Key Equations

Pressure of a sound wave	$Δ P = Δ P_{max} sin (k x \mp ω t + ϕ)$
Displacement of the oscillating molecules of a sound wave	$s (x, t) = s_{max} cos (k x \mp ω t + ϕ)$
Velocity of a wave	$v = f λ$
Speed of sound in a fluid	$v = \sqrt{\frac{B}{ρ}}$
Speed of sound in a solid	$v = \sqrt{\frac{Y}{ρ}}$
Speed of sound in an ideal gas	$v = \sqrt{\frac{γ R T}{M}}$
Speed of sound in air as a function of temperature	$v = 331 \frac{m}{s} \sqrt{\frac{T_{K}}{273 K}} = 331 \frac{m}{s} \sqrt{1 + \frac{T_{C}}{273 ° C}}$
Decrease in intensity as a spherical wave expands	$I_{2} = I_{1} {(\frac{r_{1}}{r_{2}})}^{2}$
Intensity averaged over a period	$I = \frac{〈 P 〉}{A}$
Intensity of sound	$I = \frac{{(Δ p_{max})}^{2}}{2 ρ v}$
Sound intensity level	$β (d B) = 10 {log}_{10} (\frac{I}{I_{0}})$
Resonant wavelengths of a tube closed at one end	$λ_{n} = \frac{4}{n} L, n = 1, 3, 5,…$
Resonant frequencies of a tube closed at one end	$f_{n} = n \frac{v}{4 L}, n = 1, 3, 5,…$
Resonant wavelengths of a tube open at both ends	$λ_{n} = \frac{2}{n} L, n = 1, 2, 3,…$
Resonant frequencies of a tube open at both ends	$f_{n} = n \frac{v}{2 L}, n = 1, 2, 3,…$
Beat frequency produced by two waves that differ in frequency	$f_{beat} = \| f_{2} - f_{1} \|$
Observed frequency for a stationary observer and a moving source	$f_{o} = f_{s} (\frac{v}{v \mp v_{s}})$
Observed frequency for a moving observer and a stationary source	$f_{o} = f_{s} (\frac{v \pm v_{o}}{v})$
Doppler shift for the observed frequency	$f_{o} = f_{s} (\frac{v \pm v_{o}}{v \mp v_{s}})$
Mach number	$M = \frac{v_{s}}{v}$
Sine of angle formed by shock wave	$sin θ = \frac{v}{v_{s}} = \frac{1}{M}$