waves

Question 1

1D wave equation

Answer 1

for y = f(x,t)

∂²y/∂t² = v² * ∂²y/∂²x

Question 2

general form of a 1D standing wave

Answer 2

f(x,t) = A*sin(kx-ωt)
k: wave number
A: amplitude

Question 3

wavenumber equation

Answer 3

k= 2π/λ

Question 4

1D wave velocity formula

Answer 4

v = ω/k
k: wavenumber

Question 5

1D wave on a string velocity formula

Answer 5

v = (T/µ)^½
v: velocity
T: tension
µ: density (per unit length)

Question 6

how do waves change at a boundary

Answer 6

frequency remains the same.

when transmitted through a boundary, the velocity and wave number change such that:
k₂/k₁ = v₁/v₂ = n
n: refractive index
k₂: wavenumber of the transmitted wave

Question 7

reflection coefficient equation

Answer 7

R꜀ = R/I = (k₁-k₂)/(k₁+k₂) = (1-n)/(1+n)
R: reflected amplitude
I: incident amplitude
n: refractive index
k₂: reflected wave’s wavenumber

Question 8

Transmission coefficient equation

Answer 8

T꜀ = T/I = 2k₁/(k₁+k₂) = 2/(1+n) = 1+ R꜀

Question 9

general equation of a standing wave

Answer 9

y(x,t) = Asin(kx)cos(ωt)

Question 10

the normal mode equations for a standing wave on a string

Answer 10

n is an integer (±1 , ±2, etc)

kₙ = nπ/L
ωₙ = vnπ/L

k: wavenumber
v: wave velocity
L: total string length