8. Waves and Sound

Periodic Motion

Periodic displacement and return to midpoint generating a sinusoidal curve (Pendulums, waves, spring)

Amplitude

Extend of displacement (from y = 0 to top of peak)

Period (T)

Time that separates adjacent peaks

Frequency (f)

1/T with units Hz

Phase difference

Two objects in periodic motion with the same frequency but are out of sync (peaks do not match)

At peak of motion, energy is:

All potential energy (mgh or ½kx²)

At equilibrium point of motion, energy is:

All kinetic energy (½mv²)

Restoring Force

In periodic motion, force that pulls object back to equilibrium point from each peak

Examples of restoring forces:

1. F = mg (pendulum)

2. F = -kx (spring)

As period increases…

Frequency decreases

T_spring =

2π*sqrt(m/k)

For spring, frequency increasing as…

Mass decreases and stiffness increases

T_pendulum =

2π*sqrt(L/g)

For pendulum, frequency increases as…

Length decreases (gravity is constant so g isn’t relevant)

Mechanical Waves

Involve physical motion of particles propagating through space

Transverse Waves

Displacement of particles is perpendicular to the direction the wave propagates (Ex. light)

Longitudinal Waves

Displacement of particles is parallel to the direction the wave propagates (pushing-pulling)

Wavelength

Spatial interval over which a waveform repeats itself (connecting two crests or area of compression)

Propagation Speed

Speed with which a wave propagates through space, depending on medium (v)

v =

λf

E =

hf = hv = hc/v

Does frequency change in new medium?

No, frequency doesn’t change, only wavelength/velocity does

As velocity increases…

Wavelength increases

Velocity vs. Frequency

Velocity is the speed the wave propagates, while frequency corresponds to peaks and cycles

Interference

Interactions between multiple propagating waves in the same space

Constructive Interference

Overlapping waves have amplitudes in the same directionality (add together)

Destructive Interference

Overlapping waves have amplitudes with opposite directionalities (subtract); can cancel out completely

How is sound sensed?

In terms of pressure, due to being longitudinal waves, it bounces off reflective surfaces

P_sound =

F/A

V_sound =

sqrt(B/ρ)

B = Bulk modulus (solids are more bulky)

What is sound affected by?

Mostly Bulk’s Modulus, meaning sound travels faster reflecting off solids (also in lower densities)

Intensity of Sound (I)

Measure of power delivered by sounds over area (W/m²)

Intensity vs. Loudness

Loudness is perceived, intensity is measurable

dB =

10log(I/I₀)

I₀ = 1 × 10^-12 W/m²

Distance and Sound

Sound decreases as distance increases a square because sound propagates in spheres which have area of 4πr²

Sound affected by frequency and amplitude

• Amplitude affects how we perceive loudness

• Frequency affects how we perceive pitch or quality

Doppler Effect

Effect when both sound source and observer are moving (shock waves/ultrasound); change in frequency proportional to velocity of source and observer

Doppler Effect Equation

f’ = f₀(v_sound + v_observer)/(v_sound + v_source)
f₀ = frequency without motoin

Signs matter, think common-sense:

If frequency decreases, numerator must be less (observer neg.)

If frequency increases, denominator must be less (source neg.)

Shock Wave

Doppler effect in which the speed of object is close to speed of light, constructive interference occurs, creating sudden high-pressure gradient

Ultrasound

Emitting sonic waves (above perceivable frequency) using the time it takes to bounce back to image structures

Standing Waves

Stable wave-like product due to stable patterns of interference among waves propagating in opposite directions

Nodes

Points of standing wave with 0 displacement

Antinodes

Points of maximum displacement in standing waves

Examples of standing waves

1. Taut string (ends are nodes, middle is antinode)

2. Pipes

More antinodes means…

Higher frequency and lower wavelength

First Harmonic

String with only ONE antinode, with the lowest possible frequency (fundamental frequency)

Wavelength for String Waves:

λ = (2L)/n

n = harmonic # (how many antinodes)

Pipe Harmonics

Open ends are antinodes, closed ends are nodes; harmonic number corresponds to nodes

Wavelength for Pipe Harmonics

λ = 4L/n_odd

n = # ndoes