membranes + dispersion

Question 1

the general solution to waves on a rectangular membrane

Answer 1

f = sin(nπx/a)sin(mπy/a)[Asin(ωt) + Bcos(ωt)]
n and m are different integers
a is the membrane’s width

Question 2

angular frequency and wavenumber of a rectangular membrane

Answer 2

ω = πv/a * (n²+m²)^½

k = [ (nπ/a)² + (mπ/a)² ]^½

Question 3

phase velocity and group velocity equations

Answer 3

for waves in a dispersive medium:

vₚ = ω/k

v(sub g) = dω/dk

Question 4

conditions for normal and anomalous dispersion

Answer 4

normal:
v(sub g) < vₚ
dn/dλ < 0

anomalous:
v(sub g) > vₚ
dn/dλ > 0

Question 5

how would you find the average kinetic or potential energy of an oscillator

Answer 5

KE = ½mv²
so average KE = 1/T * ∫ KE dt (limits are 0 and T)
simply sub in the expression for velocity
use v = dx/dt to calculate it

same for potential energy, U = ½kx²

For the total energy, just sum these two values

Question 6

useful integral trick for time averages

Answer 6

∫ cos²(ωt+δ)dt = ½T (between the limits of 0 and T)

the result is the same for sin²()

Question 7

how do you show the values for the wavelength of a stationary wave

Answer 7

general form:
y(x,t) = Asin(kx)cos(ωt)

apply the given boundary conditions (the string will be taught so at x=0 y=0 and at x= L, y=0. L is the length of the string)

y(L,t) = Asin(kL)cos(ωt) = 0
sin(kL)= 0
kL = nπ
k = nπ/L

k = 2π/λ
hence: 2π/λ = nπ/L
λ = 2L/n

Question 8

when asked to find the dispersion relation, what are you trying to find?

Answer 8

the angular frequency as a function of the wavenumber
ω= ω(k)

OR
vₚ = vₚ(k) can also be called the dispersion relation.

Question 9

derive the relation between the phase and group velocity

Answer 9

vₚ = ω/k
v(sub g) = dω/dk

v(g) = d(kvₚ)/dk = vₚ + k dvₚ/dk
= vₚ + k dvₚ/dλ *dλ/dk

λ = 2π/k, hence: dλ/dk = -2π/k² = -λ/k

∴ v(g) = vₚ - λ dvₚ/dλ