Key Equations

Key Equations

Pressure of a sound wave	$\text{Δ}P=\text{Δ}{P}_{\text{max}}\text{sin}\left(kx\mp \omega t+\varphi \right)$
Displacement of the oscillating molecules of a sound wave	$s\left(x,t\right)=s_{\text{max}}\text{cos}\left(kx\mp \omega t+\varphi \right)$
Velocity of a wave	$v=f\lambda$
Speed of sound in a fluid	$v=\sqrt{\frac{B}{\rho }}$
Speed of sound in a solid	$v=\sqrt{\frac{Y}{\rho }}$
Speed of sound in an ideal gas	$v=\sqrt{\frac{\gamma RT}{M}}$
Speed of sound in air as a function of temperature	$v=331\frac{\text{m}}{\text{s}}\sqrt{\frac{T_{\text{K}}}{273\text{K}}}=331\frac{\text{m}}{\text{s}}\sqrt{1+\frac{T_{\text{C}}}{273\text{°}\text{C}}}$
Decrease in intensity as a spherical wave expands	$I_2=I_1{\left(\frac{r_1}{r_2}\right)}^2$
Intensity averaged over a period	$I=\frac{\langle P\rangle }{A}$
Intensity of sound	$I=\frac{{\left(\text{Δ}{p}_{\text{max}}\right)}^2}{2\rho v}$
Sound intensity level	$\beta \left(dB\right)=10{\text{log}}_{10}\left(\frac{I}{I_0}\right)$
Resonant wavelengths of a tube closed at one end	${\lambda }_n=\frac{4}{n}L,n=1,3,5\text{,…}$
Resonant frequencies of a tube closed at one end	$f_n=n\frac{v}{4L},n=1,3,5\text{,…}$
Resonant wavelengths of a tube open at both ends	${\lambda }_n=\frac{2}{n}L,n=1,2,3\text{,…}$
Resonant frequencies of a tube open at both ends	$f_n=n\frac{v}{2L},n=1,2,3\text{,…}$
Beat frequency produced by two waves that differ in frequency	$f_{\text{beat}}=\left|f_2-f_1\right|$
Observed frequency for a stationary observer and a moving source	$f_{\text{o}}=f_{\text{s}}\left(\frac{v}{v\mp v_{\text{s}}}\right)$
Observed frequency for a moving observer and a stationary source	$f_{\text{o}}=f_{\text{s}}\left(\frac{v\pm v_{\text{o}}}{v}\right)$
Doppler shift for the observed frequency	$f_{\text{o}}=f_{\text{s}}\left(\frac{v\pm v_{\text{o}}}{v\mp v_{\text{s}}}\right)$
Mach number	$M=\frac{v_s}{v}$
Sine of angle formed by shock wave	$\text{sin}\theta =\frac{v}{v_s}=\frac{1}{M}$