Announcements 2/14/11 Prayer Exam 1: last day = tomorrow! We’re likely not going to finish...
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Transcript of Announcements 2/14/11 Prayer Exam 1: last day = tomorrow! We’re likely not going to finish...
Announcements 2/14/11 Prayer Exam 1: last day = tomorrow! We’re likely not going to finish dispersion
today, so extra time on Lab 3: now due Tues Feb 22.
Monday Feb 21 is President’s Day holiday. Tues Feb 22 is a virtual Monday
Complex numbers & traveling waves Traveling wave: A cos(kx – t + )
Write as:
Often:
…or – where “A-tilde” = a complex number, the
phase of which represents the phase of the wave
– often the tilde is even left off
( ) i kx tf t Ae
( ) i kx tif t Ae e
( ) i kx tf t Ae
Reading Quiz Which of the following was not a major
topic of the reading assignment?a. Dispersionb. Fourier transformsc. Reflectiond. Transmission
Thought Question Which of these are the same?
(1) A cos(kx – t)(2) A cos(kx + t)(3) A cos(–kx – t)
a. (1) and (2)b. (1) and (3)c. (2) and (3)d. (1), (2), and (3)
Which should we use for a left-moving wave: (2) or (3)?
a. Convention: Usually use #3, Aei(-kx-t)
b. Reasons: (1) All terms will have same e-it factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.
ˆk k i
Reflection/transmission at boundaries: The setup
Why are k and the same for I and R? (both labeled k1 and 1) “The Rules” (aka “boundary conditions”)
a. At boundary: f1 = f2
b. At boundary: df1/dx = df2/dx
Region 1: light string Region 2: heavier string
in-going wave transmitted wave
reflected wave
1 1( )i k x tIA e
1 1( )i k x tRA e
2 2( )i k x tTA e
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e 2 2( )
2i k x t
Tf A e
Goal: How much of wave is transmitted and reflected? (assume k’s and ’s are known)
x = 0
1 1 1 1 1cos( ) cos( )I I R Rf A k x t A k x t 2 2 2cos( )T Tf A k x t
Boundaries: The math
1 1 1 1 2 2( 0 ) ( 0 ) ( 0 )i k t i k t i k tI R TA e A e A e
2 2( )2
i k x tTf A e
x = 0
1 20 0B.C.1:
x xf f
1 1 2i t i t i tI R TA e A e A e
I R TA A A and 1 2
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
Boundaries: The math
1 1 2( ) ( ) ( )1 1 2
0 0
i k x t i k x t i k x tI R T
x xik A e ik A e ik A e
2( )2
i k x tTf A e
x = 0
1 2
0 0
B.C.2:x x
df df
dx dx
1 1 2i t i t i t
I R Tik A e ik A e ik A e
1 1 2I R Tk A k A k A
1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
Boundaries: The math
Like: and
How do you solve?
x = 0
1 1 2I R Tk A k A k A I R TA A A
Goal: How much of wave is transmitted and reflected?
x y z 3 3 5x y z
2 equations, 3 unknowns??
Can’t get x, y, or z, but can get ratios!y = -0.25 x z = 0.75 x
Boundaries: The results
Recall v = /k, and is the same for region 1 and region 2. So k ~ 1/v
Can write results like this:
x = 0
1 2
1 2
R
I
A k kr
k kA
Goal: How much of wave is transmitted and reflected?
1
1 2
2T
I
A kt
k kA
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
“reflection coefficient” “transmission coefficient”
The results….
Special Cases
Do we ever have a phase shift? a. If so, when? And what is it?
What if v2 = 0? a. When would that occur?
What if v2 = v1? a. When would that occur?
x = 0
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
The results….
Power
Recall: (A = amplitude)
Region 1: and v are same… so P ~ A2
Region 2: and v are different… more complicated…but energy is conserved, so easy way is:
x = 0
2 21
2P A v
2R
I
PR r
P
21T
I
PT r
P
r,t = ratio of amplitudesR,T = ratio of power/energy
Dispersion A dispersive medium: velocity is different for
different frequenciesa. Any real-world examples?
Why do we care? a. Real waves are often not shaped like sine
waves.– Non sine-wave shapes are made up of
combinations of sine waves at different frequencies.
b. Real waves are not infinite in space or in time.– Finite waves are also made up of combinations
of sine waves at different frequencies.Focus on (b) for now… (a) is the main topic of the “Fourier” lectures of next week.
Wave packets HW 17-5
Wave packets, cont.
What did we learn?a. To localize a wave in space, you need lots of
frequenciesb. To remove neighboring localized waves, you
need those frequencies to spaced close to each other. (infinitely close, really)