Compliant-wall acoustic cavity ($k{\varphi }_n+{\varphi }_{tt}=0$)

A wall displacement $\eta$ with spring law $p^{\prime }={\rho }_{0}k\eta$, combined with linear Bernoulli $p^{\prime }=-{\rho }_0{\varphi }_t$ and kinematic condition ${\varphi }_n={\eta }_t$, gives the displayed dynamic boundary condition. For a rectangular cavity of width $a$, height $b$, rigid remaining walls and sound speed $c_0$, real vertical modes obey ${\omega }^2=c_0^2[(n\pi /a{)}^2+(\beta /b{)}^2]$ and $\beta \tan \beta =-{\omega }^{2}b/k$. On each interval $((j+1/2)\pi ,(j+1)\pi )$ the tangent equation has a real root. As $c_0^2/(kb)\to \infty$ at fixed mode numbers, these branches tend to $\beta =(j+1/2)\pi$, the pressure-release upper-wall acoustic spectrum. Imaginary vertical exponents can describe additional branches and are not counted by this real-root argument.