Waves I Formulas

Question 1

Sinusoidal Waves

Answer 1

y( x, t) = ym sin (kx - ωt)

note: ym is the amplitude
k is the angular wave number
kx - ωt is the phase
ω is the angular frequency

Question 2

wavelength relationship to angular wave number (k)

Answer 2

k = 2π/λ

Question 3

Relationship of period and frequency of the wave to ω

Answer 3

ω/2π = f = 1/T

Question 4

wave speed (v) relation to k, f, T, ω, λ

Answer 4

v = ω/k = λ/T = λf

Question 5

Equation of a travelling wave

Answer 5

y(x, t) = h (kx ± ωt)

Question 6

Wave speed on stretch string

Answer 6

v = sqrt( tension/linear density)

note: density symbol = μ

Question 7

Average power (sinusoidal wave on a stretched string)

Answer 7

Pavg = 1/2 μvω^2y^2

Question 8

Interference of waves

Answer 8

y’(x, t) = [2ycos 1/2 Φ] sin (kx - ωt + 1/2 Φ)

note:
if Φ = 0 the waves are exactly in phase; interference is fully constructive

if Φ = πrad, waves are exactly out of phase; interference is fully destructive

Question 9

Standing waves (string with fixed ends)

Answer 9

y’ (x, t) = [2y sin kx] cos ωt

Question 10

Resonance frequency for a stretched string w/ fixed ends

Answer 10

f = v/λ = nv/2L

for n = 1,2,3
note: n = 1 [first harmonic], n = 2 [second harmonic]

Question 11

Speed of sound wave

Answer 11

v = sqrt(B/ρ)

note: B = bulk modulus

Question 12

Longitudinal displacement amplitude

Answer 12

s = sm cos (kx - ωt)

note: sm = displacement amplitude

Question 13

equilibrium

Answer 13

k = 2π/λ ; ω = 2πf

Question 14

pressure change caused by the wave

Answer 14

Δp = Δpm sin (kx - ωt)

Question 15

pressure amplitude

Answer 15

Δpm = (vρω)sm

Question 16

Interference two sound waves

Answer 16

Φ = (ΔL/λ) × 2π

Question 17

what Φ does fully constructive interference occur?

Answer 17

Φ = m(2π)

note: for m = 0, 1, 2,….

and ΔL/λ = 0, 1, 2, …

Question 18

what Φ does fully destructive interference occur?

Answer 18

Φ = (2m + 1)π

note: for m = 0, 1, 2,….
Φ is an odd multiple of π

and ΔL/λ = 0.5, 1.5, 2.5, …

Question 19

Sound Intensity

Answer 19

I = P/A

P is the time rate of energy transfer (power) of the sound wave, A is the area

Question 20

Relationship of Intensity to the displacement amplitude (sm)

Answer 20

I = 1/2 ρvω^2s^2

Question 21

Intensity at a distance r from a point source that emits sound waves of power Ps

Answer 21

I = P/(4πr^2)

Question 22

Sound level in decibels (β)

Answer 22

β = (10dB) log I/I0

Note: I0 = 10^(-12) W/m^2

Question 23

Standing wave patterns (pipe open at both ends)

Answer 23

f = v/λ = nv/2L

n = 1, 2, 3…

Question 24

Standing wave patter (pipe closed on one end and open at the other)

Answer 24

f = v/λ = nv/4L

n = 1, 3, 5…

Question 25

General doppler effect

Answer 25

f’ = f × [( v ± vp ) / ( v ± vs )]

note:
vp is the speed of the detector relative to the medium
vs is that of the source
v is the speed of sound in the medium

The signs are chosen such that f’ tends to be greater for motion toward and less for motion away

Question 26

Mach cone angle (shock wave)

Answer 26

sin θ = v/vs

*happens when the speed of a source relative to the medium exceeds the speed of sound in the medium

• doppler no longer applies