Waves Flashcards
What do waves do?
Transport momentum and energy without transporting matter
Transverse Wave
Definition
Disturbance is perpendicular to the direction of propagation
E.g. Waves on a string, EM waves
Longitudinal Wave
Definition
Disturbance is parallel to the direction of propagation
E.g. Sound, primary seismic
What does the speed of a transverse wave on a string depend on?
String tension
Mass per unit length
- the greater the tension, the faster the Wave
- Wave velocity is greater in a lighter string
Travelling Wave
Definition
A wave pulse that has constant shape but moves with speed v
Wave on a String
Wave Function
- string lies on the X axis at equilibrium
- the function y(x,t) defines the shape of the string at a given instant
General form of the wave function for a travelling wave on a string
y(x,t) = g(x-vt)
For a wave moving to the right with speed v
The Wave Equations
d²y/dx² = μ/T d²y/dt²
v² = μ/T
T = tension v = velocity μ = mass per unit length
Solutions to the Wave Equation
Stationary Wave
g(x) = Asin(2π/λ * x)
Solutions to the Wave Equation
Harmonic Wave Moving to the Right
g(x) = Asin(2π/λ(x-vt))
Harmonic Wave
A sinusoidal wave
Solution to the Wave Equation
Harmonic Wave
y = Asin(kx-ωt)
This assumes y=0 at x=0 and t=0
For more general initial conditions:
y = Asin(kx-ωt+d)
Kinetic Energy of a Wave Travelling Along a String
KE = 1/2 μΔxω²A²cos²(kx-ωt)
Elastic Potential Energy of a Wave Travelling Along a String
For a wave to move along a string, the wave has to stretch
EPE = 1/2 μΔxω²A²cos²(kx-ωt)
Energy of a Wave Travelling Along a String
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
Power
Definition
Energy transmitted per unit time
Power of a Wave on a String
Equations
Pav = 1/2 μω²A²v
Pmax = 2Pav
Maximum Segment Velocity in the y Direction
Vmax = ωA
Wave Impedance
Definition
Velocity response to a driving force
Large impedance means a small velocity response
What does wave impedance depend on?
String tension and mass
Wave Impedance
Equation
Z = F/v
Reflection Coefficient
r = (Z1 -Z2) / (Z1 + Z2)
Transmission Coefficient
t = 2Z1 / (Z1 + Z2)
When does complete transmission occur?
Only if impedances are matched
Transmission and Reflection
Fixed endpoint
Z2 = infinity
r = -1
t = 0
Reflected Wave has reverse direction and opposite sign (negative amplitude)
Transmission and Reflection
Freely movable end point (e.g. End of string)
Z2 = 0
r = 1
Reflected Wave is the right way up
Power Reflection and Transmission
Pr/ Pi = r² = (Z1-Z2 / Z1+Z2)²
Principle of Superposition
- 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
Reflection of Harmonic Waves
- 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