Wave PhenomenaPhysics 15c

Lecture 12Dispersion

(H&L Sections 2.6)

What We Did Last Time

! Defined Fourier integral

! f(t) and F() represent a function in time/frequency domains! Analyzed pulses and wave packets

! Time resolution t and bandwidth related by! Proved for arbitrary waveform

! Rate of information transmission bandwidth! Diracs (t) a limiting case of infinitely fast pulse! Connection with Heisenbergs Uncertainty Principle in QM

( ) ( ) i tf t F e d

=

1( ) ( )2

i tF f t e dt

=

12

t >

Goals For Today

! Discuss dispersive waves! When velocity is not constant for different ! Waveform changes as it travels! Dispersion relation: dependence of k on

! Define group velocity! How fast can you send signals if the wave velocity is not

constant?

Mass-Spring Transmission Line

! In Lecture #5, we had

! We ignored the gravity by making the strings very long

nn1 n+1

2

1 12 ( ) ( )n s n n s n ndm k kdt

+=

0nmgL

L What if we didnt make this approximation?

Wave Equation

! Equations of motion is now

! Usual Taylor-expansion trick

! Divide by (x)

! Wave equation:

2

1 12 ( ) ( )n s n n s n n ndm k kdt

mgL

+ =

2 22

2 2

( , ) ( , ( , )) ( )sx t x tm k x

t xmL

x g t =

2 2

2 2

( , ) ( , ) ( , )l lx t x tK g

tx t

x L =

2 22 2

02 2

( , ) ( , ) ( , )wx t x tc x t

t x =

wl

Kc

=

0gL

=

Natural frequency of pendulum

Solution

! Assume

! As before, we can write the solution as

2 22 2

02 2

( , ) ( , ) ( , )wx t x tc x t

t x =

( , ) ( ) i tx t a x e =2

2 2 202

( )( ) ( )i t i t i twd a xa x e c e a x e

dx = Wave eqn.

2 220

2 2

( ) ( )w

d a x a xdx c

= SHO-like if 2 20 0 >

( )( , ) i kx tx t Ae =2 2

0

w

kc

=but with

This is the difference

Dispersion Relation

! Normal-mode solutions are still! What changed is the relationship between k and

! A.k.a. dispersion relation

! NB: there are different types of dispersive waves! We are looking at just one example here

! Dispersion relation determines how the waves propagate in time and space

( )( , ) i kx tx t e =

( )w

kc =

2 20( )

w

kc

=

Non-dispersive waves

Dispersive waves

Well study how

Phase Velocity

! To calculate the propagation velocity of! We follow the point where the phase kx t is constant

! Phase velocity is the velocity of pure sine waves! Easily calculated from the dispersion relation

( )0( , )

i kx tx t e =

kx t C =C tx

k

= dx

dt k

= Phase velocity cp

( ) const.p wc c = =( )w

kc =

2 20( )

w

kc

=

Non-dispersive

Dispersive2 2

0

( )p wc c

=

No longerconstant!

Dispersing Pulses

! Imagine a pulse being sent over a distance! On non-dispersive medium, the pulse shape is unchanged

! That was because all normal modes had the same cp! On dispersive medium, the pulse shape must change

! The pulse gets dispersed! Hence the name: dispersion

! Dispersion makes poor media for communication

Dispersion Relation

! Dispersive waves have no solution for < 0! It has a low frequency cut-off at 0

! Phase velocity goes to infinity at cut-off! Wait! Isnt it unphysical? What happened to Relativity?

k

0

wc k =

2 20wc k =

pc

0

wc

2 20

wp

cc

=

Finite-Length Signal

! Phase velocity cp is the speed of pure sine waves! But pure sine waves dont carry information! Relativity forbids superluminal transfer of information

! Lets think about a finite-length pulse

! Problem: this medium cant carry waves with ! We need to make a pulse that does not contain frequencies

below the cut-off

( )f t ( )F

t

T

0

0