Minnaert resonance

The Minnaert resonance[1] is the acoustic resonance frequency of a single bubble in an infinite domain of water (neglecting the effects of surface tension and viscous attenuation). The resonant frequency is given by

${\displaystyle f={\cfrac {1}{2\pi a}}\left({\cfrac {3\gamma ~p_{A}}{\rho }}\right)^{1/2}}$

where ${\displaystyle a}$ is the radius of the bubble, ${\displaystyle \gamma }$ is the polytropic coefficient, ${\displaystyle p_{A}}$ is the ambient pressure, and ${\displaystyle \rho }$ is the density of water. This formula can also be used to find the resonant frequency of a bubble cloud with ${\displaystyle a}$ as the radius of the cloud and ${\displaystyle \rho }$ the difference between the density of water and the bulk density of the cloud. For a bubble in water at standard pressure ${\displaystyle (p_{A}=100~{\rm {kPa}},~\rho =1000~{\rm {kg/m^{3}}})}$, this equation reduces to ${\displaystyle fa\approx 3.26~{\rm {m/s}}}$, where ${\displaystyle f~}$ is the resonant frequency of the bubble. The Minnaert formula assumes an ideal gas, however it can be easily modified to account for deviations from real gas behavior by accounting for the gas compressibility factor,[2] or the gaz Bulk modulus ${\displaystyle K=\rho _{g}c_{g}^{2}}$ :

${\displaystyle f={\cfrac {1}{2\pi a}}\left({\cfrac {3K}{\rho }}\right)^{1/2}}$

${\displaystyle \rho _{g}}$ and ${\displaystyle c_{g}^{2}}$ being respectively the density and the speed of sound in the bubble.