Physics 8.03
Vibrations and Waves
Lecture 11
Fourier Analysis with traveling waves
Dispersion
Last time:
„
Arbitrary motion Î Superposition of ALL
possible normal modes
∞
∞
m =1
n =0
µ
y ( x, t ) = ∑ Am sin (k m x ) cos(ωmt + β m ) + ∑ Bn cos(k n x ) cos(ωnt + β n )
„
Orthogonal functions Î Fourier coefficients
L
2
Am =
L
∫ y ( x , t = 0 ) sin (k x )dx
2
Bn =
L
L
m
0
∫ y ( x , t = 0 ) cos (k x )dx
n
0
Fourier expansion recipé
„
„
Start with superposition of all possible modes
Determine the simplest basis functions using
Boundary conditions Æ [0, L] or [−L/2, L/2] or [−L, L]
„ Symmetry Æ f(−x, 0) = f(x, 0) or f(−x, 0) = −f(x, 0)
„ Initial condition Æ y(x, 0) = 0 or vy (x, 0) = 0
„
„
Determine the Fourier coefficients -- An and/or Bn
Use orthogonality relations with
Initial deformation y(x , t = 0) or
„ Initial velocity vy(x , t = 0)
„
„
Add the time-dependence
„
„
Fourier expansions for traveling waves
What happens if the Fourier components all
travel at slightly different speeds?
„
„
ωn Ÿ v kn Î DISPERSION
Wave equation in dispersive media
„
Phase velocity Æ velocity of a single crest of the
wave with average wave vector, k Î
ω
vp =
„
k
Group velocity Æ velocity of the slow envelope
velocity of energy transport Î
dω
vg =
dk
Corrections/comments
on today’s lecture
„
Formula for approximation of ωm was written
incorrectly on the board; the correct version is
ω 2 = c 2 k m2 (1 + α k m2 ) ⇒ ω = ck m 1 + α k m2


ω ≈ ck m  1 +
„
1

α k m2 
2

Where does the equation for a stiff string come
from?
„
For a derivation, see for example, Fetter and Walecka,
“Theoretical mechanics of Particles and Continua,”
page 221