Waves and Acoustics

Question 1

The general equation for a harmonic oscillator (Sec. 1) is:

Answer 1

m d²ψ/dt² = −sψ − b dψ/dt + F₀ cos ω_f t.

where m is the mass of the oscillator, s is a stiffness (and gives the restoring force), b is a resistance or damping and the driving force F₀ oscillates at frequency ω_f.

Question 2

Solution of the general equation:

Answer 2

Acos(wt+phi)

Question 3

A simple harmonic oscillator will respond at a frequency:

Answer 3

ω = ω₀ = sqrt(s/m)

Question 4

A driven harmonic oscillator will respond at:

Answer 4

the driving frequency ω_f in the steady state

Question 5

The impedance is defined as

Answer 5

the amplitude of the driving force divided by the complex amplitude of the oscillator velocity

Question 6

Two (or more) oscillations can be

Answer 6

added to give a resulting oscillation

Question 7

With the same frequency, the resultant can be found using

Answer 7

a phasor diagram or complex exponential arithmetic

Question 8

• With different frequencies, the phenomenon of ____ is found. What is the equation?

Answer 8

ψ(t) = A cos ω₁t + A cos ω₂t = 2A cos ωt cos ∆ωt.

where ω = (ω₁ + ω₂)/2 and ∆ω = (ω1 − ω₂)/2

Question 9

Normal modes (Sec. 1.5) are

Answer 9

collective, harmonic motions of coupled oscillators

Question 10

By considering combinations of the oscillators (for two, the sum and difference motions) we find

Answer 10

simple harmonic solutions

Question 11

The wave equation (Sec. 2) is

Answer 11

where c is the speed of points of constant phase, or phase velocity.

Question 12

Speed of a wave on a stretched string

Answer 12

c = sqrt(T / mu) with T the tension and µ the mass per unit length

Question 13

The most general solution for the wave equation is

Answer 13

ψ(x, t) = f(x − ct) + g(x + ct)

Question 14

When there is periodic motion, we write _____________ with k = ___ and λ (________) and ω =_____ is the angular frequency, f is the ____ and T ______ (or the time interval between two peaks or troughs)

Answer 14

• ψ(x, t) = f(kx − ωt) + g(kx + ωt)
• 2π/λ
• the wavelength (or distance between two peaks or two troughs)
• 2πf = 2π/T
• frequency
• time period

Question 15

What is the solution for ψ(x, t), for a general periodic wave?

Answer 15

Aei^{(kx−ωt+φ)}

Question 16

There is energy associated with a wave: for a stretched string, the potential energy is

Answer 16

1/2 T A²k²sin² (kx − ωt)

Question 17

There is energy associated with a wave: for a stretched string, the kinetic energy is

Answer 17

1/2 µA²ω²sin²(kx − ωt)

Question 18

One of the two solutions ψ =____ or ψ = ____ is called a ______ (Sec. 2.4), with the direction of travel given by ______.

Answer 18

• ψ = f(x − ct)
• ψ = g(x + ct)
• travelling wave
• the sign between x and t

Question 19

• The impedance of a stretched string,

Answer 19

Z₀ = √ T µ

Question 20

To create a wave, a ______ must be applied

Answer 20

driving force F_D = Z₀(∂ψ/∂t)

Question 21

To terminate a wave, a _____ must be applied

Answer 21

damping force or load FL = Z0(∂ψ/∂t)

Question 22

At a boundary between different ______ we can get _______and ________, with R = ______ the reflection coefficient and T = 1 + R the transmission coefficient.

Answer 22

• impedances
• reflection
• transmission
• (Z₁−Z₂)/(Z₁+Z₂)

Question 23

• Standing waves (Sec. 3.4) arise when

Answer 23

a wave is confined to a finite area with free or fixed boundary conditions

Question 24

For a stretched string of length L with fixed ends, we have ψ(x, t) = ______, with k_n =____ for n = 1, 2, 3, . . .

Answer 24

2A sinωt sin k_nx, with k_n = nπ/L

Question 25

Every point on the string moves in phase; the points with zero displacement are ____and the points with maximum displacement are ______

Answer 25

• nodes
• antinodes

Question 26

Longitudinal wave displacement:

Answer 26

displacement ψ in the direction of the wave travel

Question 27

• On an elastic rod, the wave motion consists of

Answer 27

compression and expansion of the rod

Question 28

On a elastic rod, the same wave equation is obeyed, but

Answer 28

with different speeds.

Question 29

For elastic waves on a rod with ________ A, ______ρ and ________ Y , c = ___ and Z₀ = ___

Answer 29

• cross-sectional area
• density
• Young’s modulus
• √Y/ρ
• A √ρY

Question 30

In a fluid with _____ B and _____ρ, c = ____

Answer 30

• bulk modulus
• density
• sgrt(B/ρ)

Question 31

For an intensity (_____) of i₁, the sound level in dB is defined as β = _____, with I₀ = _____

Answer 31

• (power/area)
• 10 log₁₀(I₁/I₀)
• I₀ = 10⁻¹²W/m²

Question 32

If a sound level β₂ is n dB greater than β₁, then I₂ = ______

Answer 32

• (10^n/10 )i₁

Question 33

A moving source and a moving observer will both lead to a change in the frequency observed. Moving observer: f₀ =______ for a wave moving with velocity v and an observer moving with velocity v_O

Answer 33

(1 + v_O/v)f

Question 34

• A moving source and a moving observer will both lead to a change in the frequency observed.

Moving source: ______ for a source moving with velocity v_S

Answer 34

f ₀ = f_v/(v − v_S)

Question 35

• A moving source and a moving observer will both lead to a change in the frequency observed.
• Both moving: _______

Answer 35

f ₀ = f(v + v_O)/(v − v_S)

Question 36

Wave packets can be represented as a

Answer 36

sum of harmonic waves

Question 37

• The carrier wave (the wave with the _________________) moves at the phase velocity, v_p = ___

Answer 37

• the average frequency in beats
• ω/k

Question 38

The envelope (_____________) moves at the group velocity v_g = _____

Answer 38

• slow variation at the difference frequency in beats
• dω/dk

Question 39

We can also write v_g = _______

Answer 39

v_p + kdv_p/dk

Question 40

The relationship between _______ω and ________k is called the _________

Answer 40

• angular velocity
• wavenumber
• dispersion relation

Question 41

For a non-dispersive wave, ω =___

Answer 41

ck