Travelling Waves, Standing Waves and Normal Modes

Question 1

How do you write the superposition of displacement of 2 waves?

Answer 1

u(x, t) = u1(x, t) + u2(x, t)

Question 2

How is a reflected wave different to the incident wave on a hard wall?

Answer 2

incident wave: f(x-vt)
reflected wave: -f(-x-vt)
Wave inverted and travels in opposite direction.

Question 3

What happens to a reflected wave if the boundary is a free end rather than a hard wall?

Answer 3

Incident wave: f(x-vt)
Reflected wave: f(-x-vt)
Wave travels in opposite direction. Amplitude of wave at boundary is 2x displacement.

Question 4

What should you consider for a general case of wave reflection?

Answer 4

A plane wave propagating on a string with an amplitude of 1.

Question 5

What will the amplitude be of the transmitted/reflected waves of this A=1 waves?

Answer 5

Transmitted A = t

Reflected A = r

Question 6

What is the equation for a plane wave with an amlitude of 1 in complex form?

Answer 6

y(x, t) = e^(i*(kx-ωt))

Question 7

What is the displacement at the boundary condition x=0?

Answer 7

ytot (left) = ytot (right)

Question 8

What is the force at boundary conditions x=0?

Answer 8

TL dyL/dx = TR dyR/dx

Question 9

What is the equation for r, the amplitude of the reflected wave at boundary condition x=0?

Answer 9

r = (ZL-ZR)/(ZL+ZR)

Question 10

What is the equation for t, the amplitude of the transmitted wave at the boundary condition x=0?

Answer 10

t = (2ZL)/(ZL+ZR)

Question 11

How can you use the equations for transmitted and reflected amplitudes to check limiting cases?

Answer 11

For hard wall, ZR&raquo_space; ZL

FOr free end, ZR &laquo_space;ZL

Question 12

What is the equation for displacement of standing waves?

Answer 12

utot(x, t) = utot(x) * utot(t)

Question 13

How many wavelengths can a standing wave between two rigid boundaries have?

Answer 13

L = n*λ/2

Question 14

What is an antinode?

Answer 14

Parts of maximum amplitude on a standing wave.

Question 15

What is the equation for angular frequency of a standing wave?

Answer 15

ω = npiv/L = kv

Question 16

How does an organ pipe work?

Answer 16

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.

Question 17

How does a clarinet work?

Answer 17

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.

Question 18

What is the condition for constructive interference in the general case of a cavity?

Answer 18

e^(ikL)re^(ikL)r = 1 = e^(i2npi)

Question 19

What are beats?

Answer 19

When 2 waves of same amplitude and similar frequencies travel in the same direction.

Question 20

How can you find the equation for beat frequency?

Answer 20

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.

Question 21

What is the equation including beat frequency?

Answer 21

utot = 2Acos(kx-ωt)cos(Δkx - Δωt). Second term is the beat frequency.

Question 22

What is the velocity of the beat wave?

Answer 22

v = Δω/Δk

Question 23

What determines if waves are dispersive or non-dispersive?

Answer 23

Dispersive - velocity changes with frequency or wavelength.

Non-dispersive - velocity does not change with frequency or wavelength.

Question 24

What is the idea of group velocity?

Answer 24

Wave packets travel at a certain velocity, but the individual waves within travel at their phase velocity.

Question 25

What is the doppler effect?

Answer 25

When the apparent frequency of a wave changes when the source or observer are moving.

Question 26

What is the equation for the wave in the frame of the medium?

Answer 26

f(x, t) = Ae^(i(kx+ωt)) = Ae^(i((ω/v)x + ωt))

Question 27

How can you convert the equation for the wave in the frame of the medium to the fram of the source and receiver?

Answer 27

Using Galileon transformations. x = xs + us t, where s denotes the source.Can change to r for receiver.

Question 28

What is the equation for frequency in the source and receiver frames?

Answer 28

``` ωs = ω(1+us/v) ωr = ω(1+ur/v) ```

Question 29

What are the 2 limiting cases in the Doppler effect?

Answer 29

1. Stationary receiver, moving source - set ωr = 1 | 2. Stationary source, moving receiver - set ωs = 1

Question 30

What is the equation for the relativistic doppler effect?

Answer 30

ωr/ωs = sqrt((1-u/c)/(1+uc)), where u is the speed of the source in the frame of the observer.