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

m d^{2}ψ/dt^{2} = −sψ − b dψ/dt + F_{0} 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 F_{0} oscillates at frequency ω_{f}.

Solution of the general equation:

Acos(wt+phi)

A simple harmonic oscillator will respond at a frequency:

ω = ω_{0} = sqrt(s/m)

A driven harmonic oscillator will respond at:

the driving frequency ω_{f} in the steady state

The impedance is defined as

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

Two (or more) oscillations can be

added to give a resulting oscillation

With the same frequency, the resultant can be found using

a phasor diagram or complex exponential arithmetic

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

ψ(t) = A cos ω_{1}t + A cos ω_{2}t = 2A cos ωt cos ∆ωt.

where ω = (ω_{1} + ω_{2})/2 and ∆ω = (ω1 − ω_{2})/2

Normal modes (Sec. 1.5) are

collective, harmonic motions of coupled oscillators

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

simple harmonic solutions

The wave equation (Sec. 2) is

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

Speed of a wave on a stretched string

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

The most general solution for the wave equation is

ψ(x, t) = f(x − ct) + g(x + ct)

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)

###
- 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

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

Aei^{(kx−ωt+φ)}

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

1/2 T A^{2}k^{2}sin^{2} (kx − ωt)

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

1/2 µA^{2}ω^{2}sin^{2}(kx − ωt)

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

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

• The impedance of a stretched string,

Z_{0} = √ T µ

To create a wave, a ______ must be applied

driving force F_{D} = Z_{0}(∂ψ/∂t)

To terminate a wave, a _____ must be applied

damping force or load FL = Z0(∂ψ/∂t)

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

###
- impedances
- reflection
- transmission
- (Z
_{1}−Z_{2})/(Z_{1}+Z_{2})

_{1}−Z_{2})/(Z_{1}+Z_{2})• Standing waves (Sec. 3.4) arise when

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

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

2A sinωt sin k_{n}x, with k_{n} = nπ/L

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

###
- nodes
- antinodes

Longitudinal wave displacement:

displacement ψ in the direction of the wave travel

• On an elastic rod, the wave motion consists of

compression and expansion of the rod

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

with different speeds.

For elastic waves on a rod with ________ A, ______ρ and ________ Y , c = ___ and Z_{0} = ___

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

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

###
- bulk modulus
- density
- sgrt(B/ρ)

For an intensity (_____) of i_{1}, the sound level in dB is defined as β = _____, with I_{0} = _____

###
- (power/area)
- 10 log
_{10}(I_{1}/I_{0})
- I
_{0} = 10^{−12}W/m^{2}

_{10}(I_{1}/I_{0})_{0}= 10^{−12}W/m^{2}If a sound level β_{2} is n dB greater than β_{1}, then I_{2} = ______

###
- (10
^{n/10 })i_{1}

^{n/10 })i_{1}A moving source and a moving observer will both lead to a change in the frequency observed. Moving observer: f_{0} =______ for a wave moving with velocity v and an observer moving with velocity v_{O}

(1 + v_{O}/v)f

• 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}

f _{0} = f_{v}/(v − v_{S})

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

• Both moving: _______

f _{0} = f(v + v_{O})/(v − v_{S})

Wave packets can be represented as a

sum of harmonic waves

• The carrier wave (the wave with the _________________) moves at the phase velocity, v_{p} = ___

###
- the average frequency in beats
- ω/k

The envelope (_____________) moves at the group velocity v_{g} = _____

###
- slow variation at the difference frequency in beats
- dω/dk

We can also write v_{g} = _______

v_{p} + kdv_{p}/dk

The relationship between _______ω and ________k is called the _________

###
- angular velocity
- wavenumber
- dispersion relation

For a non-dispersive wave, ω =___

ck