Waves and Sound

transverse waves

direction of particle oscillation is perpendicular to the propagation of the wave

longitudinal waves

particles of the wave oscillate parallel to the direction of propagation; wave particles are oscillating in the direction of energy transfer

propagation speed

v = fλ

f = frequency

λ = wavelength

Period

1/f

frequency (f)

number of wavelengths passing a fixed point per second

angular frequency

ω = 2πf = 2π/T

measured in radians per second

principle of superposition

when waves interact with each other, the displacement of the resultant wave at any point is the sum of the displacements of the two interacting waves

constructive interference

when waves are perfectly in phase, the displacements always add together and the amplitude of the resultant is = to the sum of the amplitudes of the two waves

destructive interference

waves are are perfectly out of phase, the displacements always counteract each other and the amplitude of the resultant wave is the difference between the amplitudes of the interacting waves

traveling wave

Traveling waves are a type of wave that propagates through a medium or space, carrying energy from one place to another without a net movement of the medium itself. These waves maintain their shape and speed as they travel.

timbre

quality of the sound that is determined by the natural frequency or frequencies of the object

audible range for humans

20 Hz - 20,000 Hz`

damping/attenuation

decrease in amplitude of a wave caused by an applied or nonconservative force

speed of sound

v = sqrt(B/ρ)

B = bulk modulus

ρ = density of the medium

solid > liquid > gas

pitch

perception of the frequency of sound

lower frequency sounds have lower pitch and higher frequency sounds have higher pitches

infrasonic waves

sound waves with frequencies below 20 Hz

ultrasonic

sound waves with frequencies above 20,000 Hz

Doppler effect

difference between the actual frequency of a sound and its perceived frequency when the source of the sound and the sound’s detector are moving relative to one another

f’ = f (v ± v_D) / (v∓ v_S)

f’ = perceived frequency

f = actual emitted frequency

v = speed of sound in the medium

v_D = speed of the detector

v_S = speed of the source

sign convention:

• top sign = toward

• bottom sign = away

Intensity

average rate of energy transfer per area across a surface that is perpendicular to the wave

Intensity is proportional to the square of the amplitude

I = P/A

P = power

A = area

sound level

β = 10 log (I/I₀)

I₀ = threshold of hearing =1 × 10^-12

β_f = β_i + 10log(I_f/I₀)

harmonic

λ = 2L/n

f = nv/2L

n = positive nonzero integer

fundamental frequency

lowest frequency (longest wavelength) of a standing wave that can be supported in a given length of a string

first harmonic

open pipe

support an antinode

open on both ends

the # of nodes = which harmonic the string is in

L = nλ/2

closed pipe

supports a node

closed at one end

closed end = node

open end = antinode

harmonic is = to ¼ of a wavelength

λ = 4L/n

f = nv/4L

n = any odd integers (1,3,5,…)