 6 = 9sÓ ` ; 032 �>ù7öNù « 032 9 9 ± 9 ´µ´¶´>#¹-02;Æ=4LNgO=_vjfhgiK[<6K6ÜPLN=_m68M<6=�Þc36=_nM5À��57LO=�µ)K¤3>;«57�N�O=��.nMKÀ<>=uf�LNgi5&¸A1436vum6n�5�;�<6=_=4;I¯gO�i57<�5un�57L¬14K7;6<68M1�8MK.;>=4L¬<6=@�)�OK7;cgO=4�O5>-*0ºn#=â´N=�vjm>nMKÕvjÝ7L¬LO=�;6148:nMnMK>Ü�143�57;><6KÀLO=Cgi8M=�;6=4;1�K.;6<68:148MK7;6=4L`<6=��)�NK.;&gi=4�O5um*=4�N8M9[<68M1457L4ÜI=4L³=4n:=4�.8:�È36;�5_��57LN=E<6=D��K736�O8:=4�4-

�A57�N=4vuK.LC5¤1�K.;&gi8M;c3�5�148M97;«36;¹=â´N=4vum6n:K¤v@3>�ÇLO8:vjm6n:=7-#z{K.;>LO8M<>=4�O=un�5VLO8:�.368:=4;&gi==�143�571�8M9.;V�68:m�=��O�*9.nM8:1s5C�ÀnM8M;>=s57nlµX=41�3�571�8M9.;¤<6=D57<>�7=4141�8M9.;�¸

��ü�$f 0 ��ü�$" µ·x>-yx>, ¸1�K.;Ç<6K.vu8M;68:KÀ=4;Æ=4n28:;cgO=4�|�7Ý�nMK "¦Ü 6 2 9 ± � ¡ ÜT1�K.;6<68:148MK7;6=4L�m*=4�N8M9[<68M1457L4Ü*��14K.;><68M1�8M9.;8:;68M1�8�57n ü76g"�9 2 ; 0 . 6g"°; -#��5¶14K.;><68M1�8M9.;¹<6=Õvu8M;68:vj8:Ís571�8M9.;¶<6=4ná�O=�LO8M<>36Kæ=4LC=4;=�LNgi=

x&�

Page 38: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

1457LOK ± � � �)Z� [ ¼ ��$f > ��$" ½  Ò ! À � � ! 6gfi; + æ ! ¾]\ + óÇæ ` ¾ * " 0B2<´ µ·x>-yx&G.¸é¢57148:=4;6<6K_nM5B<6=4�N8:��57<�5�=4LNm�571�8�57n·Ü6=4L|gÃ5_�O=4nM57148:9.;�LO=E1�K.;&�I8:=4�|gi=E=4;± � � �)Z� [  Ò ! À � ¼ * � !* f > ç z � ! ½ + æ ! ¾]\ + óÇæ ` ¾ * " 032Ú´ µ·x>-yx.x&¸z{K.vuKVn�5V��57LN=j=�LCK.�|giK.�7K.;�57n·ÜPn�5�8:;cgO=4�.�O57n2=4; " =4LDm6�NK.m�K7�O148:K.;�57nl5¤36;65¤<6=�n:gi5¤<6=Ï��OK7;6=41Ã¥�=4�4Ü[<�57;><6K@m*K.�á�O=4LN36n:gi57<6KD36;Õ1�K.;�´N36;&gOK@<6=¢=4143657148:K.;6=4Lá<68/�)=4�O=�;6148M57nM=�Lºm�57�i5n:K.L`14K[=�Ñ�1�8M=�;cgO=4L � ! Ü�5�Li5���=�� * � !* f > ç z � ! 0B2 µ·x>-yx7rc¸��5e14K.;6<>8M148:9.; 8M;68:148M57n ü76g"�9 2 ; 0 . 6g"°; Ü`;6K7L¤<65ånM57LV14K.;6<>8M148:K.;6=�L�8M;68:148�5�nM=4LÕm�57�i5=�LNgi=u14K7;�´N36;&giK¤<6=_=41�3�57148:K.;6=�L4Ü�m636=�L@57n28:;&gi=4�7�i57��nM5À=4143657148:9.;�µXx>-~rc¸�m6�O=h�[8�57vu=4;&gi==h�75�nM3�57<65B=4;¤14=4�NK6Ü�LO=Egi8:=4;6=

� ! 6 2 ; 0 � �)Z� . 68"°;°Ó ! 68"°; * " µ·x>-yx&�.¸02nEm6�OK.�>nM=4vj5e14K.;6LN8ML|gi=Æ=4;à=h��57nM3�5�� � ! 6 2 ; � �O=�LOK.n/�.=4�¤n�5�L¤=�143�5�148MK7;6=4L�<68:�)=��|¯

=�;6148M57nM=�L¤µ·x>-yx7rc¸@m�5��i5�=4;61�K.;&gi�O57�jnM57L � ! 6gfi; 5ÇgOKI<6K�gO8M=4vum*K6-lJ%57�i5Ç=4L|gi=¤=â´N=4vum6n:K6Üm*KI<>=4vuK.Lj�7=4�O8/Ñ�1s5��Àn�5Æ14K.;&�.=��O�.=�;6148M5<6=æn�5«57m6�OKskI8Mvj571�8M9.;Tܺ36LO57;6<6Km*K.�j=â´N=4vumI¯n:K6Ü%nM5æ14K.;><68M1�8M9.;8:;68M1�8�57n ü768"�9 2 ; 0 á°è)é 6g� Þnà�á "°; -2��5ÇLNK.nM3>148M97;Q=hk>571hgÃ5 ü=6g"�9hfi; 0á�è)é 68� Þ�à�á 68" � fi;:; Ü�gO8M=�;6=@=èk>m657;6LO8:9.;�<6=���K.3>�O8M=��¢1�3>�.K7L�14K[=�Ñ�1�8M=4;&gO=4LA=�LNgÃÝ�;�<�57<6K.Lm*K.�

� ! 68fi; 0¤á�è)é 6 zÇ�± ;_^ ! 68� ; + æ ! ö µ·x>-yx.�&¸<6K7;6<6= ^ ! 68fi; LNK.;n�57LC�)36;6148:K.;6=�LB<6=V»{=�LOLO=�n`<6=�K.�N<6=4; z ܧ143>�.57L�m6�OK7m68M=�<�57<6=�L_57L|¯8:;cgO97gi8:1s57L�8:vjm>nM8M1457; z ï � ! 6gfi;C� 2¤1�3�57;6<6K zE� � m�57�O5VgOKI<>K¤;65�gi36�O57n ð -�02LNgOK

x.F

Page 39: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

;6K7L¬<>8M14=�Þc36=Dn�5BLN=4�N8M=�gi�N36;61s5�<�5u<>=C��K.3>�O8M=��È14K7;c�7=4�N�.=Cvj57L`�OÝ7m68M<>KuÞc36=E1�3�57nMÞc368:=4�m*K7gi=�;6148M5BÑ�;68:gi5B<6=1 � #ÆÜ�m6�OK.m>8M=4<657<V14K.;6K[148:<�5B14K7vjK_14K7;c�7=4�N�.=4;61�8�5u=hkIm�K7;6=4;61�8�57n·-

�­Pe�]P � ����qÆ[Vq []\ ²vUNS02n�vjfhgiK[<6KE<6=³�§573u=4Lá36;�5��.=��OLN8M9.;jvjK[<68:Ñ61s57<�5A<>=4n�vjfhgiK[<6KE<6=¢£D5�nM=4�N¥c8M;TÜ.Þc36=

�N=4LO3>=4n:�7=¢3>;ÕLO8:LNgO=4vj5E3>;Õm*KI1�KC<68/�)=4�N=4;&gi=Cµ·5�n�<6=A£E57n:=4�O¥c8:;�¸%m�57�O5Dm*K[<6=4�º8M;>14K.�Nm�K.�O57�=hÑ�148:=4;&gi=�vj=�;cgO=¹nM57L�14K7;6<68M1�8MK.;>=4LV<6=Ç�)�OK.;&gi=��i5e=4; n�5LOK7nM361�8M9.;T-ÈÄ¢36=h�75�vj=�;cgO=7ܳLO=gOK.vÕ5u36;�5À��5�LO=�K.�|giK.�.K7;�57n#<6=@�)36;61�8MK.;>=4L Ó ` 68"°; Ü�=4;�=�LNgi=�vuf�gOKI<>KÀLO=B3Igi8Mn:8MÍ45j36;�5LNK.nM361�8M9.;V5�m6�OKskI8Mvj57<�5B<6=EnM5��)K7�Ovj5

üÄ c68"�9ifi; 0  =ñ �Ò ` À � � ` 6gfi;°Ó ` 68"°; µ·x>-yx&�.¸<6K7;6<6=�# =4Lu=4n³;cã6vu=4�NK«<6=Vgif��Ovu8M;6K.L_<6=¤n�5Ç=hkIm�5�;6LO8:9.;Å�`� =�Lj=�n¬;cã6vu=4�NK«<6=1�K.;6<68:148MK7;6=4L2<6=`�)�NK.;&gi=4�O5�<6=4n�m6�NK.�6n:=4vj5>-.02;un�5ALO=�141�8M9.;u<6=³=â´N=�vjm6n:K.L4Ü�3>gi8:nM8:Ís57�N=4vuK.L=�LNgi=¬vuf�giK[<6KDm�57�O5D�N=4LOK7n:�.=���nM5�=4143657148:9.;j<>=�»{3>�O�.=��OL4Ü[Þc36=¬=4Lá36;ÕvuKI<>=4nMKE�N=4<6361�8M<6K<6=EnM5B<68M;�Ý�vj8:1s5@<6=CÌ�3>8M<6K.L�Ü6���O=�m6�O=�LO=�;cgi5jK.;><�57L³Þc36=E1Ã�6KI1457;T-

�­P��1P � ����qÆ[Vq []\ ��q-�!q­R?UNR?W!XNZ02; =�LNgO=¹vuf�gOKI<6KLN=«5�LO36vu=�Þc36=¹nM5QLOK.n:36148:9.; 57m6�NK�kI8:vÕ57<65 üÄ  Li5�gO8MLN�X5�14=Çn�5

=�143�571�8M9.;�<>8:�)=4�N=4;61�8�57nPµ·xI-:, ¸�=4;_57n:�.36;6K7L§m636;&gOK.L " � 9h" � 9 ´¶´µ´ 9h"g  Ü7n:n�57vj57<6K.L��%~[÷�ü���ú¿�[õø¦�M�d��øOù[øhó���÷�<6=�n�<6K.vu8M;68:K��N=4LOm*=41hgi8/�.K6-c02Lº<6=41�8M��Ü7LO=³LN36LNgO8:gi3I�.=¬n�5�=èk>m657;6LO8:9.;¶µ·x>-~rc¸%=�;=�nPLO8:LNgO=4vj5ÍÏ 68üÄ c68"�æg;h; 0B, 68"�æ�; Ü>Þc36=�´N36;&giK�1�K.;�n�5�LÈ14K.;><68M1�8MK.;6=�LÈ<6=��)�NK.;&gi=��i5>Ü>�)K7�|¯vj57;@3>;�LO8ML|gi=�vÕ5�n:8M;6=457nc<6={=�143�5�148M97;6=4L 6 #/�� ;'¨»6 # �V ; m�5��i5�n:K.L§1�KI=hÑ�148:=4;&gi=�L � `6-

02LNgOK.LELOK.;ÆnMK.L�vuf�gOKI<>K.L�=4LNm�=�1�gi�O57nM=�LEvÕÝ7LE5�vjm6n:8�57vu=4;&gO=C36LO57<6K.L�-#�#5�L�=hkIm�57;I¯LN8MK.;6=�LCLN=Õ36LO57;«=�;«=�n�<6K7vj8:;68MK�=�LOm�571�8�57n·-�J§57�O5�nM5V=h�.K.n:36148:9.;¹gO=4vum�K7�i57n·ÜT�.=4;>=4�i5�n/¯vu=4;&gi=`LN=³36Li57;_nM57Ll<68/�)=4�O=�;6148M57L§Ñ�;>8:gÃ5�L4Ü7�@=4;&giK.;>14=4LáLO=ÈgO8M=�;6=4;�gÃ5�vB�68Mf�;�nMK.L§14K.;61�=4mI¯gOK.L�<6=214K7;6LO8:LNgi=�;6148M5È�A=4L|gÃ57�>8MnM8:<�57<TÜhÞc36=28Mvum6n:8M1s5�;�14K7vjKÈ�.5³LN=2�I8:96Ü4nM5{14K7;c�7=4�N�.=4;61�8�5>-

x.R

Page 40: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

��5jm>�O=4LN=4;&gÃ5�148M97;¶�)K.�Ovj57nT<>=@=4L|giK.L¢14K.;>14=4mIgiK.L���5uvÕ57L¬57n:n�Ý�<6=�n%m6�NK.m*9.LO8/giKu<6=@=�LNgi=gO�i57��5�´NK6-�02LNgOKjm63>=4<6=E�.=��OLO=D=4;¤=4n�1457m6o/gi36n:K_�_<>=4n��O=�148:f4;¤=4<68/gÃ57<6K_n:8M�6�NKVµily�Mm�¸h-

02;e=4nÈ1457m68:gO36nMK¶<6= �basõ!���%�d��úE�75�vjK.LB5æ8MnM3>LNgi�O57�@=�LNgO=¤vuf�gOKI<>K�57nÈ�N=4LOK7n:�.=���n�5=�143�571�8M9.;Ç<6=_J#K.8MLNLOK.;PÜP�æn:K�14K.vum�57�O57�O=�vjK.LA5À14K.;Çn�5�LALOK.n:36148:K.;6=�L�K.�>gi=�;68M<657LA14K7;<68/�)=4�N=4;61�8�57LÈÑ�;>8:gÃ5�L4-

r&Ò

Page 41: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� �����*Ø����TÙ cËedÕ��× �Ð�TÙÛÚ

�]P�OEP f WhgM\��=\�ZVR?WYUNs Î/W�Z]Wm� UNsi ���I���� jW�lk��������nmK�poq�Km��¡ä1�K.;&gi8M;c3�5�148M97;u<657�O=�vjK.Lº36;65Em*=4Þc36=�ß�5C=èkIm6nM8:1s571�8M9.;�<6=¬1497vjK�m*KI<>=4vuK.L2=hkIm6�N=4Li5��nM5�02143�5�148M97;B<6=³?A;6<�5BµX<6=�m�=�;6<68M=�;&gi=È<6=�n>LN=4�.36;><6K�K.�O<>=4;B<6=�nIgO8M=4vum*K¬�Cn�5¬m�K.LN8M1�8M9.;" ¸³=4;æ=�143�571�8MK.;>=4LAvjÝ7L�LN=4;61�8Mn:n�57LCµX<>=4m*=4;6<68:=4;&gi=�L�<6=4n%m6�O8:vj=��³K.�O<6=�;æ<6=�n%gi8:=4vum�K_�" ¸�m�57�i5¬=4;61�K.;&gi�O57�§LN3�LOK.n:36148:9.;T- 02LNgOKALO=��iÝ�m*K.LO8:�6nM=á3>gO8MnM8:Ís57;><6K¬��8:�)=��O=4;>148�5�L§�§8:;68:gi57L=�;«n�5�L�=4143657148:K.;6=4LD<6=um6�O8:vj=���K.�O<>=4;T-#02nºm6�OK.m*9.LN8:giKV<6=u�O=457nM8:Ís57��=4LNgO=Õ1457vB�68:K6ÜT=4L<6=��68M<>KE57n6LO=��.36;6<>KDK7�O<6=�;u=�;_=4n>gO8M=4vum*K6Ü�m>36=4Lº=4LºvjÝ7Ll1�K.vBã6;_�O=�LOK.n/�.=��2=4143657148:K.;6=4LÞc36=E=��7K.nM3>148MK7;�57L³=4;¤m6�O8:vj=��ÈK.�N<6=4;¤<6=4nTgO8M=�vjm*K6-��5_02143�571�8M9.;¤<6=C?A;6<�5�=4L|gÃÝ_<�57<�5�m*K.�

� � �$" � > ( � � � �$f � 0B2 µXr6-/, ¸°I8�5�m6nM8:1s57vuK.L{3>;�5Bgi�i5�;6LN�)K.�NvÕ5�148M97;�<6=E14K[K.�O<6=�;�57<�5�L¬<>=DnM5@�)K.�Ovj5»*sr" 0F* "Ã>ut * f�V* rf 0B* f Ü6n�5�;c36=h�75_=hkIm6�N=4LN8M9.;¤m�57�O5unM5B=4143657148:9.;V<6=EK.;6<�5_LO=��iÝ

r6,

Page 42: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�'�! � r" � > t��( � ���! � r" � ��± t( � �'�! � r"°� rf > ( � ���Y � rf � 0B2 µXr6-HG.¸<6K7;6<6=vr" � rf LOK.;Æn�57L�;c36=h��57LC1�KIK7�O<6=�;�57<�57L�� t 36;¹1457vB�68:KV<68ML|gi8M;&gOK¤<>=u1�=4�NK6Ü�=4n1�3�57nP�N=4m6�N=4LN=4;&gÃ5_n�5@�7=4nMK[148:<�57<À1�K.;Vn�5@Þc36=�=�nTLO8:LNgO=4vj5 68"�9hfi; �7=4�OÝ_Þc36=ELN=�vB36=��7=�=4n;c36=h�.KÕLN8ML|gi=4vj5�14K[K.�O<>=4;�57<>K6-��=�Li57�N�OK.n:n�57;><6K�n�5B=�17-%µXr6-HG.¸{<6=EnM5@�)K.�Ovj5BÞc36=CLN8M�.3>=7Ô

�� r"xw 6 > t �( � ; �% � r" � t( � �% �$f�y � �� rf w t( � �% � r" > ( � �% � rf y 0B2 µXr6-yx&¸gOK.vÕ5�;6<6K (�0�`�_K.vu8:gO8M=4;><6KAn�5�Llgi8Mn:<6=4L�Ü.<6=¬n�5A=�1Bµ)r6-Hx.¸§m�K[<6=4vuK.L28:;cgO�OK[<6361�8M�2=4L|gÃ57L��57�N8�57�6n:=4L � 0 � ö Ã>zt � ¾ �|{)0 � ¾ 02nPm>�OK.m*9.LO8/giKB=�L`=hkIm6�N=4Li5��¬nM5@=41�3�571�8M9.;�<6=�K7;6<�5Br6-HG@=4;V<6K.L{=4143657148:K.;6=4L`<>=�=��.K7nM3I¯1�8M9.;æ<>=Bm6�O8:vj=��¢K7�O<6=�;�=4;�=4n�gi8:=4vum�Kjm�57�O5Õn�57L¢;c36=���57LA��57�N8�57�6n:=4L�-���5Õm>�O8Mvu=4�O5j<6=f�LNgÃ5�L¬=�L¢8:;6vu=4<68M5�gÃ5@<>=DnM5B=41�-#µXr6-yx&¸

� ö � 0 � ¾ 6 {}� tt� ; µXr6-~rc¸¡¢LO36vu8M=�;6<6K�Þc36=�n�5�021�3�57148:9.;�<6=D?A;6<�5�=�L�} � ÜIm�K[<6=4vuK.LÈK7�>gi=�;6=4�`36;65_=èkIm6�O=�LO8M97;m�5��i5_5�=��.K7nM361�8M9.;V<6=�{ä<6=En�5BLN8M�.3>8M=4;&gO=A�)K.�Ovj5

� ö {)0 � ¾ 6�� ö ø; µXr6-H�.¸��=EnM5B<6=�Ñ�;>8M148:9.;V<6= � K7�>gi=�;6=4vuK.L

� ö 0 � � t { µXr6-y�&¸LN36LNgO8:gi3I�.=4;><6KÕnM5B=417-#µ)r6-y�&¸{=4;¤n�5�=41�-�µ)r6-H�.¸{Ñ�;�57n:vj=�;cgO=�K.�>gO=4;6=�vjK.L

� ö {)0 � ¾ 6g� � t { ; µXr6-H�.¸

rcG

Page 43: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

��=@=4LNgi5ÀvÕ5�;6=4�O5ÕK.�>gO=4;6=�vjK.L�n�57L¢<6K.LA=�143�571�8MK.;>=4LA<6=@=��.K7nM361�8M9.;eµ)r6-~rc¸³�QµXr6-H�.¸³n�57L1�3�57nM=�L{1�K.;6LNgO8:gO3>�.=�;¤nM5C=�143�571�8M9.;Õ<6=AK.;6<65>->¡¢LOoPm>36=4L�Ü657n*8M;&gi=��.�i5��{n�5D=41�3�578M97;ǵXr6-y�&¸´N36;&gOK@14K7;Õn�5�L2<6K.LÈ5�;cgO=4�N8MK.�N=4L4Ü&=�;614K7;cgO�i57�N=4vuK.Lá=4n��75�nMK.�2�63>LO1s5�<6K@<6=¬n�5���57�O8M57�6n:= -��57L`=�143�571�8MK.;>=4L@µXr6-~rc¸hÜ�µXr6-y�&¸��Ƶ)r6-Ó�7¸{LO=Em636=�<6=4;��.=4;6=��i57n:8MÍ457�È<6=En�5B�)K.�NvÕ5IÔ

� ö & 0B, 6�&<; µXr6-yF&¸<6K7;6<6= , 68&$; =4L`36;�K7m�=��i57<6K7�È=4LNm�57148M57nT5�m6nM8:1s57<6K@5@n�5D�)36;6148:9.;�Þc36=�Þc36=4�N=4vuK.LÈ8:;cgO=h¯�.�O57�4-¡¢m6�OKskI8Mvj57�O=�vjK7L{n�5�L{=41�3�571�8MK.;6=�LDµXr6-~rc¸èÜ�µ)r6-H�.¸á��µ)r6-H�.¸�36LO57;6<6KÀòEó ôsõhöiõh÷PøhóXù�ú³ûºóa÷�óaüZù�ú��gO�i5�gi57�O=�vjK.L`<>=C=�;614K.;&gO�i57�¢nMK.LÈ��57n:K.�O=�L¬<>= Üb{�� � =4;�36;¶1�K.;�´N36;&giKj<6=Dm636;&giK.L=�;�=4n§=�LOm�5�148MK�¯ýgi8:=4vum�K>-���8MLN14�O=hgi8:Ís57�N=4vuK.L�=�n§=4LNm�571�8MK�¯Zgi8M=�vjm*Ku<6=@gÃ5�n#�)K.�NvÕ5uÞc36=�n�51�KIK.�N<6=4;657<�5ægi=�vjm*K.�i5�n{=4L|gifV�O=�m6�O=�LO=�;cgi57<�5Çm�K7� f ` 0 «�~ f �nM5¶14K[K.�N<6=4;�5�<�5�=4L|¯m�5�148�5�n�m*K.� "�æ 0 ç ~ " Ül14K.; « =]çA=4;&gi=��OK.L�Ü·çN0 9 ´µ´¶´ 9]^ <6K.;><6= ^ =4L�=�nÈ;cã6vu=4�OK<6=DnMK7L¢m63>;cgOK.L�<>=@n�5_�O=�<¶=�;æn�5_<68M�N=41�148M97;�=�LOm�571�8�57n·-*J§57�O5jLO8:vjm>nM8:Ñ61s57�`nM5u;6K�gÃ571�8M9.;<6=�;6K7gi57�O=�vjK.L¿, 68f ` 9h"�æ8; m*K.�¿, `æ¡¢m6nM8:1s57;6<>K�<68:�)=��O=4;>148�5�LÈÑ�;68:gi57L³5�n�5�=41�3�571�8M9.;¹µXr>-H�&¸{K7�>gi=�;6=4vuK.L4Ô

`æ >) ` ó �æ~ f � � � t {��=�LOm*=â´O57;><6K `æ <6=En�5B=�143�571�8M9.;�57;&gi=��O8:K.�4Ü

`æ 0 ` ó �æ � ~ fY68� � t { ; µXr6-yR&¸��=Dn�5�=�143�571�8M9.;«µXr>-HR&¸Èm*KI<>=4vuK.L`�.=4�¬Þc36=Dm�57�i5_K.�>gO=4;6=��¢n:K.LÈ��57nMK7�O=4L³<6= =4;�1s57<�5m63>;cgOK¶<6=un�5V�O=�<TÜ�<6=4�*=4vuK.LC57;&gi=�LC1�K.;6K[14=4�DnMK7LE��57n:K.�O=�LE<6= � ��{½=4;«1457<�5¤m636;I¯gOK6Ü>m�5��i5B=4;>14K.;&gi�O57�`<68:1Ã�6K.L{��57n:K.�O=�L{gi=�;6=4vuK.L`Þc36=��O=�LOK.n/�.=��`n�57L�=4143657148:K.;6=4LDµ)r6-~rc¸á�µ)r6-Ó�7¸h-

r&x

Page 44: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¡¢�6K.�i5¶57m6nM8:Þc36=4vuK.L@<>8:�)=4�N=4;61�8�57LCÑ�;68:gi57LB5�n{n�5�<6Kæ<6=��O=41Ã�>KÆ<>=�n�57LC=4143657148:K.;6=4L�Þc36=Þc36=��O=4vuK.L³�O=�LOK.n/�.=��@µ)r6-~r&RB�ƵXr6-H�.¸è-

� ö � 0 { ` ó �æòñ � > { ` ó �ædó � � tC6g� ` ó �æòñ � >�� ` ó �ædó � ;± ~ " µXr6-/,�Ò&¸�

� ö {)0 � ` ó �æòñ � >�� ` ó �ædó � � tC6 { ` ó �æ�ñ � > { ` ó �ædó � ;± ~ " µXr6-/,., ¸J%57�i5¬=4;61�K.;&gi�i5��%n:K.L��75�nMK.�N=4L�<6= � ��{��O=�LOK.n/�.=4�N=4vuK.L§nM57L�=41�3�57148:K.;6=�Lȵ)r6-:,sÒ&¸P�jµXr6-/,., ¸3>gO8Mn:8MÍs5�;6<6KB=�n�vjfhgiK[<6K�<6=D²¬3>;6�.=h¯wÏ�3>gOgi5_<>=D145736�|giKuK.�O<6=�;T-��K.L`�N=4LO3>n:gÃ5�<6K.L³;c36vjf��O8:14K.L`LN=Cm6�N=4LN=4;&gÃ57;�=4;¤=4n�LN8M�.36=�;&gi=E1s57m>o:gi3>nMK6-

i ���I����� jW�lk��������nmK��������0�0�I�02;u=4LNgi5DLN=41�148M97;j=èk>m*K.;6<>�O=4vuK.Lºn�5ALNK.nM3>148M97;�<6=³n�5�=41�3�57148:9.;_<6=¬J%K.8:LOLNK.;j1�K.;u��8:�)=��|¯=�;6148M57LD�§8M;68/gÃ57L�36Li57;><6K¶n:K.L�vjfhgiK[<6K.LD®.571�K.�68·Ü�£D5�36LOLâ¯|°I=�8M<6=�nº�Ç°IK.�6�N=j²³=4nM5 ´O571�8M9.;°I3>14=4LN8:��5 Ø °6?�²�ÙÃ-��5�=41�3�57148:9.;¤<6=DJ%K.8:LOLOK7;¤=�;�<>K.L³<68Mvu=4;6LN8MK.;>=4LDµ "�9h& ¸{LN=D=èkIm6�O=�Li5j1�K.vuKuLN8M�.3>=7Ôw � ® 68"�9:&<; 0j, 6g"�9:&<; µXr6-/, G.¸<6K7;6<6= w � ® 68"�9:&<; 0 ��� ®�$" � � ��� ®��& �-¡¢m6nM8:1s57;6<>K�n�5�=41�3�57148:9.;¹GI-yR�57n§n�57<6KV8:Í4Þc368M=��O<6KV<6=_nM5V=�143�5�148M97;Ær>-:, G[ÜTK.�>gO=4;6=�vjK.L1�K.vjK��N=4LN36n:gi57<6Kun�5BLO8:�.36=�;cgO=E=hkIm6�O=�LO8:9.;

r.r

Page 45: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

® æòñ � î è ��® æµó � î è ��® ædî èhñ � �)® ædî è¦ó � >H@ ® ædî è¯ � 0j, 68"�æ�9:&©è!; ´ µXr6-/,�x&¸<6K7;6<6=�¯å=�LBgÃ57vj57ß6K¶<6=�n�57L@<68:�[8MLN8MK.;>=4LB<6=�nÈ=â´N=�=�LOm�5�148�5�nZ-%ê¬gi8:nM8:Ís57�N=4vuK.LCn:K.L@gO�O=4Lvuf�gOKI<6K7L³57�O�N8M��5Bvu=4;61�8MK.;657<6K.L`m�5��i5�<�57�`LNK.nM361�8M9.;¤5�n�5B=�143�5�148M97;�r6-/,�x>-®.571�K.�68·ÜT£E5736LNL|¯|°[=48M<>=4n§�æ°6?�²�LNK.;ÇvjfhgiK[<6K.L�LN8ML|gi=4vjÝ�gO8M14K7LA8:gO=4�i5�gi8:�7K.L�ãIgi8Mn:=4L¢m�57�i5�N=4LOK7n:�.=���;c36vj=��OK.LNK.LELO8:LNgi=�vÕ5�L�<6=_=4143657148:K.;6=4LEn:8M;6=457nM=�LA57n:�.=4�6�O578M1457L4-P02LNgiK7LDvuf�gOK�¯<6K7Ll14K.;>LO8ML|gi=�;�=4;�3>gO8MnM8:Ís57��gi57;&gÃ57Ll8/gi=��i571�8MK.;6=�L§1�K.vjK�LO=s5�;�;6=41�=4Li5��O8�5�L2m�57�i5¢K.�>gi=�;6=4�nM5CLOK7nM361�8M9.;Àvu=â´NK7�È57m6�NKsk>8:vÕ5�<�5>-I°I=Am6�NK.m*K.;6=A36;65BLOK.n:36148:9.;Õ<6=�n�5@=�143�571�8M9.;Ƶ)LOK.n:3I¯1�8M9.;<6=�m>�O36=���5&¸Bn�5¶1�3�57n`=�L�3>gi8:nM8MÍ457<�5¶m657�i5æ1457nM1�36n�57�Cn�5æLNK.nM361�8M9.;=�;en�5æm6�N8Mvu=4�8/gi=4�O57148:9.;TÜ[=4;&giK7;614=�L4Ü�=4n��75�nMK.��Þ[3>=DLN=�K.�>gO3>�.K_=4;VnM5Bm6�O8:vj=��{8:gO=4�O57148:9.;À=4L`3Igi8Mn:8MÍ457<6Km�5��i5�K.�>gO=4;6=��D=�n§��57nMK7�A<6=_n�5ÀLNK.nM361�8M9.;�=�;¹nM5ÀLO=��.36;6<�5V8:gO=4�O57148:9.;æ��=4L|gi=u5�LO3Ç�.=�Í=�L¢3>gO8Mn:8MÍs5�<6K�m�57�O5uK.�Igi=4;>=4�¬=4n���57nMK.�`=�n�n�5�gi=4�N14=��i5j8/gi=��i571�8M9.;TÜI�¤57LOo�LO3>14=4LN8:��57vu=4;&gi=��5�LNgÃ5Àn:nM=��&57�A5ÀnM5Õã6n/gi8Mvj5j8/gi=��i571�8M9.;ænM5Õ143657nl;6K.L�m>�OK.m*K.�O1�8MK.;657�iÝÕnM5ÕLOK7nM361�8M9.;ævjÝ7L57m>�OKskI8Mvj57<�5�5�n�5�LOK7nM361�8M9.;�=èk65�1�gÃ5¤571�K7gÃ57<65Vm*K.�E57n:�.36;�5�<6=hÑ�;68:148M97;Æ<>=u=��O�NK.�4-�02nm6�NKI1�=4<68:vj=�;&giK¶8/gi=��i5�gO8:�.K¤LO=Õ<68M1�=ÕLO=��@1�K.;&�.=4�N�.=4;&gO=¤1�3�57;6<>Kæn�57LD<68/�)=4�O=�;6148M57L@=�;&gi�O=nM5�LOK7nM361�8M9.;¹=hk>571�gi5¶�Æn�57L@57m6�OKskI8Mvj571�8MK.;6=�LCLN3614=�LO8/�75�L@gi8:=4;6<6=�;5¶1�=4�OK�1�3�57;6<>Kæ=4n;cã6vu=4�NKu<>=D8/gi=��i571�8MK.;6=�L³5736vu=4;&gÃ5I-��=_=èk>m>�O=4LN8MK.;>=4LD14K.vuK�r6-/,�x�m�K[<6=4vuK.L�K.�Igi=4;>=4�E36;ÇLO8:LNgi=�vÕ5�<6=_=�143�571�8MK.;>=4LE<6=_n�5LN8M�.368:=4;&gi=A�)K7�Ovj5Vµ§14K.vuK_=â´N=4vum6n:K6Ü636;VLO8:LNgO=4vj5_<>=Dru=41�3�571�8M;6=�LøhÜ

� ��� " � � � � � " � � � � ³ " ³ � � � º " º 0 # �� � � " � � � ��� " � � � � ³ " ³ � � � º " º 0 # �� ³ � " � � � ³ � " � � � ³�³ " ³ � � ³ëº " º 0 # ³� º � " � � � º � " � � � º�³ " ³ � � º�º " º 0 # º µXr6-/,src¸<6K7;6<6= � æ æ�?032_m657�i51ç70� 9 ´¶´µ´ 9:@ Ü>m�K[<6=4vuK.L`=�LO14�N8M�68:�³n�57L`=�143�571�8MK.;>=4L¬57;&gi=��O8MK7�O=4L³<6=nM5BLO8M�7368M=�;cgO=�vÕ5�;6=4�O5>Ü

rc�

Page 46: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

" � 0 � ��� 6 # � > � � � " � > � � ³ " ³ > � � º " º ;" � 0 � ��� 6 # � > � � � " � > � � ³ " ³ > � � º " º ;" ³ 0 � ³�³ 6 # ³ > � ³ � " � > � ³ � " � > � ³ëº " º ;" º 0 � º�º 6 # º > � º � " � > � º � " � > � º�³ " ³ ; µXr6-/, �.¸i ���I����� ��������mK��m�������������tw;6<>8M1s5��O=4vuK.L_n�5�Lj8/gi=4�O57148:K.;6=�Lu<>= "�æ <6=�nM5Ç�)K7�Ovj5 " � ` æ 14K7; « =4n¬;[ã>vj=��OK«<>=�8/gN¯=��i571�8MK.;6=�L4-�¡�LO3>vj8:=4;6<6KÕÞ[3>=�n�5j;[¯wf4LN8Mvj5j8/gi=4�O57148:9.;æ=�LA14K.;>KI1�8M<�5Õm657�i5ÃçC0 9 ´µ´¶´ 9:@ Ü=�;cgOK.;61�=4LB=�nÈd�fhgiK[<6K¶<>=�®.571�K.�68á=hkIm6�O=�Li5ænM5¤8/gi=4�O57148:9.; « � Õ=�;«gif��Ovu8M;6K7LC<6=jn�58/gi=4�O57148:9.;�;I¯ýf4LO8:vÕ5I-6��57L³=41�3�57148:K.;6=�L¬1�K.�O�N=4LNm�K.;><68M=�;cgO=4L¢5�=4LNgO=Cvuf�gOKI<>K�LOK.;TÔ

" ` ñ �� 0 � ��� 6 # � > � � � " ` � > � � ³ " ` ³ > � � º " `º ;" ` ñ �� 0 � ��� 6 # � > � � � " ` � > � � ³ " ` ³ > � � º " `º ;" ` ñ �³ 0 � ³�³ 6 # ³ > � ³ � " ` � > � ³ � " ` � > � ³ëº " `º ;" ` ñ �º 0 � º�º 6 # º > � º � " ` � > � º � " ` � > � º�³ " ` ³ ;02;¤�.=4;6=��i57n�m657�i5�=4n�1457LOKu<6=Ev =�143�571�8MK.;>=4L4Ü

" ` ñ �æ 0 � æ æ çø# æ°> ædó �Ò æ À � � æ è¦" � ` è > ïÒè À æòñ � " � ` è ê ç=0É 9 ´¶´µ´ 9�ð µXr6-/,��&¸¡¢m6nM8:1s57;6<>K�n�5B=41�3�571�8M9.;«µXr6-/,��&¸È5¤µ)r6-:,sx&¸

r&�

Page 47: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

® ` ñ �ædî è 0 @ 6 ® `æ�ñ � î è ��® `ædó � î è �)® `ædî èiñ � ��® `æµî èyó � > ¯ � , ædî èY; µXr6-/, �.¸i ���I� i � ��������mK��m������k��0�"�T������m����02;u=4LNgO=³vjfhgiK[<6K�=4n6��57n:K.�l<6=³n�5A8/gi=��i571�8M9.; 6?« � ; LN=4�OÝD3>gO8Mn:8MÍs5�<6K�gÃ5�;um>�OK.;&giKE1�K.vjK=�LNgif{<68MLNm�K.;>8M�6n:=7-���57L§=�143�571�8MK.;>=4Ll=�Þ[3>8:��57nM=�;&gi=4Ll5¢n�5�L§=41�L4-6µXr6-/, �.¸�m�57�i5¬=4L|gi=`vuf�giK[<6KLNK.;

" ` ñ �� 0 � ��� 6 # � > � � � " ` � > � � ³ " ` ³ > � � º " `º ;" ` ñ �� 0 � ��� 6 # � > � � � " ` ñ �� > � � ³ " ` ³ > � � º " `º ;" ` ñ �³ 0 � ³�³ 6 # ³ > � ³ � " ` ñ �� > � ³ � " ` ñ �� > � ³ëº " `º ;" ` ñ �º 0 � º�º 6 # º > � º � " ` ñ �� > � º � " ` ñ �� > � º�³ " ` ñ �³ ;02;¤=4n�1s5�LOKu<6=Ev =41�3�571�8MK.;6=�L

" ` ñ �æ 0 � æ æ ç # æ%> ædó �Ò è À � � æ è¦" ` ñ �è > ïÒè À æ�ñ � � æ è¦" � ` è ê ç70� 9 ´¶´¶´ 9�ð µXr6-/,�F&¸

¡¢m6nM8:1s57;6<>K�n�5B=41�3�571�8M9.;«µXr6-/,�F&¸È5�nM5B=4143657148:9.;Qµ)r6-/,�x&¸{K.�>gO=4;6=�vjK7L® ` ñ �ædî è 0 @ 6 ® `æ�ñ � î è ��® `ædó � î è �)® `ædî èiñ � ��® ` ñ �æµî èyó � > ¯ � , ædî èY; µXr6-/,�R&¸i ���I����� ��������mK��m����������������������5�I�l���I����kK�����0��� �n¡G��oq�£¢°I8PLO36vj57vuK.LÈ�À�N=4LNgi57vuK.L`=4nTgOf4�Ovu8M;>K " � ` æ 57nTnM57<6K�<6=��O=41Ã�>KÕ<6=�nM5B=4143657148:9.;¹µXr6-/,�F&¸;6K7L¬Þc36=�<�5_n�5�LO8:�.368M=�;&gi=�=hkIm6�N=4LN8M9.;

rc�

Page 48: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

" ` ñ �æ 0 " � ` æ �  � æ æ ïM# æ°> ædó �Ò è À � � æ è¦" ` ñ �è > ïÒ è À æ � æ è:" � ` è ð�Ä µXr6-HG7Ò&¸��5�=èkIm6�O=�LO8M97;<6=4;&gi�NKæ<6=jnMK.LC14K.�N1Ã�6=�gO=4LB=�LBn�5V14K.�N�O=�148M97;QK¶1457vB�68:K¶�6=�1Ã�6KÇ5 " � ` æm�5��i5cç70� 9 ´µ´ 9�ð m657�i5¬36;�5�8/gi=4�O57148:9.;D<6=³£E5736LOLâ¯|°I=�8M<6=�nZ-�°I8[n�57L#14K7�O�O=�141�8MK.;6=�L§LN3614=�LO8/¯��57LºLOK7;jgiK[<�57L2<>=¢36;uLOK7nMKDLO8M�7;6K6Ü.LN=4�OoM5D�O57Í4K7;�57�6n:=`=4LOm*=4�O57�áÞ[3>=¢nM5E1�K.;&�.=��O�.=�;6148M5CLO=571�=4n:=4�O=¬LO8�5C1457<�5D36;�5@<6=¢n�57Lº=�143�571�8MK.;>=4L{r>-ÓG7ÒDLO=¢nM=4Lá<�5@3>;jgif��Ovu8M;6KD<6=¢14K.�N�O=41�8M9.;vjÝ7L`�.�O57;6<6=�Þc36=D=4n�<6=hÑ�;68:<6K_=4;¤<68M1Ã��5_=41�3�571�8M9.;T-¡¢LOo�=4Lá14K7vjKELO3>�O�.=¬n�5�8:gO=4�O57148:9.;�LOK.�>�O=¬�O=4nM5 ´O57<�5�LN3614=�LO8:��5EKB°>?E²�nM5A143�5�n*LN=³<6=�Ñ�;>=m*K.�`n�5�L`=4143657148:K.;6=4LCµ)=4;¤=4n�1s5�LOKu<6=E36;�LN8ML|gi=4vj5�<6=Er_=4143657148:K.;6=4Li¸

" ` ñ �� 0 " � ` � � ¤� ��� ïM# � > � ��� " � ` � > � � � " � ` � > � � ³ " � ` ³ > � � º " � ` º ð" ` ñ �� 0 " � ` � � ¤� ��� ïM# � > � � � " � ` � > � ��� " � ` � > � � ³ " � ` ³ > � � º " � ` º ð" ` ñ �³ 0 " � ` ³ � ¤� ³�³ ïM# ³ > � ³ � " � ` � > � ³ � " � ` � > � ³�³ " � ` ³ > � ³ëº " � ` º ð" ` ñ �º 0 " � ` º � ¤� º�º ïM# º > � º � " � ` � > � º � " � ` � > � º�³ " � ` ³ > � º�º " � ` º ð ´��K7;6<6= ¤ =4LE=4nlm657�iÝ7vu=�gO�OK�<6=u571�=4n:=4�i5�148M97;�KV�X5�1�giK7�A<6=_�O=�n�5 ´O571�8M9.;PÜP=4n§143�57n%�&571�=<6=�;&gi�OK�<>=4n#�i57;6�7KÅ1A ¤ A ±I-�J§57�O5Õ=4n%1s57LNK�<6=@3>;�LO8ML|gi=�vÕ5u<6= ð =41�3�57148:K.;6=�LAn�5=�143�571�8M9.;VÞc36=D<6=4LN14�N8M�*=C5�nTvjfhgiK[<6K_<6=C°6?�²àLO=�gO8M=4;>=7Ô

" � ` ñ � æ 0 " � ` æ ��¤� æ æ ç # æ%> ædó �Ò è À � � æ èy" � ` ñ � è > ïÒ è À æ � æ è¦" � ` è ê ç70 9 ´µ´¶´ 9°ð µXr6-HGI, ¸¡¢m6nM8:1s57;6<>K�n�5B=41�3�571�8M9.;«µXr6-HGI, ¸È5�nM5B=4143657148:9.;Qµ)r6-/,�x&¸{K.�>gO=4;6=�vjK7L

® ` ñ �ædî è 0 6 > ¤ ; ® «°ædî è �¥¤ @ 6 ® `æòñ � î è ��® `ædó � î è ��® `ædî èhñ � �)® ` ñ �ædî è¦ó � > ¯ � , ædî èY; µXr6-HG.G.¸r&F

Page 49: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

02;&giK.;>14=4L�m�57�i5un�5j0º1�3�571�8M9.;�<6=BJ#K.8MLNLOK.;TÜ*n�5�L¢=�143�571�8MK.;>=4L�5u�O=4LNK.n:�7=4�¢LOK.;TÔ§µXr>-:, �7¸hܵ)r6-:,sR&¸á�Ƶ)r6-ÓG7G.¸h->��K7L`�O=4LN36n:gi57<6K.L`;c36vuf4�N8M1�K.L`<6=E<68M1Ã�657L`=41�3�57148:K.;6=�L¬LN=E=hkIm�K7;6<6�iÝ�;=�;�=�n�1s57m6o/gi36n:K�LO8:�.368M=�;&gi=7-

�]P:n1P � ����qÆ[Vq QFsj}\�R{�j�=U-�)[]\ ��q-�YqÆRTUER?W!XuZi �������� jW�lk��������nmK��������0�0�I�¡¢�6K.�i5�=èk>m*K.;6<>�O=4vuK.L¬n�5�vj57;6=4�O5�<6=E<�57�³LOK.n:36148:9.;¤5unM5�=4143657148:9.;�<>=CJ#K.8MLNLOK.;¤14K7;=�n#d�f�gOKI<6K_<6=Cz{K.n:KI1457148:9.;V02LOm*=41hgi�i5�nZ-z{K.vuKB�&5�LN=�vj=�;6148:K.;69_57;&gi=��O8:K.�Ovu=4;&gi=�Ü>Þc36=��O=4vuK.L`=�;614K.;&gO�i57�`36;65_LNK.nM3>148M97;�<6=�n�5�)K.�NvÕ5

ü76g"�9:&$; 0  Ò ` À �  Òï À � � ` ï ù ` 68"°;�ù ï 6�&<; µXr6-HG7x&¸z{K.vuKBn:K.L�gOf4�Ovu8M;>K.L ù ` 6g"°; � ù ï 6�&<; µ)m�K7nM8M;>K.vj8:K.L2<6=Dz{�6=��&�ILN�6=��>¸È�&5BLNK.;�14K.;>KI1�8/¯<6K7L4Ü6n:K�Þ[3>=D;>K.L`�O=�LNgÃ5�m*K.�`��571�=4��Ü6=4L³=4;>14K.;&gi�O57�³nMK.L{14K[=�Ñ�1�8M=4;&gO=4L � ` ï Ü>�Àm�5��i5B=4n:nMKLN36LNgO8:gi3>8M�O=�vjK7LÈn�5�=hkIm�5�;6LO8:9.;�<>=DnM5B=41�3�57148:9.;¤<6=DJ%K.8:LOLOK7;TÜw � ® 68"�9:&<; 0j, 6g"�9:&<; µXr6-HG�rc¸57LN36vu8M=4;><6KEÞ[3>=¬nM5�=4143657148:9.;_LO=¬1436vum6nM=`m657�i5�nMK7L2m636;&giK.Lá<6=³14K7nMK[1s571�8M9.;TÜ�=4;&giK7;614=�LLN36LNgO8:gi3>8MvuK.LÈn�5B=èkIm�57;6LN8M9.;V��n�5�=h�75�nM3�57vuK.LÈ=�;�<>8M1Ã�6K.L³m63>;cgOK.L³K.�>gO=4;68:=4;6<6K

 Ò ` À �  Òï À � � ` ï<¦ * �iù ` 68"�æ�;* " � ù ï 6�&�è!; � ù ` 6g"�æ8; * �hù ï 6�&©è!;* & � §©¨ , 68"�æ�9:&©è!; ª�«­¬¯®"°"±²´³­µT¶�·¹¸]·"µWº¼»´½�¸]·"µ­¾­¿À¸Á¿Â·´µ­ÃÁ½�¾­ÃÅÄÇÆÈ·"µT¶EÃ]ÆE» ª�É ÆÈ· É ·´Æ�¸Á¿À·"µÊ»´¾­»´½ É »´Æ�»>¸Ë»´¾Ê»>Ã%ÌÈÃÁÍ É ºÂ·>Ã]µÉ »+ÆȶE¿À¸Á³­ºÂ»´Æ ± ½�ÃR¶�ÃÁµ­¾�ÆEδµ 6 # �à ;Ï� ÃÁ¸]³Ê»´¸Á¿À·"µ­Ã]½:ºÂ¿Àµ­ÃË»+ºÂÃÁ½ É »´ÆE»�º¼»´½ 6 #v�à ;�� ¿Àµ­¸ÁÏ´Ð"µ­¿À¶E»´½«TÑ

Page 50: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� ` ï ¬Ò Ã|Ã]½È¶E»vÍÓ»´µ­Ã]ÆE»Ô¶EÃ]µ­¾­ÆÈÃÁ͹·"½BÕ0³�ÃWÆÈÃÁ½È·"ºÀÖ´ÃÁÆB³­µÊ»×ÃÁ¸]³Ê»´¸]¿ÂÏ"µqÍÓ»+¶EÆȿ¸Á¿Â»´º¾­Ã�º¼»ÄÇ·"ÆÈÍ©»£ ­Ø ¨ ­# ª�«­¬¯®´ÙT±¸]·"µÚÃ]º�Ö"Ã]¸]¶E·´ÆÛ¸]·"ºÂ³�Íӵʻ ­Ø ¨ 6 � ����9 � � � 9 ´©´©´ 9 �  r T;_¬ÜK»´¸Á¿ÀÃÁµ­¾­·ÛÆ�ÃhÄÇÃÁÆÈÃÁµ­¸]¿¼»Û»ÛºÂ»�Ã]¸Á³Ê»+¸Á¿ÂÏ´µ «­¬¯®"°�É ·0¾­Ã]ÍÓ·"½�Ö´ÃÁÆ�Õ0³�ÃRºÀ·"½�¶EÝÁÆÈÍÓ¿Àµ­·"½�¾­ÃIº¼»�ÍÓ»'Þ¶�Æ�¿Âß £ ÃÁ½È¶E»´µ>ÄÇ·"Æ�ÍÓ»´¾­·´½ É ·"Æ ¦�à � �]á � ¾ - à ¾ � ù ï 6�&©è!;'â­ù ` 68"�æ8; à � � / �.ã�ä à ã � §�å ¾�·"µ­¾­Ã�ºÂ» É Æ�¿ÀÍÓÃ]ÆE»æ ºÂ»W½Èé·"çè¶E¿ÂÃ]µ­Ã©¸]³Ê»´µ­¾�· ç ¨ 2 å é ¨ 2 å º¼»�½�Ã]Ð"³­µ­¾­»Ô¸Á³Ê»+µ­¾­· ç ¨ 2 å é ¨né¹ê »´½�ë½È³­¸ÁÃ]½�¿ÀÖ+»´Í¹ÃÁµT¶�à ¬

Ò ÃBºÂ»�Ã]¸Á³Ê»+¸Á¿ÂÏ´µ «­¬¯®´Ù Ö"Ã]ÍÓ·´½ìÕ5³­Ã¹½�·"ºÀ·�ÃÁ½�µ­ÃÁ¸]ÃÁ½�»´Æ�¿À·Ú¿Àµ5Ö´ÃÁÆ�¶E¿ÂÆ�ºÂ»NÍ©»+¶�Æ�¿Àß £ ê É ·+Þ¾­ÆÈÃÁ͹·"½ÃÁµ­¸]·"µT¶EÆ�»´ÆR¸Á»´¾Ê»�³­µ­·�¾­ÃKºÀ·"½¸Á·0Ã æ ¸]¿ÂÃ]µ5¶�ÃÁ½�í ` ï ê »´½�ëÊÃ]µ­¸Á·´µ5¶�ÆE»´ÆIº¼»�½�·´ºÂ³­¸]¿ÂÏ"µü îÇïsð�ñ­ò Ã]¸ ¬Ê«­¬¯®´ó�¬ô »´ÆE»�¿ÂµTÖ"Ã]Æȶ�¿ÂÆ�º¼»ÅÍÓ»+¶�Æ�¿Âß £ ½�Ã�³�¶�¿ÂºÀ¿ÂßÁÏ�ÃÁºõÍÓÝh¶E·0¾­·@¾­Ã�¾­Ã]½�¸Á·´Í É ·"½�¿À¸Á¿ÀÏ"µ §Åö ¬5÷ µ©Ã]½È¶EÃ͹Ý]¶�·è¾­·>½ÈÃìÃ]½�¸]Æ�¿Âçbà £ ¸]·"͹·>ÃÁº É ÆÈ·è¾­³�¸]¶E·¹¾­Ã�¾­·"½�ÍÓ»+¶EÆȿ¸]ÃÁ½£ ¨ùøIú ª�«­¬¯®"û"±¾­·´µ­¾­Ã ø ÃÁ½�³­µ­»PÍ©»+¶�Æ�¿Àß�¶�Æ�¿Â»´µ­Ð"³­ºÂ»´Æ¿Âµ�ÄÇÃ]Æ�¿À·"Æ ª ¶E¿ÀÃÁµ­ÃKÃÁºÂÃ]ÍÓÃ]µT¶E·"½�¾�¿ÀÄÇÃÁÆÈÃÁµT¶EÃ]½�¾­ÃŸ]ÃÁÆÈ·½È·"ºÂ·vÃÁµ`º¼»×¾­¿¼»+Ð"·"µÊ»´º ê »+çÊ»GÌÈ·v¾­ÃԺ»�¾�¿¼»´Ð"·´µÊ»´º ± ê�ú Ã]½Ó³�µÊ»üÍÓ»+¶EÆÈ¿ÂßN¶�Æ�¿¼»+µ­Ð"³­ºÂ»´Æ½È³ É Ã]Æ�¿À·"Æ ª ¶E¿ÀÃÁµ­ÃÛÃÁºÀÃÁ͹ÃÁµT¶E·"½R¾­¿ÀÄÇÃ]Æ�Ã]µ5¶�ÃÁ½R¾­Ã�¸]ÃÁÆ�·x½�·"ºÀ·ìÃ]µNº¼»�¾­¿¼»´Ð´·"µÊ»´º ê »´ÆÈÆ�¿Âç­»�¾­ÃKº¼»¾­¿Â»´Ð"·"µ­»´º ±_¬Tý ½�»´ÍÓ·´½�º¼»�¾­ÃÁ½È¸Á·"Í É ·"½�¿À¸Á¿ÂÏ´µ ª�«�¬þ®"û´±É »+ÆE»ìÆÈÃÁ½È·"ºÀÖ´ÃÁÆ�Ã]ºl½È¿Â½È¶�ÃÁÍÓ» £¦­¯ ¨ ­¥£¦­¯ ¨ î øRú ò:ÿ ­¯ ¨�ø î ú ­¯ ò ¨ ­¥Ã]µÚ¾­·´½ É »´½�·"½ ¬ ô�Æ�¿Â͹ÃÁÆÈ·>½�Ã�Æ�Ã]½�³­Ã]ºÀÖ"Ã

ø ­� ¨ ­¥° �

Page 51: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ê ¾�ÃÁ½ É ³­ÝÁ½Û½�Ã�Æ�Ã]½�³­Ã]ºÀÖ"Ãú ­¯ ¨ ­�

É »+ÆE»ÓÃ]µ­¸Á·´µ5¶�ÆE»´Æ ­¯ ¬ ��»BÖ"Ã]µ5¶E»GÌ�»©¾­ÃìÃÁ½�¶�»©¾­Ã]½�¸]·"Í É ·"½È¿Â¸Á¿ÀÏ"µÔÃ]½�Õ5³­ÃìÆ�ÃÁ½È·"ºÀÖ´ÃÁÆK³­µ<½�¿À½�Þ¶�ÃÁÍÓ»B¶�Æ�¿¼»+µ­Ð"³­ºÂ»´Æ�¾­ÃxÃ]¸Á³Ê»+¸Á¿Â·´µ­ÃÁ½ å Ã]½�ÍP³ ê ½È¿ÂÍ É ºÂà ¬ ��·"½�¾­Ãh¶�»´ºÀºÂÃ]½Û¾�Ãx¸ÁÏ´ÍÓ·¹½�ÃìÆ�ÃË»+ºYÞ¿ÀßË»©Ã]½È¶�»|¾­Ã]½�¸]·"Í É ·"½È¿Â¸Á¿ÀÏ"µ<½ÈÃ>ÃÁµ­¸]³­ÃÁµT¶EÆ�»´µvÃÁµ�ÃÁº ¸Á» É ëY¶E³­ºÀ· ® ¾­ÃÁº:ºÂ¿Âç�Æ�· Ø�� ³­Í¹ÃÁÆȿ¸Á»´º��ÃÁ¸]¿ É ÃÁ½�¿Âµ�� Ù����� ¬ ��·"½Æ�Ã]½�³­ºY¶�»´¾�·"½�µ5³­ÍÓÝ]Æ�¿À¸Á·"½½�Ã É ÆÈÃÁ½ÈÃÁµT¶�»´µÓÃÁµ¹ÃÁºÊ½È¿ÂÐ"³­Ã]µT¶EÃ�¸Ë» É ëYÞ¶�³­ºÂ· ¬������� � ��������� ���j������j� �! ²"�#i ���Û���� jW�lk��������nmK�%$ükÛ�'&���­�(I»´Í¹·"½�»>Æ�Ã]½�·"ºYÖ"Ã]Æ�º¼»PÃÁ¸]³Ê»´¸]¿ÂÏ"µ|¾­¿YÄÇÃÁÆÈÃÁµ­¸]¿¼»´ºl¾­Ã)�³­Æ�дÃÁÆ�½�³�½E»´µ­¾�·¹Ã]º�͹Ý]¶E·0¾­·>¾­Ã* »´³ ¬õ÷ ½È¶E»¹Ã]¸Á³Ê»+¸Á¿ÂÏ´µWÃ]½ +-,+'. â , +/,+ ï ¨ 0 +-,/1+ ï 1 ð 2 ï3254 é ª�«­¬¯®76T±¸]·"µÚºÂ»´½�¸]·"µ­¾­¿À¸Á¿Â·´µ­ÃÁ½�¾­Ã@ÄÇÆ�·´µ5¶�ÃÁÆ�» , î98 é ð . ò ¨ : ª�«­¬¯®´ÑT±ê ¸]·"µ­¾­¿À¸Á¿ÀÏ"µ|¿Âµ­¿À¸Á¿Â»´º , îÇïsð : ò ¨<; á°èùé>= ï ª�«­¬.ó � ±¾­·´µ­¾­Ã 0 ÃÁ½:ºÂ»ÛÖ0¿Â½È¸Á·"½È¿Â¾Ê»+¾ ¬ ô:»´Æ�»KÖ0¿Â½�¸]·"½�¿À¾Ê»´¾­Ã]½ É ÃÁÕ5³­Ã@?Ê»´½ å º¼»Û½�·"ºÀ³­¸Á¿ÀÏ"µì½ÈÃ�¾­Ã]½E»´ÆÈÆ�·"ºÀº¼»¸]·"ÍÓ·x³­µÊ»>·"µ­¾Ê»P¾�ÃBA�¾­¿ÀÃÁµT¶EÞ�Ã@½�¿ÂÃ]Æ�Æ�»DCÅÃ]µ|ÃÁº�·´Æ�¿ÂдÃÁµ ¬ ��»P½È·"ºÂ³�¸Á¿ÂÏ´µ©¶EÃ]Ï"Æ�¿À¸Ë»P¾­Ã@Ã]½È¶EÃÉ ÆÈ·"ç­ºÀÃÁÍÓ»ìÃÁ½�¸]·"µ­·0¸Á¿À¾Ê» å ºÂ»ì·"çè¶E³�Ö´·©² ¬�Òx¬ ��·"ºÀà � 6 å ê Ädz­Ã@¸]·"Í É ¿ÂºÂ»´¾Ê» É ·"Æ>)�ÃÁµT¶E·´µê ô�ºÂ»+¶Eß]Í©»´µ�µ � Ñ FE °

Page 52: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

, î ïsð . ò ¨ G = 0 [ H VI�JLKNM í IPORQ I�S Z S9T�UWVYX[Z M = ïí5\ â^] H VI_JLK í I7O Q I S Z S T�U�`ba V M = ï \ ð ª�«­¬.ó ±¾­·´µ­¾­ÃÓí I ¨ î ;ìé ò IRc I î é d ] = 0 ò à c I îfe5ò ¾­Ã]µ­·´¶�»Wº¼»´½@Ädz­µ�¸Á¿Â·´µ­ÃÁ½�¾­Ã�)�ÃÁ½È½�ÃÁºRÍÓ·0¾­¿ æ Þ¸Á»´¾Ê»´½�¾­Ã]º É ÆÈ¿Â͹ÃÁÆ ¶�¿ É · ¬Dg Ö"ÃÁ¸]ÃÁ½½�Ã É ¿ÀÃÁµ­½�»ÅÕ5³­Ã�³­µÊ»@½�·´ºÂ³­¸]¿ÂÏ"µ>»+µÊ»´ºÂëY¶E¿À¸Ë»KÃ]½�½�¿ÀÃÁÍ É Æ�ÃÍP³�¸ih­·©Í¹Ã�ÌÈ·"Æ�Õ5³­Ãì³­µÊ»Óµ5j­Í¹ÃÁÆȿ¸Á» å É Ã]Æ�·¹ÃÁ½È¶�·Nµ­·ÓÃÁ½�½È¿ÂÃ]Í É ÆÈÃ�»´½�ë å É »´ÆE»BÐ"ÆE» æ ¸Ë»´ÆȺ¼»¸]·"µ�³­µÊ»P¸Á·´Í É ³è¶�»´¾­·´ÆE» å º¼»x½È·"ºÂ³­¸]¿ÂÏ"µ ªÇ«­¬.ó ± Ã]½�¿ÂµT¶EÆ�»+¶�»´ç�ºÂÃ É »´ÆE»xÖ+»´ºÀ·"Æ�Ã]½ É Ã]Õ5³­Ãb?­·´½¾­Ã . ê 0 å ¾­·"µ­¾�à c I îkeTò å ¸Á³Ê»+µ­¾­· eml n å ½�ÃN¸]·"Í É ·"Æ�¶�»Ú»´½�¿Àµ5¶�Ï´¶E¿À¸Ë»´Í¹ÃÁµT¶�ÃÓ¸]·"ÍÓ·Obo î9] = eTò Q K ì 1 ¿Âµ�¾­Ã É ÃÁµ­¾�¿ÂÃÁµT¶�Ã�¾�à M ¬ý µ­»¹ÆÈÃ É ÆÈÃÁ½�Ã]µT¶�»´¸]¿ÂÏ"µWÍ©»+µ­Ã�Ì�»´ç�ºÂÃ@¾­Ã�º¼»P½È·"ºÂ³�¸Á¿ÂÏ´µ|ÃÁ½, îÇïsð . ò ¨<; c îÇïsð . ò^�îÇïsð . ò ª�«­¬.óT®"±¾­·´µ­¾­Ã c îÇïsð . ò ê ^�îÇïsð . ò Ã]½È¶�Î+µÔ¾­Ã æ µ­¿Â¾­»´½ É ·"Æ ª Ö"Ã]Æ ��Pb ±c îÇïsð . ò ¨ prqQ q VsX[Zt = îÇï ;vu òxwzy:îÇï ;"u ò|{~}|�ìî ;Bu 1 d G 0 . ò�� u ª�«­¬.ó"óT±� îÇïsð . ò ¨ prqQ q y î ï ;vu ò|{�}��ìî ;�u 1 d G 0 . òs� u ª�«­¬.ó´«5±¾­·´µ­¾­Ã y îfe5ò ¨ {�}��ìî ;����7� î = e d ] = 0 ò�ò�¬�÷ ½�¶�»P½�·´ºÂ³­¸]¿ÂÏ"µÓÃ���»´¸]¶E»Pµ­·"½ É Ã]Æ�͹¿À¶E¿ÀÆEÎ�¸Á·"Í>ÞÉ »+ÆE»´Æ µ5³­Ã]½È¶�ÆE»�½�·"ºÀ³­¸Á¿ÀÏ"µ>µ5³­ÍÓÝ]Æ�¿À¸Ë» ¬´ý ½E»+Æ�ÃÁ͹·"½�Ã]º­ÍÓÝh¶E·0¾­·Å¶EÆ�» É ÃÁßÁ·´¿Â¾Ê»´ºè¸Á·"µBºÀ¿Â͹¿À¶EÃ]½æ µ�¿À¶E·´½ ¬ Ò Ã]筿¾�·K»+º�h­ÃÁ¸ih­·Û¾­ÃÕ5³­ÃIºÂ·"½�¿ÀµT¶EÃÁдÆE»´µ­¾�·"½s¾­ÃÁ¸Á»´ÃÁµ�ÍP³ ê Æ�Î É ¿À¾Ê»´Í¹ÃÁµT¶EÃ�¸]³Ê»´µèÞ¾­· 2 u 2 »´³­Í¹ÃÁµT¶�» å º¼»Å¿ÀµT¶EÃÁдÆE»´¸]¿ÂÏ"µì½�Ã�hÊ»´Æ�Î�ÃÁµB³­µB¿ÂµT¶�ÃÁÆÈÖ+»´ºÀ·ÅÃÁµT¶�Æ�à t ; M ð M w ê ºÂ³­Ã]Ð"·ÅÃ]µt ; ] M ð~] M w ê ½È¿ µ­·|¸Á»´ÍP筿»©ÃÁº ÆÈÃÁ½È³­ºÀ¶E»´¾­·|¾­Ã]µ5¶�Æ�·�¾­Ã>³­µ­»|¶�·"ºÂÃ]ÆE»´µ�¸Á¿¼» É Æ�ÃhÖ0¿¼»´Í¹ÃÁµT¶EÃÃ]½È¶�»+ç­ºÂÃ]¸Á¿Â¾­» ªÇÉ ·´Æ�Ã�ÌÈÃÁÍ É ºÀ· é : Q K \ ± å ÃÁµT¶E·"µ�¸ÁÃÁ½K»7h­ë�µ�·"½�¾­Ã]¶�ÃÁµ­Ã]ÍÓ·"½ ¬�������9�>� �������������W���������� ��¢¡B�W�£��¤���¥¦¡§¥�¨©��ª��·"͹·<½�ëhÊ»<¾­¿Â¸ih�· å Ã]Í É ÃÁßÁ»´ÍÓ·´½x¾­Ã]½E»´ÆÈÆ�·"ºÀº¼»´µ­¾�· , îÇïsð . ò ÃÁµz¶EÃ]Æ�͹¿Âµ­·´½ì¾­Ã©ºÂ·"½É ÆÈ¿Â͹ÃÁÆÈ·"½­¬ â é É ·"ºÂ¿Àµ­·"͹¿Â·"½Û¾­Ã>³­µv¸Á·"µGÌȳ­µT¶E·�·"Æȶ�·"Ð"·"µ­»´º:¾­Ã>Ädz­µ�¸Á¿Â·´µ­ÃÁ½ ¬�÷ µüÃ]½È¶EøÁ»´½�·¹³­½E»+ÍÓ·"½�ºÀ·"½ É ·"ºÂ¿Àµ­·"͹¿Â·"½I¾­Ã®�¯h­ÃÁç ê ½Yh�Ã]Ö°"®

Page 53: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

,±° îÇïsð . ò ¨ °² I_J \ í I î . òx³ I îÇïlò ª�«­¬.óT°"±g æ µW¾­Ã�͹¿Âµ­¿ÀÍÓ¿ÀßË»´ÆIÃÁº�ÆÈÃÁ½�¿À¾­³­·´ îÇïsð . ò ¨ +/,±°+'. â ,L° +-,±°+ ï ; 0 + 1 ,L°+ ï 1 ª�«­¬.ó"ÙT±³­µ�·Å¾­ÃÁÍÓ»´µ­¾­»ÅÕ0³�æ´ îÇïsð . ò ½�Ã˻ŷ´ÆȶE·´Ð"·"µÊ»´ºè»´º0ÃÁ½ É »´¸Á¿À·@Ã�� É »´µ�¾­¿Â¾­· É ·"ƶµ ³�·"îÇïlòi¸ ° Q 1· J \ åÃ]½Û¾�ÃÁ¸Á¿ÀÆ p KQ K ´ îÇïsð . òx³�·"îÇïlò �"ï¹ é�; ï 1 ¨ : : 4�º�4 ¬ ; ] ª�«­¬.óTû"±Ã]½È¶�»�¿ÂµT¶EÃ]Ð"ÆE»+º å ¾­Ã]筿¾�·P»ìºÂ»@·"Æȶ�·"Ð"·"µÊ»+ºÂ¿Â¾­»´¾>¾­Ã�ºÀ·"½ É ·´ºÂ¿Âµ�·"ÍÓ¿À·"½ ³L·Tî ïlò å µ­·"½I¸Á·´µ­¾­³­¸]û »í · ¨ ¼¬ ·â 0 í|½ 1�¾· : 4�º�4 ¬ ; ] ª�«­¬.óP6T±¾­·´µ­¾­Ã ¼¬ ¨¿; îÁÀ °�+PÂ�,±° ò�· å ê í ½ 1�¾· ½È·"µqºÂ·"½P¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½B¾­Ã�º¼»ÔÃ~� É »´µ­½�¿ÀÏ"µq¾­Ã�º¼»½ÈÃÁÐ"³­µ�¾Ê»>¾­ÃÁÆÈ¿ÀÖ+»´¾Ê»P¾­Ã , îÇïsð . ò ¸Á·"µ�ÆÈÃÁ½ É ÃÁ¸h¶E·Ó»Pº¼»xÖ+»´ÆÈ¿¼»´ç­ºÀÃ�Ã]½ É »´¸]¿¼»´º ïs¬ �s»PÍ©»´µ�ÃÁÆE»ÍÓδ½�Ã æ ¸Á¿ÂÃ]µT¶EÃ�¾­Ã�¸Ë»´ºÀ¸Á³­ºÂ»´Æ�ºÀ·"½�Ã]½Û³�½E»´µ­¾�·ÓºÂ»PÆ�ÃÁºÂ»´¸Á¿ÀÏ"µ|¾­Ã�Æ�Ã]¸Á³­ÆÈÆ�ÃÁµ�¸Á¿¼»� · í|½ 1�¾· ¨ í'½ 1�¾·Äà 1 âr]­î9º�â é ò í|½ K ¾·ià K~Å ª�«­¬.ó"ÑT±÷ º�½È¿Â½�¶EÃÁÍÓ»Ô¾­Ã©ÃÁ¸Á³­»´¸Á¿À·"µ­ÃÁ½ «�¬¯óP6 Ìȳ­µT¶E·×¸Á·´µzº¼»´½ì¸Á·"µ­¾�¿Â¸Á¿À·"µ­Ã]½x¾­ÃNÄÇÆÈ·"µT¶EÃÁÆ�» , îÇï ¨;�é ð . ò ¨ , î ï ¨ é ð . ò ¨ : å ¸]·"µ­½�¶E¿À¶�³ ê Ã]µÔ³­µW¸Á·´µ'Ìȳ­µT¶E·B¸Á·´Í É ºÀÃ]¶�· É »´ÆE»>¸Á»´ºÂ¸]³­º¼»´Æ�ºÂ·"½¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½�¾­Ã@º¼»�Ã�� É »´µ­½È¿ÂÏ"µW»ì¶�·è¾�·P¶E¿ÂÃ]Í É ·x¸]·"µ|º¼»ì¸Á·"µ­¾�¿Â¸Á¿ÀÏ"µN¿Àµ­¿Â¸]¿¼»´ºõ¾Ê»´¾­» ¬ �s»´½¸]·"µ­¾­¿À¸Á¿Â·´µ­ÃÁ½�¾­Ã@ÄÇÆÈ·"µT¶EÃÁÆ�»¹Ã]µW¶�ÃÁÆÈÍÓ¿Àµ­·"½�¾­Ã�ºÀ·"½�¸Á·0Ã æ ¸]¿ÂÃ]µ5¶�ÃÁ½Û½�·"µ°² I_J \ í I ¨ °² I_J \ î ;�é ò I í I ¨ : ª�«­¬ « � ±Õ5³­Ã�½�·´µÚÃ]Õ0³�¿ÀÖ+»´ºÂÃ]µT¶EÃÁ½Û»´º É »´Æ�¾­Ã�ÃÁ¸]³Ê»´¸Á¿À·"µ­Ã]½

°´ó

Page 54: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

² áÄÆ�Çá¯È9ÉxÊ í · ¨ : ð ² áYÆ�Ëá�ÌÎÍ5ÈÁÉxÊ í · ¨ : Å ª�«­¬ « ±÷ º½�¿À½È¶EÃ]Í©» «­¬.óP6 ÃhÖ"·"ºÀ³­¸Á¿À·"µÊ»W»ÚºÂ·´½ì¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½Pí I É »´ÆE» : 4 ¬ ; ] å ê ¸Á·"µüº¼»´½¸]·"µ­¾­¿À¸Á¿Â·´µ­ÃÁ½�¾­Ã@ÄÇÆÈ·"µT¶EÃÁÆ�» «�¬.« å ½�Ã�¸Ë»+ºÂ¸Á³�º¼»´µ�í ° Q K ê í ° »>¸Ë»´¾­» É »+½�·¹¾­Ã@¶E¿ÂÃ]Í É · ¬÷ º¶EÃÁÆÈÍÓ¿Àµ­·�µ­·<ºÀ¿Âµ­ÃÁ»´º�¾­Ã «�¬¯óP6 Æ�Ã]Õ5³­¿ÂÃ]Æ�é¾�ÃÓ¶EÆ�»+¶E·ÚÃÁ½ É Ã]¸Á¿¼»+º ¬ *:»+µ5¶�· , îÇïsð . ò ¸]·"ÍÓ·+PÂ�, îÇïsð . ò ¶E¿ÀÃÁµ­Ã]µ<½�³×Ã~� É »´µ­½È¿ÂÏ"µ<Ã]µ<¶EÃ]Æ�͹¿Âµ­·"½Û¾­ÃìºÂ·"½ É ·´ºÂ¿Âµ�·"ÍÓ¿À·"½ ³ I ê ¶E¿ÀÃÁµ­Ã]µ×½�³­½¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½@Æ�ÃÁ½ É Ã]¸]¶�¿ÀÖ"·´½@í I ê í'½ K ¾I ¬ �s» É ÆÈÃÁÐ"³­µT¶E»NÃÁ½ å ¸Á³ÊδºÀÃÁ½Å½È·"µ<ºÀ·"½�¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½Ï ,L+-,ÑÐ · ¨ÓÒ · ¾­Ã�º¼»PÃ�� É »+µ­½�¿ÀÏ"µÚ¾�ÃÁº É ÆÈ·è¾­³�¸]¶E· ,±+PÂ�,,±° +/,±°+ ï ¨

°² · J \ Ò ·@³L·Tî ïlòhð ª�«­¬ «5®"±Ã]µW¶�ÝÁÆ�͹¿Âµ�·"½�¾­Ã�í · ê í|½ K ¾· ¬õ÷ µ É Æȿµ­¸]¿ É ¿À· É ·è¾­ÆÈë¼»P³­½�»´Æ�½ÈÃìºÂ»PÆ�Ã]º¼»´¸]¿ÂÏ"µÒ · ¨ Ï ,±+-,ÑÐ · ¨ é] ÔÕ ²Ö Ã'× J · í Ö í ½ K ¾× â ²Ø Ö Q × Ø J · í Ö í ½ K ¾×ÚÙÛ ª�«­¬ «TóT±Õ5³­Ã É ³­ÃÁ¾�Ãx¾­Ã]ÍÓ·"½�¶EÆ�»´Æ�½ÈÃìÄ�»´¸]¿ÂºÂ͹ÃÁµT¶�Ã@¸Á·"µÔ» ê ³­¾­»©¾­ÃxºÂ»´½ É Æ�· É ¿ÂÃ]¾Ê»´¾­Ã]½K¾­ÃìºÂ·"½ É ·"ºÂ¿ Þµ­·´ÍÓ¿À·"½�¾­ÃÜ�¯h­ÃÁç ê ½Yh­ÃhÖ ¬ÞÝ ¿Àµ×ÃÁÍPç­»´Æ�Ð"· å ÃÁ½@Íx³­¸ih­·|ÍÓδ½ÅÃ æ ¸Á¿ÂÃ]µT¶EÃB³�¶E¿ÀºÂ¿ÀßË»´Æ�³­µ�¸ËÎ+ºYÞ¸]³­ºÂ· É ½�Ã]³­¾­·+Þ)ÃÁ½ É ÃÁ¸h¶EÆ�»´º�¸Á·´µNº¼»�» ê ³�¾Ê»P¾­Ã�ºÂ»´½¶* Æ�»´µ­½ÈÄÇ·´Æ�ÍÓ»´¾Ê»´½R¾­Ã�ßÊ·"³­ÆÈ¿ÂÃ]Æ��ÛÎ É ¿À¾Ê»´½ª *�ßÞ� ±_¬Ê÷ º É Æ�·0¸ÁÃ]½�·¹¾­ÃÁº�¸ÁδºÂ¸]³­ºÂ·>½ÈÃì½È¿Â½�¶EÃÁÍÓ»+¶�¿ÂßË»>»>¸]·"µT¶E¿Âµ5³Ê»+¸Á¿ÂÏ´µ��Î+ºÂ¸Á³�ºÂ·>¾­Ã Ï ,±+PÂ�,ÑÐ · ¨ÓÒ · ¾Ê»´¾­·´½ÛºÀ·"½�¸Á·0Ã æ ¸Á¿ÂÃ]µT¶EÃÁ½Kí · ¬Êý ½ÈÃ�ºÂ»>Æ�ÃÁºÂ»´¸Á¿ÀÏ"µ|¾­Ã�Æ�Ã]¸Á³­ÆÈÆ�Ã]µ­¸Á¿Â»

� · í ½ K ¾· ¨ í ½ K ¾·ià 1 â^]­îFº�â é ò í ·ià K ªÇ«­¬.«´«5±É »´ÆE»Û¸Ë»´ºÀ¸Á³­ºÂ»´Æ ºÀ·"½�¸Á·0Ã æ ¸]¿ÂÃÁµT¶�ÃÁ½�¾­ÃRº¼»KÃ~� É »´µ­½È¿ÂÏ"µ>¾­ÃRº¼» É Æ�¿ÀÍÓÃ]ÆE»K¾�ÃÁÆ�¿YÖ+»'Þ¾­»B¸]·"µWÆ�ÃÁ½ É Ã]¸]¶�·©» ï ¾­Ã , îÇïsð . ò°+«

Page 55: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

®0¬ �s»´½à*�ßÞ� µ�ÃÁ¸ÁÃ]½�¿Y¶�»´µ>¾­Ã3A�Ã]µ5¶�ÆE»´¾­»DCRºÂ·"½�¸Á·0Ã æ ¸]¿ÂÃ]µ5¶�ÃÁ½�¾­ÃRº¼»ÛÃ�� É »´µ­½È¿ÂÏ"µ¾�Ãv³�µÊ»qÄdz�µ­¸Á¿ÀÏ"µ , î ïsð . ò å ê ¾­»´µ�¾�Ãü½�»´ºÂ¿À¾Ê»uº¼»uÄdz­µ­¸]¿ÂÏ"µ�Ö+»´ºÀ³Ê»´¾Ê»zÃÁµÖ+»´Æȿ·"½ É ³­µT¶�·"½ ª ¾­Ã�A�¸Á·´ºÂ·0¸Ë»´¸]¿ÂÏ"µ�C ± ,L° îÇï-á òh¬�g ½�ëKÕ5³­Ã å ¶�ÃÁµ­¿ÀÃÁµ­¾�·qí · êí|½ K ¾· å ³�¶�¿ÂºÂ¿À¸ÁÃ�º¼»+½�*�ßÞ� ê ·"ç�¶EÃ]µ­ÐT» ,L° îÇï-áÇò ê î +-,±° d + ïlò]îÇï-á�òKª ÃÁ½�¶E·PÃ]½ å ,ê ½�³W¾­ÃÁÆÈ¿ÀÖ+»´¾Ê»>¸Á·´µWÆÈÃÁ½ É Ã]¸]¶E·©» ï ÃÁµWÃÁº�Ã]½ É »+¸Á¿Â·BÄÇë½ȿ¸]· ±h¬í · ; l ,±° îÇï-á ò í'½ K ¾· ; l +-,±° î ï-áÇò+ ï ªÇ«­¬.«T°"±óè¬ �ÛÃÁ»´ºÂ¿À¸ÁÃźÀ·"½ É ÆÈ·è¾­³�¸]¶E·´½ÛÃ]µÚÃ]º�ÃÁ½ É »´¸Á¿À· æ ½�¿Â¸]· E,L° îÇï-á ò +-,±°+ ï î ï-áÇò ªÇ«­¬.«"ÙT±«�¬Wg ¶�ÆE»ËÖ"Ý]½x¾­Ã¹º¼»�*�ßÞ��¿ÀµTÖ"ÃÁÆȽE»´½ å ÆÈÃÁÐ"ÆÈÃÁ½Èé»´º�Ã]½ É »+¸Á¿Â·ÚÃÁ½ É Ã]¸]¶EÆ�»´º ¬ �s»*�ß � ¿Àµ5Ö´ÃÁÆȽE»Ô¶E¿ÀÃÁµ­Ã]µ É ·"ÆxÃ]µ5¶�ÆE»´¾­»ÔºÂ·"½�Ö+»´ºÂ·"ÆÈÃÁ½�¾Ê»´¾­·´½ É ·´Æ ª�«�¬.«TÙ"± å êÉ ·"Æ�½�»´ºÂ¿À¾Ê»PºÂ·"½�¸Á·0Ã æ ¸]¿Â¿ÂÃ]µT¶EÃÁ½ Ï ,±+PÂ�,âÐ · ¨ãÒ ·+/,±° îÇï-áÇò+ ï ; l Ï ,±+PÂ�,âÐ · ¨ãÒ · ªÇ«­¬.«Tû"±÷ ½È¶E»ÚA�¾Ê»´µ�ßË»DCÚÃÁµT¶�Æ�Ã�ÃÁºÅÃ]½ É »+¸Á¿Â·zÃ]½ É ÃÁ¸h¶EÆE»+º ê Ã]º�ÄÇë½ȿ¸Á· å ½ÈÃ×Æ�ÃË»+ºÂ¿ÂßÁ»u»u¸Ë»+¾Ê» É »´½È·¾­Ã|¶�¿ÂÃÁÍ É · ¬ �s»vÃhÖ"·"ºÀ³­¸Á¿ÀÏ"µz¶EÃ]Í É ·"ÆE»+º�½�ÃWÆ�ÃÁ»´ºÂ¿ÀßË»´Æ�Î<³­½E»´µ�¾­· Ã]º�ÃÁ½�Õ5³­Ã]Í©» g ¾Ê»´Í¹½�Þ)�»+½Yh�ÄÇ·"Æ�¶Äh É »´Æ�»�Ã]º�¶EÝÁÆÈÍÓ¿Àµ­·×µ­·vºÂ¿Àµ­ÃË»´º å ê Ã]º�ÃÁ½ÈÕ0³�ÃÁÍÓ»ä��ÆE»+µ'å"Þ � ¿À¸Á·"ºÀ½�·"µ É »+ÆE»×ÃÁº¶�ÃÁÆ�͹¿Âµ�·>ºÂ¿Âµ�ÃË»´º ¬�Ò Ã�Ã]½È¶EÃ�͹·è¾­·>½ÈÃ@¶E¿ÂÃ]µ­Ã�ÃÁº�Ã]½�Õ5³­ÃÁÍÓ» g )¦� �

í I î . âræ . ò ¨ í I î . ò�â æ .] t Ò I î . ; æ . ò ;èçPÒ I î . ò�wâ 0 æ .] é í ½ 1�¾I î . òsâ í ½ 1�¾I î . âêæ . ò�ë : 4 M 4 ¬ ; ] ª�«­¬ «�6T±Õ5³­Ã É ³­Ã]¾­Ã�Æ�ÃhÞ)ÃÁ½È¸ÁÆ�¿Àç­¿ÂÆȽ�Ã�¸Á·´ÍÓ·ì í ½ 1�¾I ;èí í I7î î . âêæ . ò ¨ y I î í I ð Ò I ð . ð . ; æ . ò : 4 M 4 ¬ ; ] ª�«­¬ «TÑT±°"°

Page 56: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¾­·´µ­¾­Ã í|¨ ] d 0 æ . ê ºÂ»xÄdz­µ­¸Á¿ÀÏ"µ y I Ã]½ÛÃ~� É ºÂëÀ¸Á¿À¶E»´Í¹ÃÁµT¶EÃy I ¨�;�í í I î . ò ; é0 t Ò I î . ; æ . ò ;èçPÒ I î . ò�w ; í ½ 1�¾I î . ò ª�«­¬¯° � ±Ý ¿�³�½E»´Í¹·"½�º¼»PÃ]¸Á³Ê»+¸Á¿ÂÏ´µ ªÁï|ï0± Ã]µ ªÇ«­¬ «TÑT± å ÝÁ½È¶E»�j�ºÀ¶E¿ÀÍ©»ì½�Ã�ÃÁ½È¸ÁÆ�¿Àç­¿ÂÆÈë¼»P¸]·"ÍÓ·é� ·

°²ÈxÆRðÁñ Sò ÈxñRðÁóôÈÁÉxÊ�õ î õ 1 ; º 1 ò í Ö ;èí í · ¨ yD· º ¨ : ð ÅöÅöÅ ð ¬ ; ] ª�«­¬¯° ±ý ½E»´µ­¾�· ªÇ«­¬.« ± ê ªÇ«­¬þ° ± ºÂºÀÃÁÐT»´Í¹·"½x»�³�µ`½È¿Â½È¶�ÃÁÍÓ»×¾­Ã�ÃÁ¸Á³­»´¸Á¿À·"µ­ÃÁ½BºÀ¿Âµ­ÃÁ»´ºÂÃ]½P¾­Ã�º¼»ÄÇ·"ÆÈÍ©» ÷ Ø ¨ãø ª�«­¬¯°"®"±¾­·´µ­¾­Ã ÷ Ã]½�³­µÊ»ìÍ©»+¶�Æ�¿ÀßÛ¶�Æ�¿¼»+µ­Ð"³­ºÂ»´Æ½�³ É ÃÁÆȿ·"Æ å ¸]³ ê »P½È·"ºÂ³­¸]¿ÂÏ"µÓÆ�Ã]Õ0³�¿ÂÃÁÆÈí¬ 1 · É Ã]ÆE»'Þ¸]¿Â·"µ­Ã]½ ¬bý µ É Æ�·0¸ÁÃ]¾­¿Â͹¿ÂÃ]µT¶E·B¾­Ã�½�·"ºÀ³­¸Á¿ÀÏ"µÚÍÓδ½�Ã æ ¸Á¿ÂÃ]µT¶EÃx½ÈÃx¸]·"µ­½�¿ÀÐ"³­Ã�Æ�ÃÁ»´Æ�ÆÈÃÁÐ"ºÂ»´µ­¾­·ºÂ»´½�ÃÁ¸]³Ê»´¸Á¿À·"µ­Ã]½ ªÇ«­¬ «TÑT± ¸Á·´µÔ» ê ³­¾Ê»B¾­Ã�º¼»PÆÈÃÁº¼»+¸Á¿ÂÏ´µ|¾­Ã�Æ�ÃÁ¸]³­Æ�ÆÈÃÁµ­¸]¿¼»]Pº í|½ K ¾· ¨ã� · Q K í|½ 1�¾· Q K ; í|½ 1�¾·ià K ª�«­¬¯°´óT±ºÂ»PÃÁ¸Á³­»´¸Á¿ÀÏ"µ ªÇ«­¬ «TÑT± Õ5³­Ã]¾Ê»

]Pº í|½ K ¾· ¨�� · Q K t ; yD· Q K â í í · Q K w ; t ; yD·ià K â í í ·ià K w º ¨ é ð ÅöÅöÅ ð ¬ ;èç Ū�«­¬¯°+«5±Ò Ã]½ É j�ÃÁ½ å ³­½E»+µ­¾­·�ºÂ»ÛÆÈÃÁºÂ»´¸Á¿ÀÏ"µ�¾­ÃRÆÈÃÁ¸]³­Æ�ÆÈÃÁµ­¸]¿¼» ª�ó�¬¯®"°"±�É »´Æ�»ÛºÂ» É ÆÈ¿Â͹ÃÁÆ�»�¾�ÃÁÆ�¿YÖ+»´¾Ê» å Ã]µ¸]·"ÍPç­¿ÀµÊ»´¸Á¿ÀÏ"µ�¸Á·"µìº¼» ªÇ«­¬¯°+«5± å ÆÈÃÁ½È³­ºÀ¶E»�³­µÊ»ÛÃÁ¸Á³­»´¸Á¿ÀÏ"µP¸Á·´µP½�·"ºÂ»´Í¹ÃÁµT¶Eú·´½ ¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½í · å »¹½�»´çlÃ]Æ � I Q 1 íG M î M ;�é ò í I Q 1 ; ¦ é â í-ù I]�î M 1 ;�é ò § í I â í-ù I à 1G M î M â é ò í I à 1¨ ; � I Q 1G M î M ;�é ò y I Q 1 â ù I]­î M 1 ; é ò y I ; ù I à 1G M î M â é ò y I à 1 ðúº ¨ ]èð ÅûÅ ð ¬ ª�«­¬¯°"°"±°´Ù

Page 57: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¾­·´µ­¾­Ãù I ¨ýü é : 4 M 4 ¬ ; ]: M Ð ¬ ; ] Å ª�«­¬¯°´ÙT±� ·´¶EÃ�Õ5³­ÃvºÂ·"½|¸]·èÃ æ ¸Á¿ÀÃÁµT¶EÃ]½ É »+Æ�ÃÁ½WÃ�¿ÂÍ É »´ÆÈÃÁ½�ÃÁ½�¶�δµ�¾­ÃÁ½�»´¸Á· É º¼»´¾�·"½�ÃÁµ ªÇ«­¬¯°"°"± êÃ]µ ª�«­¬ « ±_¬ ÷ º�½�¿À½È¶�ÃÁÍÓ»Ô¾­Ã|ÃÁ¸]³Ê»´¸]¿Â·"µ­Ã]½>Õ5³­Ã|½�³­ÆÈÐ"Ã É »´Æ�»<ºÂ·"½ì¸Á·0Ã æ ¸]¿ÂÃ]µ5¶�ÃÁ½ É »+Æ�ÃÁ½>ÿÀÍ É »+Æ�ÃÁ½�Æ�ÃÁ½È³­ºÀ¶E»|½�ÃÁÆ�³­µ�·Ú¸]·"µü³�µÊ»|Í©»+¶�Æ�¿Àßx¶EÆȿ¾­¿Â»´Ð"·"µÊ»+º ¸]·"µ�³­µÊ»«A%¶EÆÈÃÁµ­ßÁ»DC ª ÃÁµvº¼»É ÆÈ¿Â͹ÃÁÆ�» æ º¼» ±_¬Êý µ É ÆÈ·è¸]ÃÁ¾­¿ÀÍÓ¿ÀÃÁµT¶E·>ÍP³ ê Ã æ ¸]¿ÂÃ]µ5¶�à ª ÍP³­¸ih­·¹ÍÓδ½�Õ0³�ÃìÃ]º�¾­Ã®þ�»´³�½�½ ±½ÈÃ�Ã~� É ·"µ�ÃìÃ]µÚÃ]ºs» É Ý]µ­¾­¿À¸Áà g � Ù å ê Ã]½�ÃÁº�Õ5³­Ã�½�Ã�³­½�Ï É »´Æ�»>¿ÂµT¶EÃ]Ð"ÆE»´Æ�Ã]µÚÃ]º�¶E¿ÀÃÁÍ É ·ºÂ»WÃ]¸Á³Ê»´¸]¿ÂÏ"µü¾­Ã�)�³­Æ�дÃÁÆ�½ ¬ ô »´ÆE»WÍ©»+µ­Ã�Ì�»´Æ�Ã]ºRÃ]Æ�Æ�·´Æ쾭éA�»+ºÂ¿¼»+½�¿Âµ�ÐRCìÃ�� É ºÂ¿À¸Ë»´¾�·ÔÃÁµüÃÁº» É ÝÁµ�¾­¿Â¸]à g å ½Èé³­½ÈÏ<º¼»�Æ�Ã]Ð"º¼»�¾­Ã ] ¬ d ç ¬�÷ ½x¾­Ã]¸Á¿ÀÆ å ÃÁµüº¼»�Ã�� É »´µ­½È¿ÂÏ"µ ¾­Ã ,±° îÇïsð . ò½ÈÃP¶E¿ÂÃ]µ­ÃÁµ ó"® ¶EÝÁÆÈÍÓ¿Àµ­·"½ å ÆÈÃË»´ºÀÍÓÃ]µ5¶�ÃP½�Ã>³�½E»´Æ�δµ ® Pÿ ½�¿ ½ÈÃP¶E¿ÂÃ]µ­ÃÁµ Ù+« å ÆÈÃË»´ºÀÍÓÃ]µ5¶�ÃP½�ó­½�»´ÆEÎ+µ «5® å Ã]¶�¸ ¬

°"û

Page 58: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� ��������� � �

� ��� ���������� � � ��� ��������� �

÷ �"!$#&%'!)(+*-,/.0%'1/2435#'!768*9�:*;($3<�/3=(>!>?@2439#@?&!>#&1/20%A*96/3<#@�B1/C:D$?'E0(>3<#@FB1/!7#&!)3<GH%A1JI=E4!$?'3<�*928?&!>#&3<20I9!>?K24*9#L!>($18*9(>E03<�/!$#M6/!)NO�/68*JPB6/!7QR39E4#'#&3<�:S ) 1/?'T<!$?'#$PH($3<�"203<#MU�D$%'3H6/39#M6/!V EXWY!>?&!>�/($EZ*9# ß E0�/E0%A*9#KS\[ #&,]!$($%A?'*924!$#>^_£�>`� ���>#��®�Ma$b;c ��� dec����f �hg¶�hg�� i^��jNª§�Á���¶¡B¥Ljlkãª���¤ m|���L¥Lj��3�§�����L�¶�B¡§¥on �qpi� m'�W�§�L���@jlr«���B�ts�|�LjQR*9?A*�!>�J(>3<�u%A?'*9?v!>27I9*-243<?w6J!yxz,]3<?"C:!$6/E43{6J!y24*9#w!$(>18*9($E43<�J!>#\6/!>#&(>?&E0%|*-#v!>�}!>2(>*9,/.0%'1/243*9�u%'!>?'E03<?>P~#'!"?&!+*920E4�$�1J��,/?&3<T<?'*9C"*v!$��24!$�/T<18*��&!w�)^�[ ��D$#&%A!l,/?&3<T<?A*-C"*%A*9C�G/E0D>��#&!�(+*-24(>1J2Z*:!$2R!$?'?&3<?����73<?&C"*�G±�FB1/! h *�S�!>�u%'?'!�2Z*l#'3<201/(>E0�<���B1/ClD>?&E4(>*lS24*�#'3<201/(>E0�<�\! � *9($%A*J^�Q�*-?A*�?'!>*924E0�+*9?�203<#�(+�-24(>1J243<#�#&!�1H%AE420E4�$��!>2 I-*9203<?�6/! � 1�� �9^O � ���� �¬ ; �

° Q K² á JLK]� x�� Â��h� �Y� á � .�¡$¢ ; x I$£�ï �¤� á¥� .�¡¦¢'¢ 1 ��°H^X§G±ÝH!�1J%'E424E0�+*9?&3<��I-*9?'E03<# %AE0,]3<#L6/!�(>39�/6/E4($E43<�J!>#L6/!¨WY?'3<�u%A!$?A*JPB203<#L?'!>#&1/20%A*96/3<#@3<GJ%'!>�/E06/3<#°76

Page 59: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

($3<��(>*968*�1/�8*©6/!5D>#h%|*9#¨#&!;68*-��*©(>39�/3H($!>?¨*©(>3<�u%'E4�B18*9($E4�<� ^[ �w24*9#M($3<�/6/E0(>E03<�/!>#@6/!�WY?'3<�u%A!$?A*ª,�!>?&E4�=6/E4(>*9#L,8*9?'*;!>�J(>3<�u%A?'*9?K!$2�I-*9203<?L6/!)«oS = !$�24*9#OWY?'3<�u%'!>?A*-#�6J!>2 6/3<ClE4�JE43­¬ ®|§<PX§>¯ !$��1J�y%'E4!>Cl,�3 .�¡ #'!$T j ��24*9#°!>($18*9($E43<�/!$#:��±/^X§�²T±Sy�Y±/^X§<§G±¦P8�J!>(>!$#'EX%|*9Cl3<#¨(>3<�J3H($!>?¨243<#�I-*9203<?'!$#¨6/!;!$#&%A*9#�I-*9?&EZ*9GJ24!>#K!>��24*9#�,�3<#'E0(>E439�/!>#�Y³�´ � ¢ ! �Y³ ; � ¢ !>��1J�{%AE4!$C:,�3 .�¡ Q K ^µ[ �u%A39�/(>!$#l,�3<?;!q�&!>Cl,/203/PR,/*9?A*�!$�/(>39�B%'?A*9?203<#;I-*9203<?'!$#�6/!w24*�«�!$�¶!$2M%'E4!>Cl,�3 . K !>��24*�WY?'3<�u%'!>?A* � � ; �<P�#&!v6/!>G�!>�o($3<�/3-®($!>?"« �Y� K ¢ S¶« �¤� Q K ¢ P@,8*9?'* � � ��6/!>G�!>Cl3<#"#'*9G�!>?"203<#©I9*-243<?&!>#"6J!�« �Y� ° à K ¢ S« �Y� ° Q K ¢ !>��!$2M%'E4!>Cl,�3 .�� : ^ V !"2Z*9#;(>3<�/6JE4(>E03<�/!$#ªE4�JE4(>E4*924!$#5,]3=6/!>Cl3<#;3<GJ%A!$�/!>?203<#�I-*9243<?&!>#�6J!�« �Y� K ¢ Sw« �¤� ° Q K ¢ P8,�!>?&3©243<#�39%'?'3<#�6J3<#�I9*-243<?&!>#��/3©2439#�(>3<�/3=(>!$C:39#>^· 3�C:E0#'Cl3�#&1/(>!$6/!K($18*9�/6J3)FB1/!>?&!>Cl3<#�(>3<�/3=(>!$?R203<#�I-*9243<?&!>#¸6/!@2Z*9#�WY?&3<�u%A!>?'*9#�,8*9?A* = ^V !$G/E46J3\*"243v*-�B%'!>?&E43<?$P~#'! h E4�$3v2Z*:#&E4T<1JE4!>�u%'!�*9#'E0T<�8*9($E4�<��*w2Z*9#7,]39#'E4($E43<�J!>#OFB1/!��/3#&!�($3<�/3=(>.4*9� P

« �Y� Q K ¢�� « �Y� ° Q K ¢ = �Y� Q K ¢ � = �Y� ° Q K ¢ ��°H^º¹"±« �¤� ° à K ¢�� « �Y� K ¢ = �¤� ° à K ¢L� = �¤� K ¢

,8*-?A*�%A3=6/3�%AE0!>Cl,]3 .�Ð : ^QR*9?A* ù 1/#'*9Cl3<#¸243<# I-*9203<?'!$#�6J!M²HPu§LSO®A§<^»NOGJ%'!>�/E0!>�/6/37*9#&.u243<#�#&E4T<1/!$�u%A!>#�?'!$#'1/2X%|*96J3<#>^· *9#�¼8T<1J?A*9#K°H^X§O®�°H^º½HP8C�1/!>#h%A?'*9��24*�#'3<201/(>E0�<�\6/!52Z*�!>($18*9($E4�<�\6/!53<�/6/*l1J%'E420E4�+*-�/6/3!$�yClD$%'3H6J3v6/!©¾7¿°±\(>39�y24*9#O($3<�/6/E0(>E439�/!>#°,�!>?&E4�=6/E4(>*9#7S�243<#°6/E0#&%'E4�u%A3<#7I9*-243<?&!>#O6/!ù ^g72KE4?;*91/Cl!>�u%A*9�/6/3­!$2MI-*92439?;6/!"2Z* ¬ P�(>1/*9�/6/3­!¥�&!>(>1H%|�9G8*-C:3<#ª!>2K,J?'3<T<?'*9C:*��/3<#6/E0C:39#'#�($1/!>�u%'!�6J!�F=1J!�!>2/!$?'?'39?µFB1/!�3<GJ%'!>�/.4*9C:39#�E4G/*)6/!>($?'!$(>E4!$�/6/3/Pu($3<Cl3O#'!�!$#',�!>?h®*9G/*JP�,/1/!$#�,8*9?'*�I-*9203<?'!$#�6J! ¬ C"*+S<39?'!>#$P»!>2uI-*9243<?¸6/!¯º�!$#�Cl!>�/39?¸S524*7*-,/?'3 � E4C:*9(>E0�<�!$#@C:�9# ,/?&!>($E4#A*H^ · 3ª*9�u%'!>?'E03<?µ#&!�Cª1/!>#h%A?A*5!>�l2Z*)%A*9G/24*�°H^0§�S�24*O?'!$,/?'!$#'!>�u%A*9(>E0�<�"T<?'�À®°9Á

Page 60: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1x

0

0.002

0.004

0.006

φ

ß E0T<1/?'*<°H^X§ E�Â�Ã&Ä$Å/Æ'ÇÈuÉ5Ê~Ë4Ì8Í°ÉqÎ�Ï+Î�ÐZÑZÉqÒ�Ó¸Ô�ÕOÖe×»ØÚÙ>Û-ØLÜ�Ô�ÎyÊ~Ë4Ì/ÍOÝ�Þ�ß Â S ìtà S Æ|Ô�á�ÔÆ|Ô�Î/È»ÑZÆqÑZâ�ÎlÑãÎuÑZÆqÑ0Ç$äZå]Ï�æ'Ç$Î/ÈuÔ5Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ@ç4ÃAÔ�ÎHРÉqÃ&ǨÓ�ÉqÃ¥ÑZâ9È»ÑZÆ'Ç>æAåHè7é�êÀ×�×Kë�ì¸é¨×»Ø

−1 −0.5 0 0.5 1x

0

0.002

0.004

0.006

φ

ß E0T<1/?'*ü°=^Ú¹ E7Â�Ã&Ä$Å/Æ'Ç�ÈuÉ�Ê~Ë4Ì8ͪÉqζÏ+ζÐZÑZÉqá�Ó¸ÔlÕªÖ°×»Øîí<ï»Ø@Ü�Ô�Î�Ê~Ë4Ì/Í�ÝðÞKß Â S ìtà S Æ|Ô�á�ÔÆ|Ô�Î/È»ÑZÆqÑZâ�ÎlÑãÎuÑZÆqÑ0Ç$ä/ë\Ï�æ&Ç$Î/ÈuÔ°Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ@ç4ÃAÔ�ÎHРÉqÃ&Ç�Ó¸ÉqÃAâ-È»ÑZÆ'Ç>æ|åHè�é�êÀ×�×Më�ì¸é¨Ù»Ø

ñ ²

Page 61: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1x

0

0.002

0.004

0.006

φ

ß E0T<1/?'*ü°=^º½ E7Â�Ã&Ä$Å/Æ'Ç�ÈuÉ�Ê~Ë4Ì8ͪÉqζÏ+ζÐZÑZÉqá�Ó¸ÔlÕªÖ°×»ØóòÀ×»Ø@Ü�Ô�Î�Ê~Ë4Ì/Í�ÝðÞKß Â S ìtà S Æ|Ô�á�ÔÆ|Ô�Î/È»ÑZÆqÑZâ�ÎlÑãÎuÑZÆqÑ0Ç$ä�Ï�æ'Ç$Î/ÈuÔ5Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ@ç4ÃAÔ�ÎHРÉqÃ&ǨÓ�ÉqÃAâ9È»ÑZÆ'Ç>æAåHô�é�êÀ×�×Kë�ì"Ýöõ¨÷»Ø¼8(>*l!$�<°=^ó±/^QR*9?A*ø6/!$¼/�/E4?©2Z*-#l($3<�/6/E0(>E03<�/!>#�6J!WY?&3<�u%A!>?'*y?&!$ù8!¥�'*9�u%A!$#>P@�/3<#lE4�u%A!$?'!>#'*yFB1/!�xö#'!>*!$#&%|�À%AE4(>*w!>�y2Z*9#7WY?'3<�u%A!$?A*9#$P�!>#°6/!$(>E0?°F=1J!l24*{ú�ûú T � : Sü6/!©24*w6/!$¼8�JE4(>E0�<�­6/! = (>39�ù � : 3<GJ%'!>�/!$C:39#�FB1/! = � ú�ûú T Pó!$�B%'3<�/($!># = � : !$��2Z*9#¸WY?'3<�u%'!>?A*-#>P96J!�!>#h%A3)%'!>�/!$C:3<#FB1/! ú|ýú T � : Sy(>3<Cl3 ú|ýú T � ú|þú  P�2024!$Tu*9Cl3<#°*ÿú|þú  � : ^¸[ #ª,�3<?5!>#&3üFB1/!l,]3=6/!>Cl3<#6/!¦¼8�/E0?�2Z*9#�($3<�/6/E0(>E439�/!>#�6/!°WY?&3<�u%A!>?'*l?&!$ù8!¥�'*9�u%A!$#76/!524*�#'E4T91/E4!$�B%'!OWY3<?'C:* E

« �Y� \ ¢@� « �¤� K ¢ « �Y� °�¢L� « �¤� ° Q K ¢ = �¤� \ ¢ � = �¤� °�¢ � : ��°H^ó½T±Q�!>?'3­($3<�{!$#&%|*-#�(>3<�J6/E4($E43<�/!$# ù #&3<243�,]3=6/.Z*\%A3<C:*9?�!>2LI-*9243<?�6/!"²J^º²HPR!$�B%'3<�/($!>#�#'!G/1J#'(+*-?'3<�ü($3<�/6/E0(>E03<�/!>#76/!;WY?&3<�u%A!$?A*:?&!$ù8!¥�'*9�u%A!$#)I-�924E068*9#¨,8*9?'*:I-*9243<?&!>#�6/! ù��� : S#&!�2024!$T<�©*�243�#'E0T<1/E0!>�u%A!-P

ñ §

Page 62: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3�°H^X§ E���Î�ä0Ç�Ð�Ç��>ä0Ç�æAÉ�á;Ï9Éhæ¦ÐZÃ&Ç�Éqä)ÉqÃ¥ÃAÔ�Ã��<Ï9É�HÇ&ëöÉqÎHÐZÃAÉ:ä0Çæ|Ô�äYÏ9ÆqÑZâ�ÎöÉ�BÇÀƦÐ�Çëøä0Ç"ÎHÏ+á �qÃ¥ÑZÆ'Ç­Ô��<РÉqÎuÑZÈuÔ�Ó�Ç$Ã&Ç�Ç$ä��=Ï+Î/Ô¦æ��|Ç$äZÔ�ÃAÉhæ�ÈuÉ�ô\å@Æ|Ô�ÎoÆ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ:ÈuÉ�ç4ÃAÔ�ÎHРÉqÃ&ÇÓ�ÉqÃ¥ÑZâ9È»ÑZÆ'Ç>æMÇ�Ï+Î\ÐZÑZÉqá�Ó¸Ô�Õ¸ÖO×»ØÚ×>ò�ê-Ø� ù � ²J^ó² ù � §9P=®|§°9² °H^º¹H§�²<²9!�®q²T° §<^ó²��"°"°-!�®q²9±½<²<² ¹H^óÁu¹9Áu¹-!�®q²T° °H^Î6T°9²<Á9!�®q²T°°"°9² ¹H^º¹9²J§�Á9!�®q²T° ±/^ ±u²<²T°-!�®q²T°6<²<² §<^Î6<½ ñ �-!�®q²T° ½J^ ñ �H§ ñ !�®q²T°§�²<²9² §<^ ñ ±��9²9!�®q²T° ½J^º¹9Áu¹��-!�®q²T°

0 200 400 600 800 1000Ν

0

1e−05

2e−05

3e−05

4e−05

5e−05

6e−05

7e−05

8e−05

erro

r

ß E0T<1/?'*Ú°=^ó± E���ä ÉqÃ¥ÃAÔ�Ã5È»Ñ æqá°ÑãÎHÏ$ëBÉ�Æ|Ô�Îuç Ô�Ã¥á�É7ä0Ç�è Ç9Ï+á�ÉqÎJÐ�Ç<Ø�Ü�Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ�ç4ÃAÔ�ÎHРÉqÃ&ÇÓ�ÉqÃ¥ÑZâ9È»ÑZÆ'Ç>æ�Æ|Ô�Ωì�é¨×»ØÚ×Kë�Õ¸ÖO×»ØÚ×>ò�ê-Ø

ñ ¹

Page 63: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1χ

−0.006

−0.004

−0.002

0

0.002

φ(χ)

ß E0T<1/?'*@°H^¯° E�Â�Ã&Ä$Å/Æ'Ç�ÈuɵÊ~Ë4Ì8Í�ÉqÎlÏ+ΩÐZÑZÉqá�Ó�ÔMÕ�Ö�Ù»Ø��¦Ø���ÐZÑãäãÑ��$Ç$Î/ÈuÔ�Ê~Ë4Ì/Í�Ý{ÞKß Â S ìtà S Æ|Ô�á�ÔÆ|Ô�Î/È»ÑZÆqÑZâ�ÎlÑãÎuÑZÆqÑ0Ç$ä/ëwÆ|Ô�ÎwÆ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ@ç0ÃAÔ�ÎJРÉqÃ&Ç)ÃAÉ��/É! AÇ$ÎHРÉhæAå/ô Ý#"%$&$Oë�ì¸é¨×»ØÚ×»Ø

« �Y�(' ¢�� « �¤� K ¢ � � ;"ù 1 ¢ ´ ] ù = �¤� K ¢� � ;vù 1 ¢ = �¤� \ ¢ � ;Bù « �Y� \ ¢ ��°H^ ±5±« �Y� °M¢�� « �¤� ° Q K ¢ � � ´ ù 1�¢ ´ ] ù = �Y� ° ; � ¢

� � ;äù 1 ¢ = �Y� °K¢ � ;Bù « �Y� °K¢ ÅV !ª2Z*9#7(>3<�/6JE4(>E03<�/!$#O6/!ªWY?&3<�u%A!>?'*w,]3=6/!$C:3<#73<G/#&!>?&I-*9?)F=1J! ù �J3v,/1/!>6J!�%A3<C:*9?�243<#I-*9203<?'!$#�8©�9^ · 3<#µI-*9243<?&!>#µ,/*9?A* ù FB1/!¨#'!�1J%'E424E0�+*9?&3<�ª!$�©!>#&%'!�%A?A*-G8*»�&3°#'39� E ²J^ó²JP<²J^ ±OS®¥²J^ó±J^/QR3=6/!$C:3<#�*9,J?'!>($EZ*9?�24*�#'3<201/(>E0�<�\6/!;x �¤� ¢ (>3<�!>#h%|*9#¨(>39�/6/E4($E43<�J!>#¨6/!°WY?'3<�u%'!>?A*Sv203<#�I-*92439?'!>#�,/*9?A* ù Cl!>�J(>E439�8*96/3<#�!$��24*9#�¼8T<1J?A*9#Û°H^¯°»®�°H^ºÁH^_*),+*) -/.>#��0.Ma$b;c 1325476�a98:8�6ªcf �9�¦�hg�� i^��jNª§�Á���¶¡B¥Ljlkãª���¤ m|���L¥Lj��3�§�����L�¶�B¡§¥on �qpi� m'�W�§�L���@jlr«���B�ts�|�Lj; %AE420E4�>*9C:39# ; = 1 � M 1 ´ < 1 ¢ VsX[Z � = M � ¢ VsX[Z � = <3= ¢ S � M 1 ´ < 1 ¢ O I  à ï?> ($3<Cl3�24*9# WY1/�H®($E43<�/!$#�y �Y� � = ¢ !$��24*9#�!>($18*9(>E03<�/!$#ª�Y±/^X§@�"±¦P���±/^X§�ÁT±MSy�Y±/^º¹<¹"±¦^ñ ½

Page 64: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1χ

−0.002

0

0.002

0.004

0.006

φ(χ)

ß E0T<1/?'*ü°=^ ñ|E�Â�Ã&Ä$Å/Æ'Ç�ÈuÉ�Ê~Ë4Ì8Í;ÉqζÏ+ζÐZÑZÉqá�Ó¸ÔlÕ;ÖO×»ØÚÙ>ò-ØA��ÐZÑãäãÑ��$Ç$Î/ÈuÔvÊ~Ë4Ì8ͪÝeÞ�ß Â S ìtà SÆ|Ô�á�Ô5Æ|Ô�Î/È»ÑZÆqÑZâ�ÎlÑãÎuÑZÆqÑ0Ç$ä8ë"Æ|Ô�ÎwÔ�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉLç0ÃAÔ�ÎJРÉqÃ&ÇOÃAÉ��/É! AÇ$ÎHРÉhæAåJô ÝB"%$&$)ë�ì�é¨×»ØDC<Ø

−1 −0.5 0 0.5 1χ

−0.01

−0.005

0

0.005

0.01

φ(χ)

ß E0T<1/?'*ü°=^�� E�Â�Ã&Ä$Å/Æ'Ç�ÈuÉ�Ê~Ë4Ì8Í;ÉqζÏ+ζÐZÑZÉqá�Ó¸ÔlÕ;ÖOÙ»ØDC+íuØA��ÐZÑãäãÑ��$Ç$Î/ÈuÔvÊ~Ë4Ì8ͪÝeÞ�ß Â S ìtà SÆ|Ô�á�ÔOÆ|Ô�Î/È»ÑZÆqÑZâ�Î�ÑãÎuÑZÆqÑ0Ç$äJëlÆ|Ô�Î"Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ ç0ÃAÔ�ÎJРÉqÃ&Ç7ÃAÉ��/É! 'Ç$ÎJРÉhæAå=ô ÝB"%$&$7ë;ì¸é¨×»ØDC<Øñ ±

Page 65: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1χ

−0.002

0

0.002

0.004

0.006

φ(χ)

ß E0T<1/?'*©°H^Î6 E�Â�Ã&Ä$Å/Æ'Ç©ÈuÉ�Ê~Ë4Ì/ͨÉqÎ�Ï+ÎüÐZÑZÉqá�Ó¸Ô)ÕLÖO×»ØÚÙ�íuØ Ü�Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ)ÈuÉ�ç4ÃAÔ�ÎHРÉqÃ&Ç�ÃAÉ��/ÉFE 'Ç$ÎJРÉhæAåHô Ý#"%$&$)ë�ì¸éGEt×»ØDC<Ø

−1 −0.5 0 0.5 1χ

−0.01

−0.005

0

0.005

0.01

φ(χ)

ß E0T<1/?'*ü°=^ºÁ E�Â�Ã&Ä$Å/Æ'Ç�ÈuÉ�Ê~Ë4Ì8Í;ÉqζÏ+ζÐZÑZÉqá�Ó¸ÔlÕ;ÖOÙ»ØóòÀÙ»ØA��ÐZÑãäãÑ��$Ç$Î/ÈuÔvÊ~Ë4Ì8ͪÝeÞ�ß Â S ìtà SÆ|Ô�á�ÔOÆ|Ô�Î/È»ÑZÆqÑZâ�Î�ÑãÎuÑZÆqÑ0Ç$äºë:Æ|Ô�Î"Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉ ç0ÃAÔ�ÎJРÉqÃ&Ç7ÃAÉ��/É! 'Ç$ÎJРÉhæAåBô Ý#"%$&$�ë;ì¸éGEt×»ØDC<Øñ °

Page 66: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3P°H^Ú¹ E��¸Ã¥ÃAÔ�Ã@Ô��<РÉqÎuÑZÈuÔ)Æ|Ô�Î�Ç$ä��=Ï+Î/Ô¦æH�|Ç$äZÔ�ÃAÉhæLÈuɵô\å�Ó�Ç$Ã&Ç)Æ|Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÃAÉ��/É! 'Ç$ÎJРÉhæÆ|Ô�ÎÏ+Î\ÐZÑZÉqá�Ó¸Ô�Õ�Ö ×»ØÚ×�Ù��¦Ø� ù � ²J^ ±/PH®q²H^ó±°-² ¹H^X§�Á9±uÁ9!�®q²9±¹´°9² §<^ ±��9½P69!�®q²T°°-²<² §<^ó²<½J§�½9!�®q²T°�´°9² 6J^ó½P6u¹-±<!�®q² ñ§+²<²<² �H^º¹-±/§<§+!�®q² ñ

0 200 400 600 800 1000N

5e−06

1e−05

1.5e−05

2e−05

2.5e−05

erro

r

ß E0T<1/?'*>°H^X§�² EI��ä�ÉqÃ¥ÃAÔ�Ã�È»Ñ æqá°ÑãÎHÏ$ëBɨÆ|Ô�Îuç Ô�Ã¥á�É@ä0Ç)ô Ç9Ï+á�ÉqÎHÐ�Ç<Ø]Ü�Ô�Î/È»ÑZÆqÑZÔ�Î/Éhæ�ÈuÉLç4ÃAÔ�ÎHРÉqÃ&ÇÃAÉ��/É! 'Ç$ÎJРÉhæ�Æ|Ô�Ωì�éGEt×»ØDC�ë�Õ¸Ö ×»ØÚ×�Ù��¦Ø

ñ<ñ

Page 67: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ÝH!�?'!>*924E0�>�©1/��,/?'3<T9?A*9C:*�!>��20!>�/T918*»�&!ª�)P�,/*9?A*l3<GJ%A!$�/!>?�2Z*©#'39241/($E4�<�\6/!�2Z*9#¨!$(>18*À®($E43<�/!$# E !>(-^ �Y±/^0§&�"±>� U�D$%'3H6/3l6/! J<*9(>3<GJE ±¦P�!$(9^���±/^X§�ÁT±5� UD$%A3=6/3w6J! þ *91/#'#q®�ÝH!$E46/!$2 ±KS!$(9^4�5±/^º¹<¹"±\� U�D$%'3H6/3y6/! Ý/N°¾�±�^L[ ��2Z*ü¼8T91/?A*ü°H^X§<§�#&!�Cª1/!>#h%A?A*�x �¤� � = ¢ ,/*9?A*ø2Z*#&3<241/($E4�<�:6/!°,/?&1/!>G8* ; = 1 � M 1 ´ < 1¦¢ VYX[Z � = M � ¢ VYXûZ � = < � ¢ 1J%AE024E4�>*9�/6/3�!>2]C:D¦%A3=6/3�6/!Ý/N°¾5^[ #&%'!�,/?&3<T<?'*9C"*­%|*-C�G/E4D$�ÿ�J3<#:,J?'3<,�3<?'($E43<�/*y!$27!$?'?'39?:FB1/!�?'!>#&1/20%A*�6/!�(>3<Cl,8*9?'*9?24*�#'39241/($E4�<�{! � *9($%A*�S{2Z*#'3<201/(>E0�<�{�B1/ClD>?&E4(+*{� �739?'C:*ü¹"±;39GJ%A!$�/E468*�(>3<��203<#;%'?'!>#ClD$%'3H6/39#¨*9�u%A!>#¨Cl!>�/($E43<�/*96/3<#$P/!>2�!$?'?'39?�#&!5(+*920(>1/24*�,8*9?A*�(>*968*©E0%A!$?A*9($E4�<�~PO · � ����

K Q K²á�L ¡ JLKh� x�� Â��&� �Y� á¥� = ¡¦¢ ; x I$£�ï �¤� á¥� = ¡¦¢&¢ 1 ��°H^¯°"±�K3<� y �¤� � = ¢ � ; = 1 � M 1 ´ < 1¦¢ VsX[Z � = M � ¢ VsX[Z � = <3= ¢ 1J%AE024E4�>*9Cl3<#K($3<C:3©#&3<241/($E4�<�\6/!A ,/?'1/!$G8* C 2Z*9#~#'E4T91/!>�u%A!$#�WY1/�/(>E03<�/!$# E ²J^ó²JP>²J^þ° VsX[Z � = M � ¢ VsX[Z � = <3= ¢ S°²J^óÁ VYXûZ � = M � ¢ VYXûZ � = <3= ¢ PS©24*9# (>3<�J6/E4($E43<�/!$#LE4�/E0(>EZ*-24!>#�#'3<�NM � ; �<� = ¢µ� M � �<� = ¢µ� M �Y� � ; � ¢ � M �Y� �>� ¢ � ²J^º²g72 h *9($!>?°WY1/�/($E43<�/*9?)!$2µ,/?'39T<?A*9C:*"($3<�ø(+*968*w1J�8*v6/!�2Z*9#)#'3<201/(>E03<�/!>#)6/! A ,/?&1/!>G/* C,8*-?A*ª(>*968*�ClD$%A3=6/3/P=�/3<#K6/E4Cl3<#K($1/!>�u%A*©FB1/!)!>2 !$?'?'39?K!>?'*�!>2]C:E0#'Cl3ª,8*9?'*�2Z*9#M#'3<201H®($E43<�/!$#O6/!�,/?&1/!>G/*JP (>39�y(>*968*w1/�J36J!�243<#�%A?'!$#°C:D¦%A3=6/3<#$P~2Z* j �/E4(>*w6/E0WY!$?'!$�/(>E4*w!>?A*!$2�� j Cl!>?'3v6J!�E0%A!$?A*9($E43<�J!>#)!$�ø!>2�FB1/!�#&!�3<GJ%'1JI<3\!>2R!>?&?'3<?$P~!>#°6J!>(>E0?>P]F=1J!�(>3<�ø1/�8*#&3<241/($E4�<�o6/! A ,J?'1/!$G8* C !$2�� j Cl!>?&3�6/!E0%'!>?'*9(>E0�<��WY1/!\C"�9#©,�!>FB1/! ? 3{FB1/!�($3<�¶2Z*9#39%'?A*9#$^ · 3l*9�u%A!>?&E43<?�#&!5C�1/!>#h%A?'*l!$2�2Z*9#�%A*9G/2Z*-#���°H^ó½T±MSy� °H^ó±T±¦^[ 2<,/?&3<T<?'*9C"*�#&!L(>3<Cl,/E420��(>39��6JE4#&%'E4�u%|*-# ¬ P�,8*9?'*�I<!>?¸($3<C:��!$?A*�!>2u($3<Cl,]3<?h%|*9ClE4!$�u%A36/!$28!>?&?'3<?L,8*9?'*°(+*968*51/�8*56/!¨D>#h%|*9#$PuS�243°FB1/!�#'!�3<GJ%'1JI<35WY1/!¨FB1/!)*-2/E4?L*91JC:!$�B%A*9�/6/3!$2]I-*92439?@6J!O2Z* ¬ PH!$2~!$?'?'39?K#'! h *9(>.4*;C:�9#K,�!>FB1/! ? 3/PJD$#&%'3�#'!OC�1J!>#&%'?A*�!$�v2Z*9#L%|*9GJ2Z*9#°H^ó½vSü°H^ó±J^¸[L#h%A3<#;(+*9CªG/E43<#5#&! h E0(>E0!>?'39��,8*-?A*\2Z*9#°%A?&!>#�#'3<201/(>E03<�/!$#�6J!:,/?&1/!>G/*�(>39�(>*968*;1J�/3�6/!7243<# %A?&!>#MC:D¦%A3=6/3<#LS:!$2]?&!>#&1/20%A*96/3;WY1J!7!$2]ClE4#&C:3/^ · *�T<?A*À¼8(+*>°=^0§»¹°�/3<#C�1J!>#&%'?A*�243l*9�u%'!>?'E03<?>^* 3<C:*9�/6/3{($3<Cl3{?'!¦WY!>?'!$�/(>E4*{2Z*9#l%|*9GJ2Z*9#N°H^º½�S °H^ ±/P h *9(>!$C:3<#"1/�8*{(>39C:,8*-?A*9($E4�<�

ñ �

Page 68: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

-1-0.5

0 0.5

1-1-0.5

0 0.5

1

-1.5

-1

-0.5

0

0.5

1

1.5

’seidel.dat’

ß E0T<1/?'*�°=^0§<§ ElµÃ&Ä$Å/Æ'ÇÿÈuÉPO�Ë4ÌRQ�SBÍ�Ó�Ç$Ã&Ç�ÌTQ�SVUXWÚõ¨÷%Q�÷ZY¤å�Æ|Ô�Î Éqä°á �¦Ð Ô9ÈuÔ�ÈuÉ\[�]_^ÏuÐZÑãäãÑ��$Ç$Î/ÈuÔ}ô Ý ÷`$&$9åba Ý c Ý ÷�Ó�Ç$Ã&Ç}ä0Ç æAÔ�äYÏ9ÆqÑZâ�ÎeÈuÉøÓJÃ�Ï9Éd�»Ç õfe 1 Ë�a 1 gc 1 Íih�jlk�ËmeIa8Ì/Íihnjlk�ËmeIc�Ì/ÍAØ oñ 6

Page 69: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3x°H^ó½ E���Î;ä0ÇOÐ�Ç��>ä0Ç�æAɸáªÏ9Éhæ�ÐZÃ&Ç)Éqä=Îqp+á�ÉqÃAÔ)ÈuÉ�ÑYРÉqÃ&ÇÀÆqÑZÔ�Î/Éhæf�uÏ9ÉRæAÉRÃAÉ'Ç$äãÑ��+â�ÉqÎ�Æ'ÇÀÈ9Çá �¦Ð Ô-ÈuÔµÓ�Ç$Ã&Ç7Ô��<РÉqÎ/ÉqÃMÏ+Î�ÉqÃ¥ÃAÔ�à ÖBC<ØÚïrCÀÉFEsC)Æ|Ô�Î;ô Ý#t%$vuãÎqp+á�ÉqÃAÔ7ÈuÉLÈ»Ñs�¦Ñ æqÑZÔ�Î/Éhæ ÈuÉLÆ'ÇÀÈ9ÇÉ! |É�ÈuÉLä0ÇOÃAÉ|È&o ë"Æ|Ô�Îa:Ýwc�Ýö÷@Ó�Ç$Ã&Ç)ä0Ç>æLæ|Ô�äYÏ9ÆqÑZÔ�Î/Éhæ�ÈuÉ�ÓJÃ�Ï9Éd�»Ç�×>å�×»Øóêrhnjlk�Ë�a8Ì/Íihnjlk�Ë�a�SuÍëv×»Øyxzhnjlk�Ë�a8Ì/Íih�j{k�Ë�cSuÍAØU�D¦%A3=6/3 ²J^ó² ²J^¯° VYXûZ � M � ¢ VYXûZ � <3= ¢ ²J^óÁ VsX[Z � M � ¢ VsX[Z � <3= ¢J<*9($3<G/E |�} ç ] °"°-²P6 °<¹H§ ñþ *91/#&#h®�Ý=!>E46J!>2 ½J§9§@� ¹9Á-±/§ ¹��<¹��Ý/N°¾ Á9ÁP6 Á��9² ÁJ§<§

�K18*-6/?'3W°H^ ± E è~p+á�ÉqÃAÔ�ÈuÉ�ÑYРÉqÃ&ÇÀÆqÑZÔ�Î/Éhæ@Ó�Ç$Ã&Ç"Ï+Î�ÉqÃ¥ÃAÔ�Ã7Ö�Ù»ØÚï�ï>ÉFEsClÆ|Ô�Îwô Ý ÷`$&$;ë�a�Ýc�Ýö÷>å9Ó�Ç$Ã&Ç7ä0Ç>æ7ÐZÃAÉhæLæAÔ�äYÏ9ÆqÑZÔ�Î/Éhæ�ÈuÉRÓJÃ�Ï9Éd�»Ç<ØU�D¦%A3=6/3 ²J^ó² ²J^¯° VsX[Z � M � ¢ VYXûZ � <3= ¢ ²H^ºÁ VsX[Z � M � ¢ VsX[Z � <3= ¢J<*9($3<G/E §G°=§+±uÁ §+±��9ÁP6 §�½<ÁP6T°þ *91/#'#q®�ÝH!$E46/!$2 ÁT°À±/§ 69ÁP6<Á 6J§@�9½Ý/N°¾ ¹"°9¹H§ ¹À±u½9± ¹<¹ ñ §

ñ Á

Page 70: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 20 40 60 80 100Ν

−10

−8

−6

−4

−2

Lo

g(e

rro

r)

ß E0T<1/?'*ü°=^0§»¹ E���Ç$Ã&Ç��AÇ$äZÔ�ÃAÉhæ�á;Ä>æ��»Ã&Ç$Î/ÈuÉhæ©Èuɪô uãÎ/Ô-صÈuÉ©È»Ñs�¦Ñ æqÑZÔ�Î/Éhæ�ÈuÉ©Æ'ÇÀÈ9ÇüÉ! |É�ÈuÉä0Ç�ÃAÉ|È&o|å�ÉqäOÉqÃ¥ÃAÔ�ÃvÉhæ�á;Ä>æ�Ó¸É`�<Ï9É��/Ô�å�Éhæ"ÈuÉ|ÆqÑãÃAå@ä0Ç�æAÔ�äYÏ9ÆqÑZâ�Î�ÎHÏ+á �qÃ¥ÑZÆ'ÇøæAÉ"ÇÀÆ|ÉqÃAÆ'Ç�á;Ä>æÇ"ä0ÇwæAÔ�äYÏ9ÆqÑZâ�ζÉ�BÇÀƦÐ�ÇÀåMÆ|Ô�Î�a}Ý�c ÝOÙ»ØÚ×>å Ó�Ç$Ã&Ç"ä0Ç\æAÔ�äYÏ9ÆqÑZâ�Î�ÈuÉ°ÓJÃ�Ï9Éd�»Ç9õfe 1 Ë�a 1 gc 1 Íih�jlk�ËmeIa8Ì/Íihnjlk�ËmeIcSuÍ�o

�9²

Page 71: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3x°H^¯° E�Ü�Ô�á�Ó�Ç$Ã&ÇÀÆqÑZâ�Î:ÈuÉ�äZÔ¦æ@ÐZÃAÉhæ_�P�¦Ð Ô-ÈuÔ¦æAå=Æ|Ô�ÎlÏ+ΩÉqÃ¥ÃAÔ�Ã&Ö}Ù»ØÚï�ï�Ù>Û�ÉFEsCÀå<ô Ýö÷`$&$9åc�Ý�a:Ýö÷Më\РÔ�á;Ç$Î/ÈuÔ�Æ|Ô�á�Ô7æ|Ô�äYÏ9ÆqÑZâ�ÎvÈuÉRÓJÃ�Ï9Éd�»Ç�Ǫ׻Øyxzh�j{k�Ë�a8Ì/Íih�jlk�Ë�cSBÍAØU�D$%'3H6/3 �t%A!>?'*9(>E0�<�J<*9(>3<GJE §+½<ÁP6T°þ *91J#'#h®�ÝH!>E06/!>2 6J§@�9½Ý/N°¾ ¹<¹ ñ §

�K18*-6/?'3�°=^ ñ|E�è�p+á�ÉqÃAÔªÈuÉ�ÑYРÉqÃ&ÇÀÆqÑZÔ�Î/ÉhæLÓ�Ç$Ã&ÇlÏ+ÎÉqÃ¥ÃAÔ�Ã�Ö ò-ØÚï>ò�ÉFEt×rC�Æ|Ô�Îvô Ý��%$9åIaÝz÷ë c�Ý ÷)Æ|Ô�Ωä0Ç>æ çtÏ+Î/ÆqÑZÔ�Î/Éhæ�ÈuÉRÓJÃ�Ï9Éd�»Ç�×»ØÚ×>å~×»Øóê7ß I  à ï�> ë\×»Øyx�ß I  à ï�> ØU�D¦%A3=6/3 ²H^º² ²J^¯° O I  à ï�£ ²J^óÁ O I  à ï�>J<*9($3<G/E ¹ ñ ²<Á ¹"°´°9½ ¹À±B¹��þ *91/#&#h®�Ý=!>E46J!>2 §9§�½T° §�½9²�� §�¹-±B¹Ý/N°¾ § ñ ¹ §+±<Á §>±�6

!$�B%'?'!�243<#�%A?&!>#lC:D¦%A3=6/3<#©,8*9?'*øI9!>?:(>18*92¨6/!�D>#h%A3<#l!>#:C"�9#©?'�9,/E46J3�,8*-?A*ø3<GJ%A!$�/!>?24*�#'39241/($E4�<�{�B1/ClD>?&E4(>*�C:�9#�*9,J?'3 � E4C:*968*�*ü24*�#&3<241/($E4�<��! � *9(¦%|*üSy2424!$Tu*9Cl3<#;*�2Z*($3<�/(>201/(>E0�<�\FB1/!;!$#7!$2�U�D$%'3H6J3:6/!®Ý/N°¾ � * *9GJ2Z*¹°H^¯°"±¦P/,�3<?¨243©FB1/!5!>#�(>39��D$#&%A!�(>39�!$2µFB1/!�#'! h *-?A�w,�3<#&%'!>?'E03<?'Cl!>�u%'!�2Z*"($3<Cl,8*9?A*-(>E4�9�ü!>�u%A?&! V E0WY!$?'!$�/(>E4*9# ß E0�/E0%A*9#�S�!>2U�D¦%A3=6/3©6/!;�K3<243=(+*-(>E4�9�\[L#&,]!$($%'?A*92 ^�K18*-�/6/3­*9,J24E4(>*9Cl3<#®y �Y� � = ¢ � � M 1 ´ < 1�¢ O I  à ï�> P�1J%'E424E0�+*9Cl3<#°($3<C:3WY1/�/($E43<�/!$#ª6/!,/?&1/!>G/* E ²J^ó²JP¸²J^¯° O I  à ï?> S­²J^ºÁ O I  à ï�> S­($3<�y(>3<�/6JE4(>39�/!>#�E4�/E0(>E4*924!$#�M � ; �<� = ¢O�� M 1 ´ < 1�¢ O Q I à ï?> P:M � �<� = ¢7� � M 1 ´ < 1�¢ O I à ï�> M �¤� � ; � ¢O� � M 1 ´ < 1�¢ O I  Q ï SM �Y� �+� ¢ � � M 1 ´ < 1¦¢ O I  à ï P8!$��24*9#�%|*-G/2Z*9#�°H^ ñ SW°=^���#'!5C�1/!$#&%'?A*9�243<#�?&!>#&1/20%A*96/3<#FB1/!l#'!:3<GJ%A1HIHE0!>?&3<��,/*9?A*\!>#h%A!"!¥�&!>Cl,/243J^ · *T<?'�-¼8(>*<°H^X§�½vC�1/!$#&%A?'*�24*#&3<241/($E4�<��x,8*-?A*©²J^óÁ O ½ I  à ï?> ¾ ($3<��!$2�C:D¦%A3=6/3©6/!®Ý/N°¾�^

�H§

Page 72: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

’epoisson.dat’

-1

-0.5

0

0.5

1 -1

-0.5

0

0.5

1

0

1

2

3

4

5

6

7

8

ß E0T<1/?'*B°H^0§+½ E¸Â�Ã&Ä$Å/Æ'Ç;ÈuÉHO�Ë4ÌRQ�SBÍhå<Ó�Ç$Ã&Ç)ÌTQ�SUPWÚõ¨÷%Q�÷ZY¤å~ÏuÐZÑãäãÑ��$Ç$Î/ÈuÔ;Éqä�á �¦Ð Ô9ÈuÔ5ÈuÉ�[�]_^Æ|Ô�Îlô Ýö÷`$&$)ëa:Ý�c�Ýö÷%Q�$7ëwÆ|Ô�Ωä0Ç7çtÏ+Î/ÆqÑZâ�ÎvÈuÉ�ÓJÃ�Ï9Éd�»Ç�×»Øyx¦ß ½ I  à ï�> ¾ Ø�<¹

Page 73: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3©°H^y� E/è�p+á�ÉqÃAÔ5ÈuÉ@ÑYРÉqÃ&ÇÀÆqÑZÔ�Î/ÉhæRÓ�Ç$Ã&Ç�Ï+ÎvÉqÃ¥ÃAÔ�ÃKÖ Ù»Øóò@x+ÉFEt×>ê;Æ|Ô�Îlô Ý ÷`$&$9åqa:Ýö÷ë c�Ý ÷Më"Æ|Ô�Ωä0Ç>æ@æAÔ�äYÏ9ÆqÑZÔ�Î/Éhæ�ÈuÉRÓJÃ�Ï9Éd�»Ç©×»ØÚ×>å�×»Øóê)ß I  à ï�> ëv×»Øyx�ß I  à ï?> ØU�D¦%A3=6/3 ²H^º² ²J^¯° O I  à ïR£ ²J^óÁ O I  à ï�>J<*9($3<G/E °9Á9² ñ ° °�� ñ ½P6 °À±u½P6 ñþ *91/#&#h®�Ý=!>E46J!>2 ½<²H§�½J§ ¹9Á-±B¹9Á ¹i���9ÁP6Ý/N°¾ ½"°-±�� ½9± ñ § ½<¹9ÁT°

�K18*-6/?'3B°=^ 6 E]Ü�Ô�á�Ó�Ç$Ã&ÇÀÆqÑZâ�ÎwÈuÉ�äZÔ¦æ�ÐZÃAÉhæ_���¦Ð Ô9ÈuÔ¦æAåJÆ|Ô�Î"Ï+Î:ÉqÃ¥ÃAÔ�Ã@Ö}Ù»Øóò@x+ÉFEt×>ê�åBô Ýö÷`$&$9åc�Ýö÷>å�alÝö÷Kë\РÔ�á;Ç$Î/ÈuÔ5Æ|Ô�á�Ô7æ|Ô�äYÏ9ÆqÑZâ�ÎvÈuÉ�ÓJÃ�Ï9Éd�»Ç5Ç�×»Øyx¦ß I  à ï?> ØU�D$%'3H6/3 �t%A!>?'*9(>E0�<�J<*9(>3<GJE °À±u½P6 ñþ *91J#'#h®�ÝH!>E06/!>2 ¹i���9ÁP6Ý/N°¾ ½u¹9ÁT°

· *9#)%A*9G/24*9#©��°=^ ñ ±�So� °H^��´±7C�1/!$#&%A?'*9�­243:Su*v6/E4( h 3v!>�ø!>2R!q�&!>Cl,/203v*9�u%A!$?'E439?>P *-2�(+*-2X®($1/2Z*9?©!$27!$?'?'39?:!$�B%'?'!�2Z*�#&3<241J(>E4�9�ÿ! � *-($%|*ySo2Z*ø�=1JC:D$?'E4(>*�?&!>#'1J20%|*y#'!$?"!>2�ClE4#'Cl3,8*-?A*�%'3H6/*9#K24*9#K#&3<241/($E43<�J!>#@6/! A ,J?'1/!$G8* C (>3<�"(+*968*;1/�/3�6/!)243<#@C:D¦%A3=6/3<#$P=,�!>?&3�(>39�6/EXWY!>?&!>�u%A!$#�EX%A!$?A*9($E43<�/!$#>^�� #h%A3l#'!5C�1/!$#&%A?'*l!$��24*ª¼8T<1/?'*©°=^0§+±J^[ #v!$I=E06/!>�u%A!­FB1/!ø!$2)ClD$%A3=6/3�FB1/!­Cl!>�/39#wE0%'!>?A*-(>E439�/!>#w?&!+*920E4�>*�,8*9?'*�2420!>Tu*9?"*{2Z*#&3<241/($E4�<���B1/ClD>?&E4(>*øC:�9#©(>!$?'(>*9�8*y*ø24*­#'3<201/(>E0�<��! � *9(¦%|*�!$#©!>2�ClD$%A3=6/3ø6/!âÝ/N°¾� * *9GJ2Z*�°H^Î6T±¦P]6/!�2Z*wClE4#&C"*lC"*-�/!>?'*wF=1J!l!$�ø!>2 !q�&!$C:,J243v*9�u%'!>?'E03<?O1J%'E420E4�+*-?'!>Cl3<#�!>2ClD$%'3H6/3�6/! Ý/N°¾ ,8*9?'* h *9($!>?"24*�(>39C:,8*-?A*9($E4�<�¶(>3<� !$27ClD$%'3H6J3{6/!­�K39243=(+*9($E4�<�[ #',�!>(¦%A?'*92 ^

�9½

Page 74: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 20 40 60 80 100Ν

−10

−8

−6

−4

−2

Lo

g(e

rro

r)

ß E0T<1/?'* °H^X§+± Ef�@ä@ÑãêÇ9Ï+á�ÉqÎHРÔ>Ç$Î/ÈuÔüÉqä~�|Ç$äZÔ�Ã�Èuɪä0Çwô\åMÉqä��|Ç$äZÔ�éÈuÉqä�ÉqÃ¥ÃAÔ�êæAÉ�ÃAÉ|È=Ï9Æ|É|åul���¦Ð Ô9ÈuÔ°ÈuÉ�[�]_^}Æ|Ô�Î*a:Ý�c�ÝOÙMë�ç Ï+Î/ÆqÑZâ�ÎwÈuɵÓJÃ�Ï9Éd�»Ç�×»Øyx¦ß I  à ï�> o

�-±

Page 75: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

f �9�¦�9�>� i^��jNª§�Á���¶¡B¥Lj��¯ª���¤ m'���L¥@j��3�����������§¡B¥è�W��� ¤��|¥¦¡B¥ä¡§���m¥¶��¥�s�������� ����\j5� �W�/�Hm|���QR*9?A*o!>#h%A!�ClD$%'3H6J3o%A*9C�G/E0D>� %A?'*9G8*»�'*-?'!>Cl3<#\(>39�z2Z*-#\WY1J�/(>E03<�/!>#ây �¤� � = ¢ F=1J!�#'!,/?&!>#&!>�u%|*9��!>2¨2Z*­!>($18*9($E4�<� ��±J^0§»¹´±¦P ; = 1 � M 1 ´ < 1¦¢ VYX[Z � = M � ¢ VYXûZ � = <3= ¢ S � M 1 ´< 1 ¢ O I  à ï�> P8,/*9?A* � � =�� t ; �<�+�bw ^g72 E4T<18*-2µFB1/!"($3<� V E0WY!>?&!>�/($EZ*9# ß E0�/E0%A*9#>P]%|*9C�GJE4D>�ø#&!:?&!+*920E4�>�\1/��,/?'39T<?A*9C:*w!>�{�¬ ï<¯ P/(>39��!$2�F=1J!;3<GH%A!>�J!>Cl3<#�2Z*�#'39241/($E4�<�v�B1/Cl!>?&.4(+*H^V !$�u%A?'3"6/!>2�C:E0#'Cl3:,J?'3<T<?'*9C:*l#&!ª(+*-24(>1J2Z*©!>2�!>?'?&3<?)!$�u%A?'!;2Z*l#'3<201/(>E0�<���B1/ClD>?&E4(>*lS24*�#'3<201/(>E0�<�\! � *9($%A*J^

O � p KQ K p KQ K � x�� Â��h� �Y� � = ¢ ; x I>£mï �Y� � = ¢'¢ 1 � � � = ��°H^ ñ ±[ �"!$28(+*9#&3ª6/!�2Z*OWY1/($E4�<�«y �¤� � = ¢�� ; = 1 � M 1 ´ < 1�¢ VYXûZ � = M � ¢ VsX[Z � = <3= ¢ Pu24*9#L(>3<�/6JEX®($3<�/!>#¨6/!°WY?&3<�u%A!>?'*l#&3<� E, � ; �<� = ¢µ�ã, � �9� = ¢R�ã, �Y� � ; � ¢ �ã, �¤� �+� ¢ � ²J^ó²[ ��24*ø%A*9G/24*�� °H^ºÁ"±©#&!�C�1/!$#&%'?A*9�ÿ243<#l!>?&?'3<?&!>#lF=1J!ü#'!�3<GJ%A1HIHE0!>?&3<��*-2�!¥�&!>($1J%|*9?:!>2,/?&3<T<?'*9C"*�($3<�\6/E4#h%AE0�B%'3<#�I-*92439?'!>#�,/*9?A*�2Z* ¬ ^Q�3H6/!$C:39#lI9!>?l6/!�24*ø%A*9G/24*ÿ� °H^óÁT±�F=1J!ü*92�*91/Cl!>�u%A*9?©!>2�I-*9243<?�6J!�24* ¬ I-*9C:39#:*3<GH%A!>�J!>?¨1/�!>?'?&3<?¨(+*96/*�I9!>��C"�-#�,]!$F=1J! ? 3/P�!$�B%'3<�/($!>#7%'!>�/6J?'!>Cl3<#¨FB1/!52Z*�#'39241/($E4�<��B1/ClD>?&E4(+*�#&!�*9,/?'3 � E4C:*�C�1JSw?A�9,/E06/3©*�2Z*�#&3<241/($E4�<�v! � *9($%A*l($3<�JWY3<?&C:!°*-1/C:!$�u%|*�!>2I-*9203<?�6/!52Z* ¬ �¤¼8T/^b°H^X§G°"±�^* ?A*9G/*»�'*9�/6/3�($3<�v2Z*�WY1/�/(>E0�<� � M 1 ´ < 1�¢ O I  à ï�> PH1J%AE024E4�>*9Cl3<#L(>3<Cl3�(>39�/6/E4($E43<�J!>#�6/!WY?&3<�u%A!>?'* E , � ; �<� = ¢M� � M 1 ´ < 1 ¢ ORQ I à ï�> � , � �<� = ¢L� � M 1 ´ < 1 ¢ O I à ï?> � , �Y� � ; � ¢M�� M 1 ´ < 1 ¢ O I  Q ï = , �¤� �+� ¢ � � M 1 ´ < 1 ¢ O I  à ï

�"°

Page 76: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3¹°H^óÁ Eq�¸Ã¥ÃAÔ�Ã�Ô��<РÉqÎuÑZÈuÔ5Æ|Ô�Î"È»Ñ æ�ÐZÑãÎHРԦæ��|Ç$äZÔ�ÃAÉhæKÈuÉ@ô�ë"Æ|Ô�Îa©Ýwc�Ýö÷>å Ó�Ç$Ã&Çä0Ç7ç Ï+Î/ÆqÑZâ�ÎN��Ë4ÌRQ�SB͸Ýöõfe 1 Ë�a 1:g c 1 Í_h�jlk�ËmeIa8Ì8Í�o$Ø� [ ?'?&3<?¹ §<^º²9²<²<²<²<²-!&�°²<²6 ñ ^y��� ñ ½<½��-!�®q²9±§>± ¹H^Î6 ñ °9¹����-!�®q²<Á¹-² ¹H^º¹76u¹=§»¹��-!�®|§G°¹ ñ 6J^ ñ §<§_69± ñ !�®|§ ñ

0 10 20 30Ν

−40

−30

−20

−10

0

Lo

g(e

rro

r)

ß E0T<1/?'*Ó°=^0§G° ER�öá;Ç&ëBÔ�Ã��AÇ$äZÔ�õÓ�Ç$Ã&Ç5ôå�Éqä_�AÇ$äZÔ�Ã�ÈuÉqä¸ÉqÃ¥ÃAÔ�Ã7È»Ñ æ¥á°ÑãÎHÏ$ëBÉ|å~Æ|Ô�Î�avÝ�c ÝOÙ>åÓ�Ç$Ã&Çvä0ÇçtÏ+Î/ÆqÑZâ�Îw��Ë4ÌTQ�SuÍ:Ý�õfe 1 Ë�a 1 g c 1 Í�h�j{k�ËmeIa8Ì/Íih�jlk�ËmeIcSBÍ�o$Ø?� Ç$Ã&Ç�ô Ö��%$øÉqÎÇÀÈuÉqä0Ç$ÎHРÉ7Éqä ÉqÃ¥ÃAÔ�ÃA�<Ï9ÉMä0Ç0�ãá;Ä �<Ï+ÑãÎHÇi¡MÆ'Ç$äZƦÏ+ä0Ç;Éhæ�Æ|ÉqÃAÔ-Ø� ñ

Page 77: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3�°=^0§�² E���Ã¥ÃAÔ�Ã�Ô��<РÉqÎuÑZÈuÔwÆ|Ô�Î�È»Ñ æ�ÐZÑãÎHРԦæ �|Ç$äZÔ�ÃAÉhæ5ÈuÉOô ëøÆ|Ô�ÎPa�Ý ÷OëPc�Ý ÷>åÓ�Ç$Ã&Ç)ä0Ç7çtÏ+Î/ÆqÑZÔ�Î�Ë�a 1 g c 1 Ítß I  à ï�> Ø� [ ?'?&3<?¹ ¹H^º¹<¹9½9Á<²J§+!�®q²J§± ½J^º¹H§�²76J§�69!�®q²<½6 °H^¯°H§+±u¹��H§+!�®q²P6§�² §<^ ñ ½T°-Á��9²9!�®|§�½§+± ½J^y� ñ ±<²T°"°-!�®|§G°

[ ��24*l%A*9G/2Z*¹°H^X§�²JP�#&!ª,/1/!$6/!;I9!>?O($3<Cl3"*92¸E0?7*-1/C:!$�u%|*9�/6J3"!>2¸I-*92439?�,/*9?A*©2Z* ¬ P�!>2!$?'?'39?�(+*968*�I<!$�;#&! h *9(>!5C:�9#�,�!>FB1/! ? 3/^f �9�¦����� �m¥����§� m'������ �� �W���=m'�¢��¥Lj¢� ¤���¥¦¡§¥Lj£�m¥���¥¦�����L�� �� �\jus� �W�/�Hm|���*¤ ¡§� n �qpi� m'�W�B�����Lj�r«���B�x���Ljè��� m'� ���ú�W��ªB�¶�L�� ¶�¡§�¦¥�¥��qjHjN¥��[ �©!>#h%|*°#'!$(>(>E0�<�l#'! h *9?'�°1/�8*O(>39C:,8*-?A*9($E4�<��!$�B%'?'!�203<#µ!>?&?'3<?&!>#L3<GJ%'!>�/E06/3<# ($3<�©(+*968*1/�J3v6/!�2439#5U�D$%'3H6J3<#>^~QL?&E4Cl!>?&3w1J%AE024E0�+*9�/6J3:24*w�73<?&C"*w¹",8*9?A*:?'!+*-24E4�>*9?)!$2�(>�924($1/2436/!$2~!$?'?'39?�!>�\2439#KClE4#'Cl3<#@,/1/�u%A39#�,8*9?A*�203<#K6J3<#�U�D$%'3H6J3<#>^ V D>#&,/1/!>#�#'!°1H%AE420E4�>*9?A�52Z*��3<?'C:*�¹�#'�9243�!>� V E0WY!>?&!>�/($EZ*9# ß E0�/E0%A*9#µS©,8*9?'*;!$2~UD$%A3=6/3�6/!°�K39243=(+*9($E4�<� P9!>2�!>?&?'3<?#&!�39GJ%A!$�/6/?A�©(>39��24*�!>(>1/*9(>E0�<�<°=^ ñ ^· *9#�%|*9G/24*9#@°=^0§<§;S×°H^X§»¹©C�1/!>#h%A?'*9�­243<#7!>?&?'3<?&!>#O3<GH%A!>�JE46/3<#7,8*9?'*w243<#�6/3<#5UD$%A3=6/3<#($3<��24*:��3<?'C:*JPJ%A*9C�G/E0D>�#'!ª20!56/E4!$?'3<�243<#¨ClE4Cl3<#�I-*92439?'!>#�*l2Z* ¬ ^�Q�*9?'* V EXWY!>?&!>�H®($EZ*9# ß E4�JE0%|*-#�24* ¬ !>#;!>2@� j C:!$?'3ü6/!:,8*9?h%AE4($E43<�J!>#56/!"(>*968*�!¥�&!"6/!:2Z*\?'!$6�Sy,8*9?A*[ #',�!>(¦%A?'*924!$#�24* ¬ !$#¨!>2�� j Cl!>?'3©6/!°%'D>?'ClE4�J3<#�6/!52Z*�! � ,/*9�/#'E0�<� ^

���

Page 78: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

0 5 10 15Ν

−40

−30

−20

−10

0

Lo

g(e

rro

r)

ß E0T<1/?'*@°H^X§ ñ|Ei��ä8ÉqÃ¥ÃAÔ�ÃµÈ»Ñ æ¥á°ÑãÎJÏ$ëBÉ@Æ|Ô�Îuç Ô�Ã¥á�ɵÇ9Ï+á�ÉqÎHÐ�ÇOÉqä(�AÇ$äZÔ�õÈuɸä0Ç�ôå<Æ|Ô�Î�a:Ý�c�ÝOÙu{��Ë4ÌRQ�SBÍ�Ý Ë�a 1�g c 1 Ítß I  à ï�> o�ul����o$Ø�K18*-6/?'3©°H^X§<§ E���Ã¥ÃAÔ�ÃAÉhæ�Ô��<РÉqÎuÑZÈuÔ¦æ¨Æ|Ô�Î:äZÔ¦æLá �¦Ð Ô9ÈuÔ¦æG§KÑãç ÉqÃAÉqÎ/ÆqÑ0Ç>橨~ÑãÎuÑYÐ�Ç>æ�u![�]_^foLë*�¸ænEÓ�É|ƦÐZÃ&Ç$äZÉhæRÓ�Ç$Ã&Ç;Éqä�É! �Éqá�ÓJäZÔ5ÈuÉ�õfe 1 Ë�a 1 g c 1 Íih�jlk�ËmeIa8Ì8Íih�jlk�ËmeIcSBÍAØ��]æ'Ç$Î/ÈuÔ7è¨Ô�Ã¥á;ÇOíuØ

UD$%A3=6/3 [L?&?'3<? �V E0WY!>?&!>�/($EZ*9# ß E4�/EX%|*9# ¹H^¯°9½P6<²-!¦®q²9½ ¹ ñ�K39243=(+*9($E4�<�v[ #&,]!$($%A?'*92 ±/^Î6 ñ §�6-!¦®|§ ñ ¹ ñ

�K18*-6/?'3Ú°H^X§»¹ ET�¸Ã¥ÃAÔ�ÃAÉhæ°Ô��<РÉqÎuÑZÈuÔ¦æOÆ|Ô�ÎväZÔ¦æ�á �¦Ð Ô9ÈuÔ¦æOÈuÉ�§KÑãç�ÉqÃAÉqÎ/ÆqÑ0Ç>æ�¨]ÑãÎuÑYÐ�Ç>æ u![�]_^fo�ë�¸æ�Ó¸É|ƦÐZÃ&Ç$äZÉhæRÓ�Ç$Ã&Ç;Éqä É! �Éqá�ÓJäZÔ5ÈuÉ)Ë�a 1 g c 1 Ítß ½ I  à ï�> ¾nªI«�¬d­ a(® ¯+ô�¯9°%c ­ �UD$%A3=6/3 [L?&?'3<? �V E0WY!>?&!>�/($EZ*9# ß E4�/EX%|*9# �H^¯°"°9Á ñ !¦®q²-± §+±�K39243=(+*9($E4�<�v[ #&,]!$($%A?'*92 §<^y�9²<²u¹À!¦®|§ ° §+±

�76

Page 79: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-6/?'3©°H^X§�½ E���Ã¥ÃAÔ�ÃAÉhæ�Ô��<РÉqÎuÑZÈuÔ¦æ¨Æ|Ô�Î:äZÔ¦æLá �¦Ð Ô9ÈuÔ¦æG§KÑãç ÉqÃAÉqÎ/ÆqÑ0Ç>橨~ÑãÎuÑYÐ�Ç>æ�u![�]_^foLë*�¸ænEÓ�É|ƦÐZÃ&Ç$äZÉhæRÓ�Ç$Ã&Ç;Éqä�É! �Éqá�ÓJäZÔ5ÈuÉ�õfe 1 Ë�a 1 g c 1 Íih�jlk�ËmeIa8Ì8Íih�jlk�ËmeIcSBÍ ªU�D¦%A3=6/3 [ ?&?'3<? �V EXWY!>?&!>�/($EZ*9# ß E0�/E0%A*9# §<^ ñ<ñ §�6��i�-!¦®t± §�²<²�K3<203H(>*9(>E0�<�\[ #',�!>(¦%A?A*-2 6J^ ñ §<§�69± ñ !¦®|§ ñ ¹ ñ

�K18*-6/?'3Ú°H^X§+± ET�¸Ã¥ÃAÔ�ÃAÉhæ°Ô��<РÉqÎuÑZÈuÔ¦æOÆ|Ô�ÎväZÔ¦æ�á �¦Ð Ô9ÈuÔ¦æOÈuÉ�§KÑãç�ÉqÃAÉqÎ/ÆqÑ0Ç>æ�¨]ÑãÎuÑYÐ�Ç>æ u![�]_^fo�ë�¸æ�Ó¸É|ƦÐZÃ&Ç$äZÉhæRÓ�Ç$Ã&Ç;Éqä É! �Éqá�ÓJäZÔ5ÈuÉ)Ë�a 1:g c 1 Ítß ½ I  à ï�> ¾ ªU�D¦%A3=6/3 [ ?&?'3<? �V EXWY!>?&!>�/($EZ*9# ß E0�/E0%A*9# §<^º½9Á<ÁP69± ñ !¦®q²"° §�²<²�K3<203H(>*9(>E0�<�\[ #',�!>(¦%A?A*-2 ½J^�� ñ ±u²T°"°À!¦®|§ ° §+±

[ � 2Z*-#l%A*9G/2Z*-#|°H^X§�½yS °H^X§+±{#&! h *9($!�1/�8*{($3<Cl,8*9?A*-(>E4�9�ÿ6/!�6/!>27!>?'?&3<?"39GJ%A!$�/E46/3($3<�ö2439#"ClD$%A3=6/3<#l6/! V E0WY!>?&!>�/($EZ*9# ß E4�JE0%|*-#ü�YC:D¦%A3=6/3�6/!�Ý/N°¾�±�1J%AE024E0�+*9�/6J3���3<?h®C:*v¹"S­[ #',�!>(¦%A?'*924!$#°1J%AE024E4�>*9�/6/3"2Z*"!$(>18*9($E4�<�ü°H^ ñ P],8*9?A*w24*9#)WY1/�J(>E439�/!># ; = 1 � M 1 ´< 1 ¢ VsX[Z � = M � ¢ VsX[Z � = <3= ¢ S � M 1 ´ < 1 ¢ O ½ I  à ï?> ¾ ^8[ �!>#h%A!5(+*9#&3:24* ¬ %A3<C:*�!>2 I-*9203<?($3<��!$2�FB1/!5#'!53<GJ%'E4�/!$�/!5!>2�!>?&?'3<?�C:�9#�,�!>FB1/! ? 3",8*-?A*�(+*96/*lClD$%'3H6/3J^_*)F±�) -/.�²�³0.Ma$b;c 1325´µ²�¶?·02�¶_8

· *l!$(>18*-(>E4�9�6J!b¸K1J?'T<!$#�($3<�2Z*9#¨(>39�/6/E4($E43<�J!>#�6/!°WY?'39�B%'!>?'*lFB1/!�1/#A*-C:3<#$P/%AE0!>�/!1/�/*l#&3<241/($E4�<�\FB1/!;,8*9?'!$(>!�1/�6/E4!$�B%'!;6J!;#&E4!>?&?A*J^|g)#'.¸FB1/!;!>�!>2¸(>!$�B%'?'3ü��!>� � �¦¹ ±h *+Sy1/�8*�,]!$�/6/E4!$�u%A!wFB1/!:(>?'!$(>!w($3<�{!$2M%AE0!>Cl,]3��Y¼/T/^ °H^0§&�"±¦^ · 3<#ªClD$%'3H6/39#;!$#',�!>(¦®%'?A*920!>#LWY1/�/(>E03<�8*9�"C:!¥�&3<?M($18*9�/6J3�#'!)%'E4!>�J!>�"WY1/�J(>E439�/!>#�GJE4!>�"(>3<Cl,]39?&%|*-68*9#>^=��3�3<GH®#h%|*9�u%A!-P�*91/�\(>1/*9�/6/3©#'!°,J?'!>#&!>�u%|*l!>#h%A!°T<?'*96/E4!$�u%A!°(>?&!>($E4!>�u%'!;*-24?'!$6/!>6J3<?�6/!$2�(>!>�u%'?'3/P!$2�C:D¦%A3=6/3»ºR*91�WY1/�/(>E03<�8*üG/E0!>�¶*9T<?&!>Tu*-�/6/3­1/�oC:*�S93<?�� j Cl!>?'3ø6/!w%'!>?&C:E0�/3<#�!$�24*"! � ,8*9�J#'E4�9�­6/! ¼ �Y� �Z½ ¢ ^ · *"¼8T<1J?A*­� °H^0§%¾T±�C�1J!>#&%'?A*v2Z*l#'3<201/(>E0�<�ü�B1/ClD>?&E4(+*:6/!�2Z*�9Á

Page 80: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1x

−0.7

−0.2

0.3u(

x,y)

t=0.24

t = 0.2

t = 0.32

ß E0T<1/?'*q°H^0§&�q¿ �Mä�Ç9Ï+á�ÉqÎHÐ�Ç$Ã\Éqä5ÐZÑZÉqá�Ó¸Ô\ä0ÇwÓ¸ÉqÎ/È»ÑZÉqÎHРÉ\ÆqÃAÉ|Æ|É$ØG[=Ô�äYÏ9ÆqÑZâ�ζÎHÏ+á �qÃ¥ÑZÆ'Ç�ÉqÎÈ»Ñ æ�ÐZÑãÎHРԦæ7ÐZÑZÉqá�Ó�Ô¦æ�Æ|Ô�Î�Àwé}×»ØÚÙ>åHô�é©C9×»Ø!$(>18*9($E4�<�v6J!Á¸K1/?&T<!>?&#�(>3<� ¬ � ç ]=PJ,8*9?'*ª1/�3½ �¹ �Ã}�]�Ä/^ · 3�F=1J!°2Z*9#M#'E4Cª1/2Z*9($E43<�J!>#�B1/ClD>?&E4(+*-#�C:3<#h%A?'*9?'3<�o!$#lFB1/!-P@*øCl!>6JE468*­FB1/!*91/Cl!>�u%|*-C:3<#�!$2�I-*9243<?ª6/! ¬ P 2Z*#&3<241/($E4�<���=1JC:D$?'E4(>*"#'!�*9,/?'3 � E4C:*w*"24*"#'39241/($E4�<�ü! � *9($%A*J^ [ #°,�3<#'E0G/24!;6/E4#&C:E0�=1JE4?¨!>2,/?&3<G/20!>C:*�(>3<��!>2�T<?'*96/E4!$�u%A!5!>��!>2¸3<?'E0T<!>�~P/E4�u%A!$T<?A*-�/6/3l2Z*�!>($18*9(>E0�<�!>� t ¹ �>�bw P8!$#&%'3!$#;,�3<#'E0G/24!�,�3<?°24*\#&E4Cl!$%'?'.Z*w6J!>2L,/?'39G/24!$C"*JP~,]!$?'3!>#&%'3S<*��/3243 h E0(>E0C:3<#)!>��!$#&%A*%'!>#'E0#>^· 3:FB1/!�%|*9CªG/E4D$��($3<�/#&%A*-%|*-C:3<#¨($3<��24*9#¨#'E4Cª1/2Z*9($E43<�J!>#>PJ!>#�F=1J!;!$2�T9?A*96/E0!>�u%A!�#'!h *-(>!5C:!$�/3<#�,/?&3<�B1/�/($EZ*96/3l#'E�24*ªI=E4#'($3<#'E068*96�*91/Cl!>�u%|*H^

¾<²

Page 81: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

−1 −0.5 0 0.5 1x

−1.005

−0.505

−0.005

0.495

0.995

ß E0T<1/?'*Ú°=^0§&¾q¿ [=É�á;Ï9Éhæ�ÐZÃ&Ç$Î�ä0Ç�æAÔ�äYÏ9ÆqÑZâ�Î�É�=ÇÀƦÐ�ÇÅuãäãÑãÎ/É'Ç©æ|â�äãÑZÈ9Çro�ë\ä0Ç�æAÔ�äYÏ9ÆqÑZâ�Î�ÎHÏ+á �qÃ¥Ñ{EÆ'Ç�uãä�ÆãÎ/É'Ç©Ó Ï+ÎJРÉ'ÇÀÈ9Çro:Æ|Ô�ÎwÀ/é¨×»ØÚ×>ò`Ç e¸Ø:�MäLÑãêÇ9Ï+á�ÉqÎJÐ�Ç$Î/ÈuÔüÉqä��AÇ$äZÔ�éÈuɪô ä0ÇvæAÔ�äYÏ9ÆqÑZâ�ÎÎqp+á�ÉqÃ¥ÑZÆ'Ç5æAÉKÇAÓJÃAÔn<Ñãá;Ç�Ç7ä0Ç)æAÔ�äYÏ9ÆqÑZâ�Î\É�BÇÀƦÐ�Ç<Ø�[=Ô�äYÏ9ÆqÑZâ�Î:ÎHÏ+á �qÃ¥ÑZÆ'ÇÁÈ~Ë4ÌRQ¥ÕqÍMÆ|Ô�Ωô Ý�É@$BØ

¾J§

Page 82: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

Ê�ËfÌ Í�ÎÐÏAÑTÒ%ËfÌ*ÓIÑ�Ô[ ��!$#&%'!�%'?A*9G8*��&3l#&!;! � ,/1/#'E0!>?&3<��24*9#�T<!>�J!>?A*-24E46/*96/!>#�6J!�203<#�U�D¦%A3=6/3<#¨6/! V E0WY!$?h®!$�/(>E4*9# ß E4�/EX%|*9#�� U V�ß ±OSü243<#�[ #',�!>(¦%A?A*-24!>#:� U�[I±¦P�S­#'!lE4201/#&%'?'�v#&1�1/#&3�!$��I-*9?'E03<#!¥�&!>Cl,/243<#d¿¸2Z*\!>(>1/*9(>E0�<��6/!:(+*9203<?>P�24*!$(>18*-(>E4�9�{6/!"39�/68*JP¸2Z*\6/!N¸K1/?'T9!>?'#$P�Sy24*6/!Q�3<E4#&#'3<� P�%A3=68*9#7D>#h%|*9#O6J!�*9Cl,/24E03©1/#'3v!$� ß .4#&E4(>*J^ · 3<#5U V�ß #'3<�ü203<#)C:�9#�(>3<�J3H($EX®6/39#LS�1J%AE024E4�>*96/3<#�6/!>#&6/! h *9(>!�C�1/( h 39#M*�Õ/3<#>^ · 3<#MU�[{!$C:,�!>�>*9?'39�:*5,�3<,/1/24*9?'E0�+*9?&#'!6/!$#'6/! h *-(>!�½<²�*iÕ/39#�S h *9�\!$C:,�!>�>*96/3l*�1/#A*-?'#'!5!$��C:�9#�?'*9C"*-#�6/!°2Z* ß .4#'E0(+*ªC:�9#*9202Z�w6/!:#'1{1J#'3�E0�/E4($EZ*92�!>��24* V E4�8�9ClE4(>*v6/!"203<# ß 241/E06/3<#$^�[ �{!>#h%|��?&!$I=E4#&E4�<�~P�($3<�H®#h%|*-%A*9C:39#�F=1J!ª2439#)U�[ö68*9�?'!>#&1/20%A*96/3<#�C:�9#�,/?'!$(>E0#'3<#�(>18*-�/6/3"#&!;1J#A*:1J��� j Cl!>?'3¬ 6/!K%AD$?'ClE4�/39#R!$��2Z*7! � ,8*9�/#&E4�<��6J!�2Z*�#'3<201/(>E0�<��*9,J?'3 � E4C:*968*~¼ ° �Y� �F½ ¢ (>3<Cl,8*9?'*968*($3<�w!>2]C:E0#'Cl3;� j Cl!>?&3�6/!7,8*9?&%'E4($E43<�/!$#M6J!>2~6J3<C:E0�/E43�!>#&,8*9($EZ*92�1/#A*-6/3�!>�v203<#KU V�ß ^g71/�/FB1/!\243<#�U�[�#&E�#'3<�oC:*9#�24*9G]39?'E439#'3<#;6/!vE4Cl,/24!$C:!$�u%|*9?$PR!$2K%A?'*9G8*»�&3­! � %'?A*­#'!($3<C:,�!>�J#A*�,/1/!$#724*�(>3<�uI<!$?'T<!$�/(>E4*:FB1/!�,/?'!$#'!$�B%A*9��!$#¨! � ,]39�/!>�/($EZ*92 ^[ #&%'!M%A?'*9G8*»�&37#'E0?&I=E4��(>3<Cl3�1/�ª!$�B%'?'!$�8*9ClE4!>�u%'3),8*9?'*7!$2=!$#&%'1/6/E4376/!K24*�#&3<241J(>E439�/!>#�B1/ClD>?&E4(+*-#�6/!5!>(>1/*9(>E03<�/!>#¨6/EXWY!>?&!>�/($EZ*920!>#�FB1/!�*9,8*9?&!>($!>�!>� ß .4#&E4(+*HPHSv6/!¥�'*l!$#',8*-(>E43,8*-?A*{!>#h%A1/6JE43<#\C:�9#w6/!¦%|*9202Z*96/39#wFB1/!ø?'!$I=E0#'!>� *9#&,]!$($%'3<#\6/!ø2Z*{E4�Hù81/!>�J(>EZ*{6J!­243<#!$?'?'39?'!>#)6/!0Öh*920EZ*9#&E4�/T ×|P/!$#&%|*-G/E420E468*96S�($3<�uI<!$?'T<!$�/(>E4*JP FB1/!�!>�­!$#&%'!�%A?A*-G8*»�&3/P�,]39?O2Z*! � %'!>�/#&E4�<�6/!$2�C:E0#'Cl3/PHS<*l�/3©#&!�?&!+*920E4�$��(>3<�C"�-#�6/!$%'!>�/E0C:E0!>�u%A3/^[ 2O!>#h%A1/6JE43{6/!­!$#&%'3<#vC:D¦%A3=6/3<#:,/?'3�I<!$!ø1/�8* h !$?'?A*-C:E0!>�u%|*�FB1/!­#&!>T<1J?A*9Cl!>�u%A!!$#&%|*-?'!>Cl3<#l1/#A*-�/6/3{!$�ö243<#l,/?&� � E0C:3<#l6/3<#:*iÕ/39#"!>��,/?'3<GJ24!>C:*9#©6/!�*9#&%'?'39¼8#&E4(>*yS?&!>2Z*À%AE0I=E068*96y�=1JC:D$?'E4(>*JP�FB1/!"!>#�1/�8*ü6J!w2Z*9#ª*9?'!>*9#�6/!"?'!>($E4!$�B%'!\*9,/20E4(+*-(>E4�9��6/!"243<#U�[�^

Page 83: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

Ø � �ÚÙ������ � Ø� ����� � � � � Û � ������ �ö� Ü � ������ �ö�

Ý )>`Þ) ßX6�¶áà ³Þ8â132Xãä2�.©å:6�¶_2�8· *5�/3<?'C:*°6/!)1J�lI9!>($%'3<?©æ�!>#@1/�"��ç/Cl!>?&3;?&!+*92�,�3<#'EX%AEXI<3/Pu!$28(>18*92�,/?&3<,�3<?'($E43<�8*51/�8*C�1J!>#&%'?A*°6/!$2Iè è�½né < é�êë � è èH6J!>2/I<!$($%'3<? S�#'!¨6/!$�/39%|*5,�3<?�ì � ì»^ · *��J3<?'C:*O6/!¨1/�©I<!>(¦%A3<?#'*-%AE0#&W�*9($!°243<#�#&E4T<1/E0!>�u%A!$#�* � E43<C:*9#�YE ±Þì`æ�ì Ð�¹ #'Eíæ ��Âî Sïì`æ�ì �/¹ #&Eíæ �/î ^�YE4E ±Áì � æHì ��ð � ð ì`æ�ì�ì7,8*9?A*�1/���ç/Cl!>?'3©!$#'(+*-2Z*9?�?&!+*92�3©(>39C:,/20!q�&3 � ^�YE4E4E¼±Áì`æ ´Bñ ì~ò¢ì`æ�ì ´ ì ñ ì^ÝHE¨!$2¨I9!>(¦%A3<?væ %AE0!>�/!\(>39C:,�3<�/!$�B%'!># µ �íó � �(ô � ÅöÅûÅ � ��õ ¸BP !>�u%A3<�J(>!>#"2Z*9#©%A?&!>#l�/3<?'C:*9#C:�9#�1/#'*968*9#¨(>39C�1/�/Cl!>�u%A!5#&!�6J!$¼8�/!$��6/!524*©#&E4T<1JE4!>�u%'!OWY3<?'C:*ö¿· *\§$®¥�/39?'C:*©6J!0æ P�!>#�2Z*©#'1JC"*©6/!ª203<#¨C:�=6/1/203<#�6/!�2Z*9#¨($3<C:,�3<�/!$�u%A!>#76/!>2�I9!>($%'3<?>P!$#�6J!>(>E0?>P

ì`æHì ó �µð �íó ð ´ ð �(ô ð ´ ÅöÅöÅ ´ ð ��õ ð9� õ² á�÷ ó ð � á ð ��øª^X§G±¾<½

Page 84: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

· *��/3<?&C"*�E0�J¼8�/EX%|*�6/!Áæ P8!$#�!$2�C:�=6/1/203�C"� � E0C:3ª6/!52Z*9#�($3<C:,�3<�/!$�u%A!>#¨6/!Áæì`æ�ì q � < é � Ì ð � á ð ��øª^º¹"±· *y¹»®¥�J3<?'C:*ü6/!3æ P !>#©24*ø?'*9E4�"(>18*96J?A*968*ø6/!\2Z*­#&1/C"*­6/!\203<#�(>18*-6/?A*96J3<#©6/!\243<#Cl�H6J1/243<#K6/!52Z*9#�($3<C:,�3<�/!$�u%A!>#¨6/!$2 I<!$($%A39?>^

ì`æ�ì ô � � ð �íó ð ô ´ ð �(ô ð ô ´ ÅöÅöÅ ´ ð ��õ ð ô ¢ óÐùúô �üû õ² ál÷ ó ð � á ð ôZý óÐùúô ��øª^ó½T±Ý ),+*) ßX6�¶áà ³Þ8â132þ²\cP³ ÿ ³Áå�¶�a��· *\�/3<?'C:*v6/!©1/�8*\C"*À%A?'E0� � !$#;1J����ç/Cl!>?'3?'!>*92 ,�3<#'EX%AE0I93/P !$2@($18*92 �/39#�6/*�1/�8*C�1J!>#&%'?A*�6/!$2:è èy½né < é�êë � è è]6J!�24*�C"*À%A?'E0�)Sv#'*-%AE0#&W�*9($!>�243<#�#&E4T<1/E0!>�u%A!$#¨* � E43<C:*9#�YE ±Þì � ì Ð ¹ #'E � ��Âî S�ì � ì �¹ #&E � ��î ^�YE4E ±Áì � � ì �Vð � ð ì � ì7,/*9?A*©*920T�ç/�\��ç/C:!$?'3©?'!>*92�3©(>3<Cl,/20!q�&3 � ^�YE4E4E¼±Áì � ´ � ì~ò¢ì � ì ´ ì � ì»^�YE0I�±0ì � � ì�ò¢ì � ì�ì � ì»^Ý )F±�) ßX6�¶áà ³Þ8�� 6bà �P³Áå¸a��r2�8� !$($%A39?'!>#7S�C:*-%A?&E4($!>#>P<�&1/�u%A39#O#A*-%'E4#hW�*9(>!$�­1/�8*w(>39�/6/E4($E4�<��!>FB1/EXI9*-24!>�u%'!�*"2Z*:(>3<�/6JEX®($E4�<� �YE0I�±:6/!ø2Z*{�J3<?'C:*�6/!øC:*-%A?&E4(>!$#>^��K3<C:3�($3<�/#&!>(>1J!>�/($EZ*JP¨2Z*��/3<?'C:*{6/!ø1/�8*C:*-%A?&E4�)Sw2Z*�6/!51/�\I<!$($%A39?�#&!;6JE4(>!$�\#'!>?�(>39C:,8*À%AE4GJ24!>#K#'E

ì � æ�ì~ò¢ì � ì�ì`æ�ì»� æ ��¹ Å ��øª^ ±5±

¾9±

Page 85: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

Ý ) �3) ßX6�¶áà ³Þ8â132þ²\cP³ ÿ ³Áå�¶�a�� ��²��»6�¶á1va$cP³Þ1�³ÝH!>* � 1/�8*�C:*-%A?&E4��6/!l� ë��Pë ±@Svæø1/�\C:E0!>C�G/?&3ª6/!$2 (>39�À�&1/�u%A3���6/!O203<#KI<!$($%'3<?'!$#6/!��J3<?'C:*��<P E ^]!-^0PTæ � �}#&E�ì`æ�ì � �<^ [L�øT<!$�/!>?'*92 P]2Z*"�J3<?'C:*w6/!>2RI<!>(¦%A3<?�ì � æHìI-*9?&.Z*�*-#'.K($3<Cl3Åæ I-*9?'.4*JP�æ � ��^�ø��J3<?A*ü#&1/,]39�/Tu*9Cl3<#ªFB1/!�æ���!>#�ClE4!>CªG/?'3�6/!>2($3<�À�&1/�u%A3���P¸!$2M($18*92��8*9(>!"F=1J!ïì � æ��iìl*-24(+*-�/(>!l#'1�C:� � E4Cl3vI9*-243<?$^¸[L�u%'3<�/(>!$#�2Z*�/39?'C:*�6/!52Z*�C"*À%A?'E0�)#&!;6/!¦¼8�/!�,]39?

ì � ì � ì � æ�ì � < é ������ ÷ ó ì � æHì Å ��øª^¯°"±[ #&%A*��/3<?'C:*�6/!­24*�C:*-%A?&E4��#&!­6/E4($!­#'!$?v#'1/G�3<?'6JE4�8*96/*{6/!­2Z*��J3<?'C:*{6/!>27I<!>(¦%A3<?So*-1J%A3<C:*-%'E4(+*-C:!$�B%'!"#'*-%AE0#&W�*9($!24*ü(>3<�J6/E4($E4�<�o6/!\(>39C:,8*À%AE4GJE424E068*96 P�,J1/!>#l#'E�æ ó !>#($18*924FB1/E0!>?�3-%A?'3�C:E0!>C�G/?&3ü6/!���P~ì � æ ó ì\ò ì � æ���ì � ì � ì � ì � ì�ì`æ�ì©S<*øFB1/!ì`æ�ì � �<^ V !M243O*-�B%'!>?&E43<?�#'!�#&E4T<1J!MFB1/!-P9,8*9?'*7%A3=68*��/3<?'C:*76J!KC:*-%A?&E4�L#'1/G�3<?&6/E4�8*-68*JP

ì! �ì � < é ������ ÷ ó ì! zæ�ì � < é ����"� ÷ ó ì`æ�ì � � ��øª^ ñ ±6/39�/6/!# �!>#�24*ªC:*-%A?&E4�¨1/�/EX%|*9?&EZ*J^ · *9#K�J3<?'C:*9#)�9�!$;!&% ($3<��ì`æ�ì � �<PH#'!O6/!¦¼8�/!$��*($3<�u%AE4�B18*-(>E4�9� ÿ· *�§$®¥�J3<?'C:*O6/!�2Z*°C"*À%A?'E0� � !>#@2Z*°#'1/C:*°C"* � E0C"*O6J!�203<#LC:�=6/1/203<#µ6/!¨203<#L!>24!$C:!$�H®%'3<#�6/!5(>3<201/Cl�8*�6/!52Z*�C:*-%A?&E4� �· *��/3<?&C"*�E4�H¼8�/E0%A*�6/! � !$#�24*�#'1/C:*�C"� � E0C"*�6/!5203<#�C:�=6/1/203<#�6/!�2439#�!>24!$C:!$�u%A3<#6/!5?&!>T<20�<�v6/!52Z*�C:*-%A?&E4�9^· *l¹»®¥�J3<?'C:*�6/!52Z*�C:*-%A?&E4� � !>#¨2Z*�?'*9.4�°($18*96/?'*968*�6/!>2¸?'*96/E43�!$#',�!>(¦%A?A*-2�6J! ��'�� P6/39�/6/!ª!$2R?'*96/E03:!$#',�!>($%'?A*92)( � ��'�� ¢ !>#O!$2�Cl�=6/1/243lC:*�S93<?�6/!ª203<#�I9*-243<?&!>#�,/?'3<,JE43<#6/!524*�C"*-%'?'E0� � ' � ^

¾T°

Page 86: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

�K18*-�/6/3 � !$#�?&!+*92 Sw#'E4ClD$%'?'E0(+*�#'!5($1/C:,J24!°FB1/! �+* � � P8!>�u%A3<�J(>!>#ì � ì ô � t ( � � ô ¢ w óÐùúô � t ( ô � � ¢ w ó-, $ � ( � � ¢L� < é �³

ð í á ð Å ��øª^y�"±Ý )A_*) ÿ .�å:6v1�6 1320/ ²\c3·022143ü²3å�å�³[ �©!>#h%|*5#'!>($(>E0�<�©#'!¨6/!>#&(>?&E4G�!�!>2/ClD$%'3H6/3°6J!7¾¨1/�/T9!¦®q¿°1J%&%|*O6/!�39?'6/!$�:±l��¾7¿°±5±�^65H!>*1/�,/?&3<G/24!$C"*�6J!5I-*9203<?�E0�/E4($EZ*92�6/!52Z*�#&E4T<1/E0!>�u%A!°WY3<?&C"*

= è �87 � ½�� = ¢ � = � ½9� ¢ � = � ��øª^þT±[ �u%A3<�J(>!>#K!>2�C:D¦%A3=6/3;6J!O¾¨1/�/T<!�®q¿°1J%'%A*;,8*-?A*;!$#&%'!O,/?'39G/24!$C"*5!>#M68*96/3;,]3<?@2Z*�#'E0T<1H®E0!>�u%A!°!$(>18*9($E4�<� ¿

= õ;:Tó � = õ�´<} � = ó ´ $ = ô�´ $ =?> ´ ="@ ¢ ��øª^óÁT±

6/39�/6/! = ó � 7 � ½ õ � = õ ¢= ô � 7BA ½ õ�´ < $ � = õ¨´ < $ = ó-C=?> � 7BA ½ õ�´ < $ � = õ¨´ < $ = ô!C="@ � 7 � ½ õ�´ < � = õ�´ <D=?> ¢ø7#'. PH!>2�I9*-243<? = õ;:Tó !$#�6/!$%'!>?&C:E0�8*96/3�,�3<?M!>2~I-*9203<? = õ C:�9#�!$2~,/?&3H6/1J($%A3 < �s½né < é�êë�E6/!$2RE0�u%A!>?hI-*9243<?E±�,]3<?�1/�8*l,]!$�/6/E0!>�u%A!�!$#&%'E4C:*968*J^ V E0(F�8*",�!>�/6JE4!>�u%'!ª!>#7!>2�,/?'39C:!$6/E436/!524*9#�#'E0T<1/E4!$�u%A!>#�,�!>�J6/E4!$�B%'!>#d¿= ó P8!$#�24*�,]!$�/6/E0!>�u%A!�!$I-*92018*968*�!$��!$2�,/1/�u%A3lE4�/E0(>E4*92]6J!>2�E4�u%'!>?&I-*9203/^¾ ñ

Page 87: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

= ô P/!$#�2Z*�,�!>�/6JE4!>�u%'!°!>�\!>2 ,J1/�u%A3lC:!$6/E43�6J!>2 E0�B%'!>?hI9*-243�1/#'*9�/6/3©!$2~I-*9243<?K6/! = ó FB1/!6/!¦%A!$?'ClE4�8*�!$2 I9*-243<?�6J! = !>�!>2�,/1/�u%'3v½ õ�´ < , $©1J#A*9�/6J3:!$2�C:D¦%A3=6/3�6/!�[ 1/24!$?>^=?> P8!$#)6/!��B1/!$I93"2Z*©,�!>�/6/E0!>�u%A!�!>��!>2¸,/1/�u%'3:Cl!>6/E03©6/!>2¸E4�u%'!>?&I-*9203/P/,�!>?&3w*"�/39?A*�1/#h®*9�J6/3©!>2 I-*92439?�6/! = ô !$2�(>18*92�6J!$%A!$?'ClE4�8*�!$2 I-*9243<?K6/! = ^="@ Pu!$#@24*°,]!$�/6/E4!$�u%A!¨!>�l!>2/I-*9203<?�¼8�8*-286/!>28E0�B%'!>?hI9*-243<?�S�1/#A*9Cl3<#L!>2/I-*9203<?µ6/! =?> ,8*9?A*6/!¦%A!$?'ClE4�8*9?K!>2 I-*92439?�6/! = ^�K18*-�/6/3ª24*9# (>1/*-%A?&3;,�!>�/6JE4!>�u%'!>#K#&3<�l,/?'3<Cl!>6JEZ*968*-#>Pu#'!�24!$#L68*�C:�9#L,�!>#'3;*52Z*9#L,�!>�H®6/E0!>�u%A!$#¨6/!>2�,/1J�B%'3:Cl!>6JE43

G � = ó ´ $ = ôR´ $ =?> ´ ="@} H �søª^X§�²T±I 1/!°!$2 C:D¦%A3=6/3©¾�¿5±©#&!+*�6/!°3<?&6/!>�(>18*À%A?'3JP/#'E0T<�/E0¼/(+*�FB1/!°!>2�!$?'?&3<?�,]39?�,8*9#&3l!$#&%A�!$��!$2�3<?'6/!$��6J! <4J P8ClE4!$�u%A?A*-#�F=1J!�!$2�!>?&?'3<?�%A3-%|*92�*-(>1/C�1J2Z*96/3�!$#¨6/!53<?'6/!$� < @ ^Ý )-K�) Lï³ -NMO�P³;c�8�a¦bªc P a98:.©¶?2�å�³ 132RQ�6b²�¶Rad2�¶[ ��C�1J(F�8*9#µ*-,/24E0(+*9($E43<�/!$#¸,/?A�9(¦%AE0(+*9#$P-�/37,]3=6/!$C:3<#�E4Cl,/20!>Cl!>�u%|*9?¸ClD$%'3H6/39#R�B1/ClD>?&EX®($3<#5G8*9#A*-6/3<#5#'3<GJ?'!©#'!$?'E4!$#56/!�S�391/?'E0!>?>^Tø)20T<1/�8*9#°6J!l24*9#O6/EX¼8(>1J20%|*-6/!>#°#&3<�í¿�2439#°(>3-®!¦¼8(>E0!>�u%A!$#�6/!TS83<1/?&E4!$?w,8*9?'*o1J�8*{WY1/�/(>E0�<� *9?'GJE0%A?'*9?'E4*��/3o#'!ø($3<�/3=(>!>� !>�}WY3<?&C"*($!>?'?'*968*;Sl6/!$G]!$�v,]39?K243�%|*9�u%'3ª#'!$?�*9,/?&3VUJE0C"*-6/3<#@6/!O*920T<1/�8*�C"*-�/!>?'*JPBF=1J!O#'!>*�!$¼J®($E4!>�u%'!ª,8*9?'*l?&!>($1/,]!$?A*9?7!>�ü!$2R!$#',8*-(>E43lWY.4#&E4(>3l2Z*©E4�HWY3<?'C:*9(>E0�<�vFB1/!ª!$#)(>*924($1/2Z*96/*l!$�!$2~!>#&,8*9($E43ª%'?A*9#hWY3<?'C:*96/34WB,�!>?&3�2Z*9#M#'E4Cl,/20!>#@�/3<24E0�/!+*-?'E46/*96/!>#@#'39�v2Z*9#MF=1J!O(>3<�J6/1/(>!$�*�2Z*-#�(>3<Cl,/24E0(+*9E03<�/!$#K!�U=%'?'!>C:*9#$^· *lWY39?'C:*l6/!;#'1/,�!>?'*9?�!>#&%A*9#O6/EX¼8(>1J20%|*-6/!>#�!>#7!>2�1/#'3:6/!ª24*l%A?'*9�/#hWY3<?'C:*968*©6/E4#&(>?&!$%|*6/!XS83<1/?&E4!$?KSv2Z*-#�?'!>24*9(>E03<�8*96/*9#�#'!$?'E0!>#¨6/E4#&(>?&!$%|*-#�6J!YS83<1J?'E4!$?>^

¾��

Page 88: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

QR*9?A*�(>1/*924FB1/E0!>?�!>�u%'!>?'3OZ\[ ¹ P�*92~(>3<�À�&1J�B%'3:6J!�,J1/�u%A3<#� ¡�� $^]`_Z _ �¹ � HaHaH �FZcb �9� �søª^X§<§ed

#&!720!>#@242Z*-C"*O�/3=6/3<#@3�,/1/�u%A3<#M6/!�2Z*5?'!>6~^ · 3<#@(>3=!$¼/(>E4!$�u%A!>#M6/E4#&(>?&!$%A39#M6/!�S83<1/?'E0!>? 6/!1/�/*�WY1/�/($E4�<�lI-*92418*-68*�!$�w2439#M($3<C:,J24!q�&39#G¼v6J!>�u%A?&3�6/!gf ¹ �!$e]�h�(>3<�:?'!>#&,]!$($%'3©*;!>#h%A3<#,/1J�B%'3<#¨#'3<�

ê¼`i � �Z jlk óm ¡ ÷4� ¼ �¤� ¡>¢on kqp isrot buZ , $ ò = òvZ , $wb � H �søª^X§»¹?dV !$G/E46J3:*©2Z*�?&!>24*9(>E0�<�v6/!53<?&%'3<T<3<�/*924E068*96

�Z jxk óm ¡ ÷4� n pzy rot �|{ � } ³ G � Z < � < �¹ �!~©�9�!~Y$=� H�HaHaH¹ E ½9� E }T�dé�} E } ��� ¢%'!>�/!$C:3<#¨24*ªWY�<?'C�1J2Z*�6/!5E4�uI9!>?'#&E4�<�

¼ �¤� ¡>¢ �j ùúô k ómi,÷ k�j ùúô ê¼`i n p i-rot _ �¹ � H�H �FZcbö� �søª^X§�½6d

�K3<�J#'!>($1/!>�u%'!>Cl!>�u%A!-P~!>2�,�3<20E4�/3<ClE43� j ¼ �¤� ¢ � j ùúô k ómi`÷ k�j ùúô ê¼`i n p isr �søª^X§+±qd!$#ª2Z*E4�u%A!$?',�3<2Z*-(>E4�9�y%'?'E4T93<�/3<ClD$%'?'E4(>*v6/!"T<?'*96/3�Z , $�6/!�¼¶!$�{2439#;�/3=6/3<#w��ø;^0§»¹"d¦PE ^ó!-^0P � j ¼ �¤� ¡>¢L� ¼ �Y� ¡$¢ ��_ �¹ � HaHaH �FZ�b �<^�[ #h%A!�,]3<20E4�/39C:E03ª!>#¨#&3<243©(>39�/3H($E46/3©($3<C:324*�#'!>?&E4!56/E0#'(>?&!$%A*©6J!YS83<1J?'E4!$?�6/!�¼¸^

¾�¾

Page 89: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

· 3<#¸I-*9243<?&!>#¸6/!0ê¼ è i }M6J!>,�!>�/6/!$��#'3<203¨6/!K203<#�ZeI-*9203<?'!$#�6J!©¼�!$�ª203<#¸�/3H6J3<#��søª^X§»¹?d�^ · *%'?A*9#hWY3<?'C:*968*¨6/E4#&(>?&!$%|*76/!�S83<1/?&E4!$?�� V S_ºBd�!$#µ!>2=C:*9,]!$3)!$�B%'?'!�203<#�Zð��ç/Cl!>?'39#µ(>3<C�®,/20!q�&3<# ¼ �¤� ¡>¢ P�_ �/¹ � HaH�H �FZTb;� SO203<#�Z ��ç/Cl!>?'39#�(>3<Cl,/24!¥�&3<#Aê¼`i-P = � b�Z , $H� HaH�H Z , $�b�<^ · *9#M6/3<#MWY39?'C:*9#L(>3<�uI<!$�/(>E03<�8*920!>#�,/*9?A*�2Z* V S_º #'39�w68*968*-#K!>�­��øª^X§»¹?dµS���ø;^0§�½?d¦P($3<�:24*ÞçJ20%AE0C"*7!>(>1/*9(>E0�<�w*°I<!$(>!>#K?'!$WY!$?'E068*°(>3<Cl3;!$2�E4�uI<!$?'#'3 V S_º°^�� #&%A*9#MCª1/!>#h%A?A*-�FB1/!:2Z*v%A?A*-#&WY3<?&C"*96/*6/E0#'($?'!$%A*�6/!OS83<1/?'E0!>?5!>#;1/�8*\%A?A*-�/#&WY3<?&C"*-(>E4�9��3<?h%A3<T93<�8*92 !$�� j ^[ 2�39,]!$?A*96/39?)6/!�E0�u%A!>?&,]392Z*9($E4�<� � j ,/1/!>6J!�#'!$?OIHE0#&%'3w(>3<Cl3w1/�/*",/?'3�S9!>(>($E4�<�ø#&3<G/?'!!$2�!$#',8*-(>E43O� j 6/!5243<#¨,�3<24E0�/3<ClE43<#@%A?&E4T<3<�J3<C:D¦%A?&E4(>39#�6/!�T<?A*96J3OZ , $=P8(>3<��?'!>#&,]!$($%'3*�2Z*�*-,/?'3VUHE4C:*9(>E0�<�v6/E0#'(>?&!$%A*©6J!>2¸,/?'3=6/1/(¦%A3©E4�u%'!>?'�J3

� ¼¸��� ¢µ����� ô ýV�� ¼ �Y� ¢ � �¤� ¢-� � �søª^X§e�?d¾¨!+*920C:!$�u%A!9PJ2Z*ªWY3<?&C"*�G/E024E0�/!+*92

� ¼¸��� ¢ � $e]Z jlk óm ¡ ÷4� ¼ �¤� ¡>¢ � �Y� ¡¦¢ �søª^X§ ñ d($3<E4�/($E46/!M(>3<��!$2J,/?&3H6J1/($%'3°E4�u%A!$?'�/3©�søª^X§e�?d¸#'Eö¼©SY��#&3<��,�3<24E0�/3<ClE43<#�6/!�T9?A*96/3#Z , $H^

� ¼¸��� ¢ j � � ¼¸��� ¢ � ¼��s� � � j H �søª^X§@�?d�K3<Cl3w1/�/*\($3<�/#&!>(>1J!>�/($EZ*JPL�søª^X§ ñ d7!$#°1/�ø,/?'3=6/1/(¦%A3vE0�B%'!>?&�/3v#'3<GJ?'!�� j ^ · *wE4�u%A!$?h®,�3<2Z*-(>E4�9� � j ¼�6/!51/�/*©WY1/�J(>E4�9�($3<�u%AE0�=1/**¼�#A*-%'E4#hW�*9(>!52Z*�E06/!>�u%'E468*96

� � j ¼��s� ¢ j � � ¼¸��� ¢ j � � � � j H �søª^X§&¾6d· 3<#µ($3H!¦¼8(>E0!>�u%A!$# 6/E4#&(>?&!$%A39#µ6/!�S83<1/?&E4!$?�,/1/!$6/!>�©#&!>?µ!!UJ,J?'!>#'*96/3<# %A*9C�G/E0D>�ª!$��%AD>?&C:E ®�/39# 6/!¨243<#µ($3H!¦¼8(>E0!>�u%A!$#M6J!�S�391/?'E0!>?µ!!U/*-($%A39#@6J!�¼¸^q5=E`� £ ��� £ !$#L2Z*O#'!$?'E0!�6/!BS�391/?'E0!>?6/!�¼�dK(>39�BI9!>?&T<!ª*¼�!>�(+*968*©�/3=6/3���ø;^0§»¹"d¦P/!$�B%'3<�/($!>#),�3<?��søª^X§�½6dK(>39�/#'!$T<1/E4Cl3<#¾<Á

Page 90: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ê¼`i ���¼`i ´ :��m �¡6¢;£ �¤¡e¥ �¼`i : j�¦ = � b§Z , $H� HaH�H �FZ , $#b � �søª^X§�Á6d· *KWY�<?&C�1/2Z*O��øª^X§�Á6d�C�1/!>#h%A?'*¨FB1/!L!$2 = ®¥D$#'E0C:3�Cl3=6/3�6/!L2Z*ME4�u%A!$?',�3<2Z*9($E4�<��%A?&E4T<3<�J3<C:D¦%A?&EX®(>*76/!�¼�6J!>,�!>�/6/!K�/3)#&3<24376/!>2 = ®¥D$#'E0C:3�C:3=6/376/!H¼¸P»#'E0�/37%A*9C�G/E0D>�ª6J!M%A3=6/3<#�2439#RCl3-®6/39#@6J!�¼:!>28($18*92ö×q*-24EZ*-#n×M!>2 = ®tD>#&E4Cl3�6J!�24*O?'!$6:6/E0#'($?'!$%A*J^ · *5WY?&!>($1/!>�/($EZ* � = ´ Z < ¢ õ×¥*924E4*9#'!$#nר24*;WY?'!$(>1/!$�/(>E4* = õ #&3<G/?&!°2Z*;?&!>6�WH!>#h%A3<#�#&3<�vE4�/6JE4#&%'E4�/T91/E4G/20!># !$�v(+*968*��J3H6/3S<*lFB1/!;xDi : j)¦ �Y� ¡$¢L� xDi �Y� ¡¦¢ ^Ý )�¨*) �v6X9²�.Ka¦bªc 132 ²\c �la98íå:2�à ³R©�¶Ra91vad³Þ·Þ6ªc�³&�.H6;c 9³�3¶Ra9à 2�¶cª�d³ 1va98íå¸a$c*å�³ 132 .©2�¶_65H!>* «­¬

� M6/39�/6/!

«�

®¯¯¯¯¯¯¯¯¯°�!� � ó � ô � > ±;±;± ±;±;± � j² ó ³@ó � ó² ô ³dô � ô ¹

^ ^ ^ ^ ^ ^ ^ ^ ^¹ ² jlk ó´³ jlk ó � jxk ó² j ³ j

µ·¶¶¶¶¶¶¶¶¶¸ �¬

®¯¯¯¯¯¯¯¯¯°¹ �¹ ó¹ ô^^^¹ jlk ó¹ j

µ·¶¶¶¶¶¶¶¶¶¸ �`M �

®¯¯¯¯¯¯¯¯¯°º7 �7 $^^^7 jxk ó7 j

µ·¶¶¶¶¶¶¶¶¶¸»�¼-½�¾�»¿¼�À

�!� ¹ � ´ � ó ¹ ó�´ ±;±;± ´ � j ¹ j � º �søª^º¹9²6dÁ<²

Page 91: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

² p ¹ pÁk ó ´Â³ p ¹ p ´ � p ¹ p :Tó ��7 p ³ � �<� H�HaH ��Zcb � �søª^º¹H§ed² j ¹ jlk ó ´Â³ j ¹ j ��7 j �søª^º¹<¹?d� à ½�Ä-¾�ÅÇÆ �¿È¿É�ÊÁÉ Äs» ÊÁÉ"È¿Ë0�?à Å4»XÄs» È Æ4Ä-Äs» ÃÇÈ¿ËÌÉ\¬ ñ ¯Ð¿¹ p :Tó � � p ¹ p ´ = p ³ �¹ � HaH�HaH �FZcbö� �søª^º¹9½6dÍ#Î ½s» à »�Ï+¾?¼ÐÆ Ã4É » U�Ñ Ä-»¿¼ Ë4�?ÃÒÑ�É Ä É � pBÓ = p ^@� ¾ à ¼ Ë Å4»¿Ä-» ÊÁÉ ¼g» È Æ É"È�Ë ¾ à »�¼ �søª^º¹H§ed Ó�søª^º¹9½6dr¿

² p ¹ pÁk ó ´Â³ p ¹ p ´ � p ¹ p :Tó � 7 p¹ p :Tó � � p ¹ p ´ = p» ÊÁË Ï ËÁÃ4É"à Å4¾ ¹ p :Tó ² p ¹ p�k ó�´u³ p ¹ p ´ � p �¤� p ¹ p ´ = p ¢L�87 pÔ Æ4»&Äs» ® »�¼ È Ä Ë Î Ë Ï+¾"¼ È ¾?Ï�¾��³ p ´ � p � p ¢ ¹ p ´ ² p ¹ pÁk ó �­7 p bÕ� p = p �søª^º¹-±qdÅ ËaÖ�Ë Å Ë » à Å4¾g» à ½�Äs» ³ p ´ � p � p ¼-»X½ Ë » à »¹ p ´ ² p³ p ´ ² pÁ×Ðp ¹ pÁk ó � 7 p bØ� p = p³ p ´ � p � p �søª^º¹?�?dÅ4» ÊÁÉ » È9^4��ø;^Ú¹-½6d À4½s» à »�Ï+¾?¼ ¹ p � � p�k ó ¹ pÁk ó ´ = p�k ó �søª^º¹ ñ dÙ Ã ½�¾ ÃÇÈ »¿¼�À È ¾?Ï Ñ�É Ä É"à Å4¾�Ú ø;^Ú¹"�?d Ó Ú ø;^Ú¹ ñ d ¼s»w½ Ë » à » Ô Æ4» ¿

ÁJ§

Page 92: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

� pÁk ó�Û b ² p³ p ´ � p � p � = pÁk óÜÛ 7 p bÝ� p = p³ p ´ � p � p ³ÞÛàß?á HaH�H á Zcb ß H Ú øª^º¹��?dâ ¾?Ï�» ÃÇãVÉ"à ÅÇ¾ä» Ê)ÈVå"Ê�È Æ Ê ¾+ÅÇ»çæ p À = p É+Ñ�É Äo½ Ë Ä§Å4»Xæ jlk ó Ó = jxk ó ¼s»çÆ4¼ ÉOÊÁÉ » È Æ É"È�ËÁè?ÃÚ øª^êé?ë?d2Ñ�É Ä É » Ê�ÈVå�ÊÁÈ Æ Ê ¾ ��É"È�ËÌÉÐÉ ½sÄ å ¼�À Ó É ¼-ì�¾ Î ½�» à »¿Ä�½s¾�Å É ¼ ÊÌÉ ¼�æ p » = pí É Ä É ¾ Î ½s» à »�ÄBæ jxk ó À = jxk ó ÆǼ É Ï�¾?¼ ÊÌÉ » È9^ Ú ø;^îé"é?d Ó Ú ø;^îé�ï6d2È ¾ Ã�ð Û Z�b ß ¿{ ² j ¹ jlk ó�ñ³ j ¹ j Û 7 j¹ j Û æ jxk ó ¹ jlk ó�ñ = jxk óÅ4»&Å4¾ à Å4»&¼s»&¾ Î ½ Ë » à » ¹ j Û b ² j³ j ¹ jlk ó�ñ 7 j³ j H Ú øª^êéi¾6dí ¾?Ä ËÁà ¼ Ñ » È¿È¿Ë�è?à Å4»+Ú ø;^îéi¾?d Ó Ú ø;^îé"ò?d ¾ Î ¼s»�Ä Ö�É Ï�¾?¼ Ô Æ4»æ jxk ó�Û b ² jº j = = jxk óÜÛ 7 jº j Ú øª^êé"ó6dø ¼sìôÀ È ¾ à ¾ È¿Ë » à ÅǾÐæ jxk ó Ó = jlk ó À ÊÌÉ » È Æ É"È�ËÁè?Ã Ú øª^êé?ë?d2à ¾?¼ Ñ »¿Ä-Ï Ë ½�» È¿É"ÊÁÈ Æ ÊÌÉ Äõæ jlk ô À= jlk ô À ^0^X^ Àöæ � À = �+^Ù Ã ½�¾ ÃÇÈ »¿¼�ÀÜ» ÃÇÈ ¾ à ½�Ä É Ï�¾?¼ ¹ ó À ^X^0^ À ¹ j È ¾ à ½ É"Ê Å4» Ô Æ4»O» ÊlÖ"É�Ê ¾?ÄXÅ4» ¹ � ¼s» È ¾ à ¾ ã�ÈVÉJ^Ù Ã ¼s»�÷?Æ Ë Å ÉOËÁà ŠËÁÈ¿É Ï+¾"¼ È�è Ï+¾+Äs» É"ÊÁË�ãVÉ ÄB» Ê�ÈVå"Ê�È Æ Ê ¾OÅ4» ¹ �>^ â ¾ à ¼ Ë ÅÇ»¿Äs» ÊÌÉ » È Æ É"È�ËÁè?ÃÚ øª^êé"ø6dr¿ �!� ¹ � ñ � ó ¹ ó�ñ H�HaH ñ � jlk ó ¹�ù k ó�ñ � j ¹ j Û º HÙ ÊÁË Ï Ë�à » ¹ ó À ^0^X^0^ À ¹ j » U�Ñ Ä-»¿¼ å"à Å4¾ ÊÁÉ ¼w» à ½sú¿Ä-Ï Ë�à ¾?¼�ÅÇ» ¹ � ÀûÆǼ É"à ÅǾ ÊÌÉ Æ Ã4ÉOü�è ÄsÏýÆ ÊÌÉÅ4»&Ä-» È ÆÇÄsÄs» Ã4È¿ËÁÉ ÅÇ» ÊÁÉçü ¾?ÄsÏ Éö¿ ó6é

Page 93: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

¹lþ Û�ÿ þF¹ � ñ�� þ _ Û ¹ á¿ß?á HaH�H á Z Ú øª^ ï?ø6d5�Ë À _ Û ¹ ½s» à »�Ï+¾"¼ ¹ � Ûvÿ � ¹ � ñ�� � À É ¼sì�Àÿ � Ûàß?á � � Û ¹ H Ú øª^ ï � dø§� ¾?Ä É À4¼sÆ4¼o½ Ë ½�Æ Ó » à Å4¾ ÊÌÉ » È Æ É"È�ËÁè?Ã Ú ø;^êï"ø6d » Ã�ÊÌÉ » È Æ É"È�ËÁè?Ã Ú ø;^îé�ï6d À4¼-» Ê�Ê »�÷ ÉýÉö¿¹lþ Û æ þ k ó ÿ þ k ó ¹ � ñ æ þ k ó�� þ k ó�ñ �Vþ k ó H Ú øª^ ï6é?dâ ¾?Ï Ñ�É Ä É"à Å4¾�Ú øª^ ï6é?d2È ¾ Ã Ú øª^ ï?ø6d ¾ Î ½s» à »�Ï+¾"¼{ ÿ þ Û æ þ k ó�ÿ þ k ó

� þ Û æ þ k ó�� þ k ó�ñ �Vþ k ó _ Ûàß?á HaH�H á Zø§� ¾?Ä É À4¼sÆ4¼o½ Ë ½�Æ Ó » à Å4¾ Ú ø;^êï?ø?d » Ã Ú øª^êé"ø6d�!� ¹ � ñ � ó � ÿ ó ¹ � ñ���ó � ñ HaH�H ñ � j � ÿ j ¹ � ñ�� j � Û º¾4ÀÇ» Ãü ¾?ÄsÏ ÉýÈ ¾?Ï Ñ�É"È ½ É

� ¹ � ñ Û º Ú øª^ ï?ï6dÅ4¾ à Å4»� Û jm þ ÷4� � þ ÿ þ á Û jm þ ÷ ó � þ á� þ Ú øª^ ï��qdÓ� Ã�É"Ê Ï�» à ½s»"À ¹ � Û º b � H Ú øª^ ï6�?d� »¿¼-Æ4Ï Ë Äs»�Ï+¾?¼2» Ê�Ñ Ä-¾ È »¿¼s¾�Å4»&¼s¾ Ê Æ È�ËÁè?à Å4» «­¬

Û � » à » Ê ¼ Ë ÷?Æ Ë » à ½s» É�Ê ÷?¾?Ä Ë ½sÏ+¾ó?ï

Page 94: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

����������� ��������� ��!#"$�&%� ^ æ p À � p Ñ�É Ä ÉYð Û ¹ á HaH�H á Z b ß È¿É"ÊÁÈ Æ ÊÌÉ Å É ¼2ÅÇ» ÊÌÉ ¼2» È Æ É"È�Ë ¾ à »�¼&Ú ø;^îé"ë?dõÉ"Ã Å Ú ø;^îé"ó?d¦^éH^ ¹ � â É"Ê�È Æ ÊÁÉ Å4¾gÅ4» É"È Æ4»¿Ä-Å4¾ Éö¿Ë�d ÿ � Û ß À � � Û ¹ ^ËÁËÌd ÿ þ À � þ Ñ�É Ä É _ Û ß?á HaH á Z´ÈVÉ"Ê�È Æ ÊÁÉ Å4¾?¼�ÅÇ»OÚ øª^êë?d¦^ËÁË�Ë�d � É"à Å(' ÈVÉ�ÊÁÈ Æ ÊÌÉ Å4¾gÅÇ» ü Ä-¾?Ï�Ú øª^ ï��qdË�ÖÇd ¹ � ¾ Î ½s» Ã4Ë Å4¾ÐÅ4»+Ú øª^ ï6�?d�^ïJ^ ¹lþ Ñ�É Ä ÉX_ Ûàß?á HaHaH á Z\È¿É"ÊÁÈ Æ ÊÌÉ ÅǾgÅ4»+Ú øª^ ï?ø6d�^Ý*)�+,) ÿ .�å.-0/1- /3254v76dà 6981:<;�6>=?8 /32 @A:<BDC.CÙ ¼-½s»õÏ�ú�½s¾�ÅǾ Ê ¾ Ö�É Ï�¾?¼ É » U�Ñ4ÊÁË�ÈVÉ Ä�Æ�½ ËÁÊ�ËÁã¿É"à Å4¾ ÊÁÉ ¼�» È Æ É"È¿Ë ¾ à »¿¼ Ö�Ë ¼o½ É ¼)» à » ÊqÈVÉ�Ñ ì�½sÆ Ê ¾Å4» ÊFE ú�½s¾�ÅǾ0G Ï Ñ4Ê ì È¿Ë ½�¾ ^â Æ É�à Å4¾w» ÃgÊÁÉ ¼Þ» È Æ É�È¿Ë ¾ à »¿¼ÜÅÇ» â Ä É"ÈIHKJ�L§ËÁÈ ¾ Ê ¼-¾ Ã Ú éH^ �NM d)��É Ó ZNb ß Ñ Æ Ã ½�¾?¼ ËÁà ½�»�Ä Ã ¾?¼ÉgÊ ¾ ÊÁÉ Äs÷"¾gÅ4» ÈVÉ Å É ÊÁÉ À Ñ ¾�ÅÇ»¿Ï�¾?¼�»¿¼ È Ä Ë Î Ë Ä ÊÌÉ ¼�ÅÇ» ÊÁÉçü ¾?ÄsÏ É ÷?» à »¿Ä É"Ê À

ó��

Page 95: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

ñ O ó P ó bÝ� ó P ô Û � ó!áb�é ô P ó�ñ O ô P ô bÝ� ô P > Û � ôVá^^^ ^^^b~é p P pÁk ó�ñ O ð P p bÕ� p P p :Tó Û � p á±;±;± ±;±;±b~é õ k ó P jlk ô�ñ O jlk ó P jxk ó Û � jxk óÅ4¾ à Å4» ÊÌÉ ¼ é À O À � Ó � ¼-¾ Ã È ¾ à ¾ È¿Ë Å É ¼ ^RQ » ÊÌÉ » U�Ñ Ä-»¿¼ ËÁè?à É"à ½�»�Ä Ë ¾"Ä¿À Ñ ¾�Å4»¿Ï�¾?¼wÆǽ ËSJÊ�ËÁãVÉ Ä ÊÌÉ Ñ Ä Ë Ï�»¿Äç» È Æ É"È¿Ë�è?Ã(T Æ Ã ½�¾ È ¾ ÃuÊÁÉ ¼s»¿÷"Æ Ã Å É » È Æ É"È¿Ë�è?ÃuÑ�É Ä É » ÊÁË Ï Ë�Ã�É Ä É P ó¾ Î ½�» ÃÇË » à Å4¾gÆ Ã�Égà Æ4» Ö�É » È Æ É"È�ËÁè?Ã Ó »�¼-½ É ÆǼ É Ä ÊÌÉUT Æ Ã ½s¾ È ¾ ÃÊÌÉ » È Æ É�È¿ËÁè"à ½sÄs»�¼ Ñ�É Ä É» ÊÁË Ï ËÁÃ4É Ä É P ô Ä-»¿¼sÆ Ê ½ å�à Å4¾ à ¾"¼w¾"½�Ä É » È Æ É"È¿Ë�è?à ÊÌÉOÈ Æ É"ÊVT Æ Ã ½s¾ È ¾ à ÊÌÉ » È Æ É"È¿Ë�è?Ã È Æ JÉ ½sÄs¾ à ¾?¼B¼ Ë Ä Ö » Ñ�É Ä É » Ê Ï Ë�Ã4Ë Ï É Ä É P > Ó É ¼sì)¼sÆ È »¿¼ Ë�Ö�É Ï�» à ½�»�À ��É ¼-½ É Ô ÆÇ» ÊÌÉ*ç4Ê ½ Ë Ï É» È Æ É"È�ËÁè?Ã Ô Æ4»X¾ Î ½s» à ÷ É Ï+¾?¼ Ô Æ4»�Å4»X¼s¾ Ê ¾g» Ãü Æ Ã4È¿Ë�è?à Å4» P jxk ó ^L ¾"½�»�Ï+¾?¼ Ô ÆÇ»�» ÊÜÈ ¾�» È¿Ë » à ½�» � » Ã È¿É Å É�à Æ4» Ö�É » È Æ É"È¿Ë�è?à »¿¼&» Ê Ï Ë ¼sÏ�¾ Ô Æ4»g» ÃTÊÌÉ» È Æ É"È�ËÁè?Ã�É"à ½�»�Ä Ë ¾?Ä È ¾?Ä-Äs»�¼ Ñ ¾ Ã Å Ë » à ½�» ^ Ù Ã ½s¾ Ã4È »�¼¿ÀDÅ4»¿¼ Ñ Æ4ú�¼�Å4» Ô Æ4»XÄs» É�ÊÁËÁã¿É Ï�¾?¼�Æ Ã�É» ÊÁË Ï ËÁÃ4É"È¿Ë�è?à Ä-»¿¼sÆ Ê ½ É À

W pYX P pÁk ó bu� pÁk ó P p Û � p�k ó!á Ú øª^ ï?ò6db�é p P p�k ó�ñ O p P p bu� p P p :Tó Û � p áÅ4¾ à Å4» W óxÛ O ó À � óxÛ � ó ^�ø�Ê » ÊÁË Ï Ë�Ã�É Ä P p�k ó Ê�Ê »�÷ É Ï�¾?¼ ÉA O p b é p � pÁk óW pÁk ó C P p bu� p P p :TóxÛ � p ñ W p � pÁk óW pÁk ó»�¼BÅÇ» È¿Ë Ä¿À ó6�

Page 96: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

W p P p bÕ� p P p :TóxÛ � p á Ú øª^ ï6ë?dÅ4¾ à Å4»W p Û O p é p � pÁk óW pÁk ó � � p Û � p ñ W p � pÁk óW ð�b ß � ð Û $ á Z á HaH�H � HÙ Ê�ç4Ê ½ Ë Ï�¾ Ñ�É Ä�Å4»&» È Æ É"È¿Ë ¾ à »¿¼�¼-¾ ÃR[

W jxk ô P õ k ô bu� jxk ô P jlk ó\Û � jlk ô Ú øª^ ï M db~é jlk ó P jlk ô�ñ O jxk ó P õ k ó´Û � õ k ó HÙ ÊÁË Ï Ë�Ã�É"à ÅǾ P jlk ô à ¾?¼�Å É ÀA O jlk ó b é jlk ó � jlk ôW jxk ô C P jlk óÜÛ � jxk ó�ñ é jlk ó � jlk ôW jlk ô á»�¼BÅÇ» È¿Ë Ä¿ÀW jlk ó P jxk ólÛ � jlk ó H Ú øª^ ï?ó6d\ É ¼�» È Æ É"È¿Ë ¾ à »�¼wÚ ø;^êï?ë?d Ó Ú øª^ ï?ó6d ÏýÆ4»¿¼o½�Ä É"Ã Ô Æ4» ÊÌÉ ¼-¾ Ê Æ È¿ËÁè"Ã+Ñ ÆÇ»¿Å4»§¼s»¿Ä ÈVÉ"Ê�È Æ ÊÁÉ Å ÉÅ4»

P jlk ó\Û � jlk óW jxk ó Ú øª^]�6ø6dP p Û ßW p � � p ñ � p P p :Tó � � ð Û Z�bu$ á Zcb Z á HaH�HaHaH áVß �Å4¾ à Å4» ÊÌÉ ¼ W Ó � ¼s¾ Ã Å É Å É ¼ Ñ ¾"Ä [

ó?ò

Page 97: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

W ó Û O p�^ W p Û O p b é pW pÁk ó á Ú øª^]� � d� ó Û � ó ^ � p Û � p ñ é pW p�k ó � pÁk ó HÙ Ã ÏçÆ ÈF� ¾?¼ Ñ Äs¾ Î Ê »�Ï É ¼ W pqÓ é p , W pÁk ó ¼-¾ ÃçË�à Å4» Ñ » Ã Å Ë » à ½�»¿¼)Å4» Ê ½ Ë »�Ï Ñ ¾ Ó ¼s¾ ÊÁÉ Ï�» à ½�»¼-¾ Ã�È¿É"ÊÁÈ Æ ÊÌÉ Å4¾?¼�Æ Ã4ÉÐÖ » ã À Ë�à Å4» Ñ » Ã Å Ë » à ½s»¿Ï�» à ½�»&Å4» Ê�Ñ�É ¼-¾+Å4» Ê ½ Ë »¿Ï Ñ ¾ ^

ó6ë

Page 98: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

_ `Racbd`Regfihkjglnmdj

o �9p,q S ìÁ¼ Ë�ÈVÉ?r ¾ Ê ^ �sq ^ � ¾ Î »¿Äo½ � »�¼ Ã4Ë�ÈIH À â Ù â 5ut Àwv�»�Ä È »¿Ä É Ù Å ËÁÈ�ËÁè?Ã Ú � ó?ó?ò6d�^o é pyx â É Ï Ñ ¾?¼ Ù Ê » È ½sÄs¾?Ï É ÷ à ú�½ Ë�È ¾?¼ x ^ � ¾ É"Ê Å�z É�à ÷?¼ à »¿¼s¼�À \ G Ek{#5|t Ú � ó?ó?ó6d�^o ï pyx L Æ4Ï�»¿Ä ËÁÈVÉ�Êx5 ¾ Ê Æǽ Ë ¾ à ¾ ü í É Ä-½ ËÌÉ"Ê}Q§ËY~ »¿Äs» à ½ ËÌÉ�Ê Ù Ô Æ É ½ Ë ¾ à ¼?�����������?��� �y�>�I�>���I�� �>�����9��� x ^w��^�Qª^D5 Ï Ë ½���À ÍU� S Í � Q À�v�� Ë ÄsÅ Ù Å Ë ½ Ë ¾ Ã Ú � ó?ó?ó?d¦^o � pyx v��4» z ÉVÖ » Ù Ô Æ É ½ Ë ¾ à ËÁà �����sq ^ í ^ Q�Ë » à »�Ä¿À SK^ 5]^ � Æ ã Ï å"à À�6½s½ ÑR[��K��à Æ4Ï�Äs» Ê ^êÉ »�½ ^ Ï Ñ ÷ ^ Å4» � Ù Å4Æ ÈVÉ ½ Ë ¾ Ãw� v�Æǽs¾?Ä ËÁÉ"Ê ¼ � À Ú 5 » Ñ ½s»¿Ï Î »¿Ä �K� Àé"ø"ø?ï6d¦^o � pyx t�� ¾�¾ H ¾ à â x ^�t§Êy  » Ê�Ê » Ó À�G Ä É í ¾K� Ê À t ÅÇÅ Ë ¼-¾ Ã|J z »¿¼ Ê » Ó À S ¾?ÆÇÄ-½� Ù Å Ë ½ Ë ¾ ÃÚ � ó?ó M do ò pyx 5�Ñ » È ½sÄ É"Ê¡E »�½�4¾�Å4¼ É�à Št§Ñ4Ñ4ÊÁË�ÈVÉ ½ Ë ¾ à ½s¾ â ¾ Ã6Ö » È ½ Ë ¾ à x ^ � ¾?÷"»¿Ä í » Ó ½s»¿Ä�À {§ÃÇËSJÖ »¿Äs¼ Ë ½ Ó LBËÁÈ »"À â » à ½�»¿Ä ü ¾?Ä SÞÊ Æ Ë Å E » È � É"Ã4Ë�È ¼ \ » È ½�Æ4Ä-» 5 »¿Ä Ë »"À � Äs¾£¢ ä{§ÃÇË�Ö »�Ä J¼ Ë ½ Ó À í Ä-¾ Ö�Ë Å4» Ã4È »äÚ � ó MKM do ë p¦¥ 5�Ñ » È ½sÄ É"ÊFE »�½�4¾�Å4¼�À S Æ Ã Å É Ï�» à ½ É"Ê ¼ ËÁÃ�5�Ë�à ÷ Ê » Q ¾?Ï É"ËÁà ¼ x â ^ â É"à Æǽs¾4À E�^¨§�^'BÆ4¼s¼ É"ËÁÃ4Ë À tª^ I Æ É Ä-½�»�Äs¾ ÃÇË À v ^ª©�É"à ÷4À 5�Ñ Ä ËÁà ÷?»¿Ä J�r »¿Ä ÊÌÉ ÷�Ú é"ø?ø?ò?d¦^o M«p¦¥ Í Ã É Ô Æ É ¼ ˬJ Ê�ËÁà » É Ä Ñ�É Ä É Î ¾ ÊÁË�È » Ô Æ É ½ Ë ¾ à ¾ È Æ4Ä-Ä Ë�à ÷ ËÁÃTÉ »¿Ä-¾�Å Ó Ã4É Ï Ë�È ¼ x0­ ^.Qª^â ¾ Ê » ^ I Æ É Äo½ ^#t§Ñ4Ñ4Ê ^wE�É ½�� ^4Ö ¾ Ê ^�ó À Ñ4É ÷?¼ ^Dé"é?��Joé�ï?ò Ú � ó?� � d¦^o ó p Ù ^ � ^®� » à ½�¾ Ã0É"à Š�©^¯L;^ í ÊÁÉ ½ ã Ï É�Ã4à À ¥ t ½ É Î Ê » ¾ ü ¼s¾ Ê Æǽ Ë ¾ à ¼�¾ ü ¾ à » JÅ Ë Ï+» à ¼ Ë ¾ Ã�É"Êu� ÆÇÄs÷?»�Äs¼x» Ô Æ É ½ Ë ¾ à x I Æ É Ä-½ ^«t§Ñ4Ñ4Ê ^�E�É ½�� ^�Ö ¾ Ê ^?é"ó À Ñ�É ÷?¼ � ó6��J9é � éÚ � ó6ë?é?d�^ ó M

Page 99: İndice general 1. Introducción 7 1.1. Ecuación de Onda ...

o � ø pyx L Æ4Ï�»¿Ä ËÁÈ¿É"Ê � » È�ËÁÑ »¿¼ Ë�à â x z ËÁÊÁÊ�ËÌÉ Ï°' ^ í Ä-»¿¼s¼�À 5ÇÉ Æ Ê|tª^ v�»�Æ H ¾ Ê ¼ H Ó ÀKz Ë�ÊÁÊ�ËÌÉ Ïv ^Vr »�½s½s»¿Ä ÊÁËÁà ÷ É"à Š� Ä ËÁÉ"à í ^ûS�ÊÌÉ"Ã4à »¿Ä Ó À â É Ï Î Ä Ë Å4÷?» {§ÃÇË�Ö »�Äs¼ Ë ½ Ó í Ä-»¿¼-¼¿À 5 » È>J¾ Ã Å Ù Å Ë ½ Ë ¾ Ã Ú � ó?ó?ë?d¦^o �K�9pyx 5�Ñ » È ½sÄ É"Ê�É�à ŠSÞËÁÃ4Ë ½�» Q§ËY~ »¿Äs» Ã4È » 5 ¾ Ê Æǽ Ë ¾ à ¾ ü ½��4» � Æ4Ä-÷?»¿Ä-¼ Ù Ô Æ É ½ Ë ¾ à xâ ^¯��É ¼-Å4» Ö�É"à ½§»�½ ^ É"Ê ^ â ¾?Ï Ñ Æǽ�»�Äs¼ É"à ŠSÞÊ Æ Ë Å4¼ Ö ¾ Ê � � À Ñ�É ÷?¼ ^Dé"ï�J�� � Ú � ó M ò6d¦^

ó?ó