Waves in 2D

Question 1

What heppends in the simplest case when adding two components

Answer 1

y1 + y2 = 2A [cos {(w1-w2)/2 t - (k1-k2)/2 x}][cos {(w1+w2)/2 t - (k1+k2)/2 x}]

Question 2

Waves in 2D restoring force

Answer 2

T𝛿y𝛿x ∂^2z/∂x^2 + T𝛿y𝛿x ∂^2z/∂y^2

=> T𝛿y𝛿x ∂^2z/∂x^2+ T𝛿y𝛿x ∂^2z/∂y^2 = ρ𝛿x𝛿y ∂^2z/∂t^2

Question 3

For an infinite membrane

Answer 3

z = Aexp{i(ωt - k͟·r)}

Question 4

For a finite membrane

Answer 4

z = Aexp{i(ωt - (kₓx + kᵧy)} + Bexp{i(ωt - (kₓx - kᵧy)}

z = 2Asin{ωt - kₓx}sin{nπy/b}

Question 5

cut off frequency

Answer 5

fmin = ωmin/2π

where ωmin = πc/b

Question 6

group velocity

Answer 6

vg = c √(1 - ωmin^2/ω^2)

Question 7

time taken

Answer 7

t = x/vg

Question 8

c =

Answer 8

√Y/ρ

Question 9

fmax =

Answer 9

1/π √K/m

Question 10

K =

Answer 10

YA/a

and for one atom per lattice point

K = a^2 Y/a

Question 11

A 2D membrane diagram

Answer 11

see notes

Question 12

How an equation for waves transmitted on the membrane can be determined.

Answer 12

1) boundary conditions

2) initial membrane conditions

boundary conditions will determine whether we need sin or cos functions for each dimension, including time

Question 13

cut - off frequency

Answer 13

k^2 = kx^2 + ky^2

for a wave to propogate kx must be real

k = w/c and ky = pi/b

therefore wmin = pi c /b

Question 14

2D Membrane stretched between two parallel supports diagram

Answer 14

for a finite membrane diagram