Waves I Formulas Flashcards
Sinusoidal Waves
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
wavelength relationship to angular wave number (k)
k = 2π/λ
Relationship of period and frequency of the wave to ω
ω/2π = f = 1/T
wave speed (v) relation to k, f, T, ω, λ
v = ω/k = λ/T = λf
Equation of a travelling wave
y(x, t) = h (kx ± ωt)
Wave speed on stretch string
v = sqrt( tension/linear density)
note: density symbol = μ
Average power (sinusoidal wave on a stretched string)
Pavg = 1/2 μvω^2y^2
Interference of waves
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
Standing waves (string with fixed ends)
y’ (x, t) = [2y sin kx] cos ωt
Resonance frequency for a stretched string w/ fixed ends
f = v/λ = nv/2L
for n = 1,2,3
note: n = 1 [first harmonic], n = 2 [second harmonic]
Speed of sound wave
v = sqrt(B/ρ)
note: B = bulk modulus
Longitudinal displacement amplitude
s = sm cos (kx - ωt)
note: sm = displacement amplitude
equilibrium
k = 2π/λ ; ω = 2πf
pressure change caused by the wave
Δp = Δpm sin (kx - ωt)
pressure amplitude
Δpm = (vρω)sm
Interference two sound waves
Φ = (ΔL/λ) × 2π
what Φ does fully constructive interference occur?
Φ = m(2π)
note: for m = 0, 1, 2,….
and ΔL/λ = 0, 1, 2, …
what Φ does fully destructive interference occur?
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, …
Sound Intensity
I = P/A
P is the time rate of energy transfer (power) of the sound wave, A is the area
Relationship of Intensity to the displacement amplitude (sm)
I = 1/2 ρvω^2s^2
Intensity at a distance r from a point source that emits sound waves of power Ps
I = P/(4πr^2)
Sound level in decibels (β)
β = (10dB) log I/I0
Note: I0 = 10^(-12) W/m^2
Standing wave patterns (pipe open at both ends)
f = v/λ = nv/2L
n = 1, 2, 3…
Standing wave patter (pipe closed on one end and open at the other)
f = v/λ = nv/4L
n = 1, 3, 5…