= Compliant-wall acoustic cavity
{title2=$k\phi_n+\phi_{tt}=0$}
A wall displacement $\eta$ with spring law $p'=\rho_0k\eta$, combined with linear Bernoulli $p'=-\rho_0\phi_t$ and kinematic condition $\phi_n=\eta_t$, 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 $\beta\tan\beta=-\omega^2b/k$. 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.
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