Waves

Question 1

What do waves do?

Answer 1

Transport momentum and energy without transporting matter

Question 2

Transverse Wave

Definition

Answer 2

Disturbance is perpendicular to the direction of propagation

E.g. Waves on a string, EM waves

Question 3

Longitudinal Wave

Definition

Answer 3

Disturbance is parallel to the direction of propagation

E.g. Sound, primary seismic

Question 4

What does the speed of a transverse wave on a string depend on?

Answer 4

String tension
Mass per unit length

• the greater the tension, the faster the Wave
• Wave velocity is greater in a lighter string

Question 5

Travelling Wave

Definition

Answer 5

A wave pulse that has constant shape but moves with speed v

Question 6

Wave on a String

Wave Function

Answer 6

• string lies on the X axis at equilibrium

- the function y(x,t) defines the shape of the string at a given instant

Question 7

General form of the wave function for a travelling wave on a string

Answer 7

y(x,t) = g(x-vt)

For a wave moving to the right with speed v

Question 8

The Wave Equations

Answer 8

d²y/dx² = μ/T d²y/dt²

v² = μ/T

T = tension
v = velocity
μ = mass per unit length

Question 9

Solutions to the Wave Equation

Stationary Wave

Answer 9

g(x) = Asin(2π/λ * x)

Question 10

Solutions to the Wave Equation

Harmonic Wave Moving to the Right

Answer 10

g(x) = Asin(2π/λ(x-vt))

Question 11

Harmonic Wave

Answer 11

A sinusoidal wave

Question 12

Solution to the Wave Equation

Harmonic Wave

Answer 12

y = Asin(kx-ωt)
This assumes y=0 at x=0 and t=0

For more general initial conditions:
y = Asin(kx-ωt+d)

Question 13

Kinetic Energy of a Wave Travelling Along a String

Answer 13

KE = 1/2 μΔxω²A²cos²(kx-ωt)

Question 14

Elastic Potential Energy of a Wave Travelling Along a String

Answer 14

For a wave to move along a string, the wave has to stretch

EPE = 1/2 μΔxω²A²cos²(kx-ωt)

Question 15

Energy of a Wave Travelling Along a String

Answer 15

KE and PE of the string segment are equal

E = μΔxω²A²cos²(kx-ωt)

But this energy is constantly varying with t and x so we average the energy over 1 Period

Eav = 1/2 μΔxω²A²

Question 16

Power

Definition

Answer 16

Energy transmitted per unit time

Question 17

Power of a Wave on a String

Equations

Answer 17

Pav = 1/2 μω²A²v

Pmax = 2Pav

Question 18

Maximum Segment Velocity in the y Direction

Answer 18

Vmax = ωA

Question 19

Wave Impedance

Definition

Answer 19

Velocity response to a driving force

Large impedance means a small velocity response

Question 20

What does wave impedance depend on?

Answer 20

String tension and mass

Question 21

Wave Impedance

Equation

Answer 21

Z = F/v

Question 22

Reflection Coefficient

Answer 22

r = (Z1 -Z2) / (Z1 + Z2)

Question 23

Transmission Coefficient

Answer 23

t = 2Z1 / (Z1 + Z2)

Question 24

When does complete transmission occur?

Answer 24

Only if impedances are matched

Question 25

Transmission and Reflection

Fixed endpoint

Answer 25

Z2 = infinity
r = -1
t = 0
Reflected Wave has reverse direction and opposite sign (negative amplitude)

Question 26

Transmission and Reflection

Freely movable end point (e.g. End of string)

Answer 26

Z2 = 0
r = 1
Reflected Wave is the right way up

Question 27

Power Reflection and Transmission

Answer 27

Pr/ Pi = r² = (Z1-Z2 / Z1+Z2)²

Question 28

Principle of Superposition

Answer 28

• obeyed by linear waves
• if two or more waves are moving through a medium, the resultant wave function at any point is the sum of the wave functions of the individual waves

Question 29

Reflection of Harmonic Waves

Answer 29

• if a harmonic wave travels on a string fixed at the end, it is reflected, incident and reflected waves superimpose
• 2 sinusoidal waves with same amplitude, frequency and wavelength but travelling in opposite directions

y = 2A0 sin(kx) cos(wt)

Every part of the string vibrates in SHM with the same f and phase unlike a travelling harmonic wave where amplitude is constant but phase is different