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

ab

v

Direction: negative x direction

Reminder Harmonic waves summary

1

v

)(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

v

always >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: 2

2

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 ieeee iiii

2sin ,

2cos

Magnitude (modulus, absolute value): zzryxz ~~~ 22

Complex conjugate: ii rereiyxiyxz ~

212121~~ yyixxzz Math:

212121

~~ ierrzz

21

2

1

2

1~~

ierr

zz

2121~~~~ zzzz eee

xz ee ~

12 ie1 ie

ie i 2/

ziz ee~2~

zzz ~~21~Re 1Im

2z z z

i

PhasorLets rotate the arrow in Argand diagram at angular frequency :

t

tkxAtx sin,moving left

This rotating arrow is called phasor A

CCW 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 ˆˆˆ

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

2

k

Vector k is called propagation vector

Plane waves: equation

rkiAer This is snap-shot in time, no time dependence

To 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:

k

vThat 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

02

2

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!