Travelling Waves, Standing Waves and Normal Modes Flashcards
How do you write the superposition of displacement of 2 waves?
u(x, t) = u1(x, t) + u2(x, t)
How is a reflected wave different to the incident wave on a hard wall?
incident wave: f(x-vt)
reflected wave: -f(-x-vt)
Wave inverted and travels in opposite direction.
What happens to a reflected wave if the boundary is a free end rather than a hard wall?
Incident wave: f(x-vt)
Reflected wave: f(-x-vt)
Wave travels in opposite direction. Amplitude of wave at boundary is 2x displacement.
What should you consider for a general case of wave reflection?
A plane wave propagating on a string with an amplitude of 1.
What will the amplitude be of the transmitted/reflected waves of this A=1 waves?
Transmitted A = t
Reflected A = r
What is the equation for a plane wave with an amlitude of 1 in complex form?
y(x, t) = e^(i*(kx-ωt))
What is the displacement at the boundary condition x=0?
ytot (left) = ytot (right)
What is the force at boundary conditions x=0?
TL dyL/dx = TR dyR/dx
What is the equation for r, the amplitude of the reflected wave at boundary condition x=0?
r = (ZL-ZR)/(ZL+ZR)
What is the equation for t, the amplitude of the transmitted wave at the boundary condition x=0?
t = (2ZL)/(ZL+ZR)
How can you use the equations for transmitted and reflected amplitudes to check limiting cases?
For hard wall, ZR»_space; ZL
FOr free end, ZR «_space;ZL
What is the equation for displacement of standing waves?
utot(x, t) = utot(x) * utot(t)
How many wavelengths can a standing wave between two rigid boundaries have?
L = n*λ/2
What is an antinode?
Parts of maximum amplitude on a standing wave.
What is the equation for angular frequency of a standing wave?
ω = npiv/L = kv
How does an organ pipe work?
Has two open ends of a tube, and there are standing waves inside with pi phase change from normal standing waves, so fundamental has 1 node rather than 1 antinode etc.
How does a clarinet work?
One open end and one closed end in a tube. Closed end is a node in displacement, and openend is antinode indisplacement. Only odd nodes are allowed due to this.
What is the condition for constructive interference in the general case of a cavity?
e^(ikL)re^(ikL)r = 1 = e^(i2npi)
What are beats?
When 2 waves of same amplitude and similar frequencies travel in the same direction.
How can you find the equation for beat frequency?
Similar frequencies, so k1 = k-Δk and k2 = k+Δk, and ω1 = ω-Δω and ω2 = ω+Δω. Substitute these into the equation anf use superposition.
What is the equation including beat frequency?
utot = 2Acos(kx-ωt)cos(Δkx - Δωt). Second term is the beat frequency.
What is the velocity of the beat wave?
v = Δω/Δk
What determines if waves are dispersive or non-dispersive?
Dispersive - velocity changes with frequency or wavelength.
Non-dispersive - velocity does not change with frequency or wavelength.
What is the idea of group velocity?
Wave packets travel at a certain velocity, but the individual waves within travel at their phase velocity.
What is the doppler effect?
When the apparent frequency of a wave changes when the source or observer are moving.
What is the equation for the wave in the frame of the medium?
f(x, t) = Ae^(i(kx+ωt)) = Ae^(i((ω/v)x + ωt))
How can you convert the equation for the wave in the frame of the medium to the fram of the source and receiver?
Using Galileon transformations. x = xs + us t, where s denotes the source.Can change to r for receiver.
What is the equation for frequency in the source and receiver frames?
ωs = ω(1+us/v) ωr = ω(1+ur/v)
What are the 2 limiting cases in the Doppler effect?
- Stationary receiver, moving source - set ωr = 1
2. Stationary source, moving receiver - set ωs = 1
What is the equation for the relativistic doppler effect?
ωr/ωs = sqrt((1-u/c)/(1+uc)), where u is the speed of the source in the frame of the observer.
