Chapter 2Wave Motion
Lecture 3
Phase and phase velocity The superposition principle Complex representation 3D waves: plane waves
ExampleGiven expression: , where a>0, b>0 2, cbtaxtx Does it correspond to a traveling wave? What is its speed?
Solution:1. Function must be twice differentiable
acbtaxx
2
22
2
2ax
bcbtaxt
2
22
2
2bt
2 2
2 2 2
1x t v
2. Speed:
22
2 212 bav
abv
Direction: negative x direction
Reminder Harmonic waves summary
1v
)(sin txkA v
2k
Functional shape: Wave parameters:k - propagation number - wavelength - period - frequency - angular temporal frequency - wave number
v
Alternative forms:
txA 2sin
txA 2sin tkxA sin
txA v
2sin
- for wave moving right+ for wave moving left
mostlyused
These eq-ns describe an infinite monochromatic (monoenergetic) wave.Real waves are not infinite and can be described by superposition of harmonic waves. If frequencies of these waves cluster closely to a single frequency (form narrow band) the wave is called quasimonochromatic
single frequency
22 1
Harmonic wave: Initial phase
tkxAtx sin,Consider wavetkx phase:
When written like that it implies that 0,00 txtx
With a single wave we can always chose x axis so that above is trueBut in general case 0,
00 txtx
x
This is equivalent to the shift of coordinate x by some value a a
taxkAtx sin, katkxAtx sin, tkxAtx sin, - initial phase tkxphase:
Harmonic wave: Phase
x
tkxA sinCan use cos():
tkxAtx sin, 2cos, tkxAtx
equivalent equations
Special case: = = 180o phase shift
x
tkxAtx sin, kxtAtx sin, 2/cos, kxtAtx
Note: sin(kx-t) and sin(t-kx) both describe wave moving right, but phase-shifted by 180 degrees ().
Harmonic wave: Phase derivatives
tkx Phase:Partial derivatives:
xtrate of change of phase with time is equal to angular frequency (=2)
kx t
rate of change of phase with distance is
equal to propagation number
tkxAtx sin,
phase velocity of a wave
Harmonic wave: Phase velocity tkx Phase:
What is the speed of motion of a point with constant phase?
v
kxt
tx
t
x
from the theory of partial derivatives
sign gives direction
In general case, for any wave we can find the phase velocity:
t
x
xt
valways >0by definition Add sign to give direction:
+ in positive x direction- in negative x direction
Phase (red) vs. group (green) velocity(to be discussed later)
The superposition principle
Consider differential wave equation: 22
22
2 1tx
v
If 1 and 2 are both solutions to that equation, then their superposition (1+2 ) is also a solution:
2
212
2221
2 1tx
v
22
2
221
2
222
2
21
2 11ttxx
vv=
=
Proof:
The superposition principle
Superposition principle: the resulting disturbance at each point in the region of overlap of two or more waves is the algebraic sum of the individual constituent waves at that location.
Note: once waves pass the intersecting region they will move away unaffected by encounter
Superposition of traveling waves: http://vnatsci.ltu.edu/s_schneider/physlets/main/waves_superposition.shtml
The superposition principle: example
Note: the resulting wave is still a harmonic wave (the same k)
The superposition principle: special casesTwo waves are in-phase:
tkxA
tkxA
sinsin
22
11
1 2 sinA A kx t Amplitude of the resulting wave increases: constructive interference
Two waves are out-of-phase:(=180o=)
tkxAtkxA
sinsin
22
11
tkxA sin22
1 2 sinA A kx t Amplitude of the resulting wave decreases: destructive interference
The complex representation
Complex numbers: 1 where,~ iiyxz
Argand diagram
In polar coordinates:
sincos~
sin ,cosirz
ryrx
Euler formula: sincos iei irez ~Any complex number:
Wave: tkxAtkxAtx cos'sin,can use sin or cos to describe a wave
tkxAAetx tkxi cosRe,Convention - use cos: itkxi AeAetx ,Usually omit Re:wave equation using complex numbers
The complex number math
sincos iei sincos ie i i
eeee iiii
2sin ,
2cos
Magnitude (modulus, absolute value): zzryxz ~~~ 22Complex conjugate: ii rereiyxiyxz ~
212121 ~~ yyixxzz Math: 21
2121~~ ierrzz
212
1
2
1~~ ie
rr
zz
2121~~~~ zzzz eee
xz ee ~
12 ie1 ie
ie i 2/ziz ee ~2~ zzz ~~21~Re 1Im 2z z zi
PhasorLets rotate the arrow in Argand diagram at angular frequency :
t
tkxAtx sin,moving left
This rotating arrow is called phasor ACCW rotation - wave moves leftCW rotation - wave moves right
Phasor: superposition
Adding two waves can be done using phasors
212121
ii eAeA iAe
Complex numbers can be added as vectors
Phasor addition: http://vnatsci.ltu.edu/s_schneider/physlets/main/phasor1.shtml
Phasor: superposition
Adding two waves can be done using phasors
212121
ii eAeA iAe
Complex numbers can be added as vectors
Example: out-of-phase waves
tkxAA sin211 Amplitudes subtractPhase does not change
3-D waves
Surfaces joining all points of equal phase are called wavefronts.
Example:Wavefronts of 2-D circular waves on water surface (superposition where waves overlap)
http://www.falstad.com/wavebox/
3-D waves: plane waves(simplest 3-D waves)
All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction
3-D waves: plane waves(simplest 3-D waves)
All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction
Unit vectors
An equation of plane that is perpendicular to kji zyx kkkk
aconstrk All possible coordinates of vector r are on a plane k
Can construct a set of planes over which varies in space harmonically:
rkAr sin rkAr cosor rkiAer or
Plane waves
rkr sin The spatially repetitive nature can be expressed as:
kkrr
In exponential form:
kirkikkrkirki eAeAeAer /For that to be true: 12 ie 2k
2k
Vector k is called propagation vector
Plane waves: equation
rkiAer This is snap-shot in time, no time dependenceTo make it move need to add time dependence the same way as for one-dimensional wave:
trkiAetr , Plane wave equation
Plane wave: propagation velocity
Can simplify to 1-D case assuming that wave propagates along x:
trkiAetr , i||r
tkxiAetr ,We have shown that for 1-D wave phase velocity is:
kv
That is true for any direction of k+ propagate with k- propagate opposite to k
More general case: see page 26
Example: two plane wavesSame wavelength: k1= k2=k=2/,Write equations for both waves.Solution:
Same speed v:1=2==kv
trkiAe
Dot product:zkykxkrk zyx
Wave 1: 1 1k r k z kz
1 1
i k z tA e direction
Wave 2: zkykrk cossin 222
tzykieA cossin22 2 tkzA cos11 tzykA cossincos22
Note: in overlapping region = 1 + 2
ExampleGiven expression ,where a>0, b>0: btaxtx 2,Does it correspond to a traveling wave? What is its speed?
Solution:1. Function must be twice differentiable
32 x
x
42
2
6 ax
x
bt
022
t
2
2
22
2
2
2
2
2 1tzyx
v
2. Wave equation:
06 4 ax Is not solution of wave equation!This is not a wave traveling at constant speed!