Chapter 5 - Transverse Wave Motion

Question 1

Name the three distinct wave velocities associated with transverse waves.

Answer 1

1. The particle velocity: Which is the simple harmonic velocity of the oscillator about is equilibrium position. (∂y/∂t)
2. The wave or phase velocity: The Velocity with which planes of equal phase, crests or throughs, progress through the medium. (∂x/∂t)
3. The group velocity: A number of waves of different frequencies, wavelengths and velocities may be superimposed to form a group.

Question 2

Give the wave equation.

Answer 2

∂²y = 1 ∂²y

∂x² c² ∂t²

Where

c = T/p, (Tension on string/linear density)

Question 3

What is the solution of the wave equation?

Answer 3

y = a sin(ωt – ϕ) = a sin (2π/λ) (ct – x)

Where:

1. ω = 2πc/λ
2. ϕ = 2πx/λ (The phase lag between the oscillator at x = 0 and x)
3. λ is defined as the wavelength (Distance between x when the phase difference in 2π rad).
4. Note that c can also be written as c = vλ.(The product of frequency and wavelength)

Question 4

Give 4 equivalent expressions for y = f(ct-x) (Could be sine of cosine)

Answer 4

y = a sin (2π/λ) (ct-x)

y = a sin (2π) (vt – x/λ)

y = a sin ω(t – x/c)

y = a sin (ωt – kx)

y = ae^{i(ωt – kx)}

Where k = 2π/λ = ω/c which is called the wave number.

Question 5

What is the wave number?

Answer 5

The wave number is:

k = 2π/λ = ω/c

Question 6

Define the impedance of a string.

Answer 6

Z = transverse force = F

     transverse velocity       v

OR

Z = T/c = pc (Since T = pc²)

Question 7

What would happen when a wave travelling along a string with impedance Z₁ travel across a section with impedance Z₂?

Answer 7

Part of the wave will continue in the region of impedance Z₂ and part would be relected into the region Z_1.

Question 8

1. What is the reflection coefficient amplitude
2. Give the equation of the reflection coefficient amplitude
3. What is the transmission coefficient amplitude
4. Give the equation od the transmission coefficient amplitude

Answer 8

1. The reflection coefficient amplitude is the ratio of the reflected wave amplitude to the incident wave amplitude.
2. B₁/A₁ = (Z₁-Z₂)/(Z₁ + Z₂)
3. The transmission coefficient amplitude is the ration of the transmitted wave amplitude to the incident wave amplitude.
4. A₂/A₁ = (2Z₁)/((Z₁ + Z₂)

Question 9

1. Explain what would happen when the end of the string is fixed.
2. Explain what would happen when the end of the string is loose.

Answer 9

1. If the end of the string is fixed, then Z₂ = ∞, so that B₁/A₁ = -1, thus the wave is completely reflected but has a phase change of π.
2. If the end of the string is loose, then Z₂ = 0, so that B₁/A₁ = 1 and A₂/A₁ = 2, which explains the ‘flick’ of the loose end. (double the amplitude).

Question 10

1. What is the reflected intensity coefficient.
2. Give the equation of the reflected intensity coefficient.
3. What is the transmitted intensity coefficient.
4. Give the equation of the transmitted intensity coefficient.

Answer 10

1. The reflected intensity coefficient is the ratio of the reflected wave energy to the incident wave energy.
2. Reflected energy = Z₁B₁² = (B₁/A₁)² = [(Z₁ – Z₂)/(Z₁ + Z₂)]²

Incident energy Z₁A₁²

3. The transmitted intensity coefficient is the ration of the transmitted wave energy to the incident wave energy.
4. Transmitted energy = Z₂A₂² = (2Z₁Z₂)/(Z₁ + Z₂)²

Incident energy Z₁A₁²

Question 11

Describe how you would math two impedances of Z₁ and Z₃ for a wave of wavelength λ.

Answer 11

The impedances are mathed when a object with impedance:

Z₂ = sqrt(Z₁Z₃)

with a length of

l = λ\4

is inserted between Z₁ and Z₃.

This is true for waves in all media.

Question 12

How is the normal mode frequencies or modes of vibration defined on a string of length l?

At which points will the string remain at rest?

Answer 12

sin (ω_nx/c) = sin (nπx/l)

The string will remain at rest where:

sin (ω_nx/c) = sin (nπx/l) = 0

or

nπx/l = rπ ( r = 0,1,2,3,…,n)

Question 13

1. When will a standing wave ratio be applicable.
2. Give the equation for the standing wave ration.
3. How can the reflection coefficient be calculated from the minumum and maximum amplitudes.

Answer 13

1. If energy is lost between the incident and reflected waves, then the amplitudes will not completely cancel resulting in a standing wave. (no nodes).
2. Standing wave ratio = (A₁ + B₁)/(A₁ - B₁) = (1 +r)(1-r), where r is the reflection coefficient (B₁/A₁).
3. r = B₁/A₁ = (SWR - 1)/(SWR + 1)

Question 14

1. Give the equation of the observed frequency when the source is moving and the observer stationary.
2. Give the equation of the observed frequency when the observer is moving and the source stationary.
3. Give the combination if both observer and source are moving.

Answer 14

1. v’ = vc/(c-s)
2. v’’ = v(c-o)/c
3. v’’’ = v(c-o)/(c-s)