 Solution

Retain the rigid-wall cosine in $x$ and the lower-wall cosine in $y$:The wave equation gives ${\omega }^2=c_0^2[(n\pi /a{)}^2+(\beta /b{)}^2]$. The upper-wall condition gives $-k\beta \sin \beta /b-{\omega }^2\cos \beta =0$, henceFor nonzero real $\beta$, eliminating $\omega$ gives $-\tan \beta =M(\beta +{\kappa }_n^2/\beta )$ with $M=c_0^2/(kb)>0$, ${\kappa }_n=n\pi b/a$.

On every interval $((j+1/2)\pi ,(j+1)\pi )$, the left side decreases from positive infinity to zero, while the right side is finite and positive. Continuity gives at least one intersection per interval for each $n$. The functions are analytic between their poles, so roots are isolated there; their union over countably many intervals is a countably infinite set. Reflection gives the redundant negative-$\beta$ solutions. For $n=0$, the limiting formal root $\beta =0$ is the zero-frequency gauge mode, not one of these positive-frequency branches.

The graph uses representative dimensionless values $M=1$ and $b/a=1$; the interval argument applies for all positive values. For fixed $n$ and fixed branch $j$, take $M\to \infty$. The positive right side becomes large, forcing the intersection towards the tangent pole from the right:More precisely, if ${\beta }_j^0=(j+1/2)\pi$, then ${\beta }_j-{\beta }_j^0\sim {\beta }_j^0/[M(({\beta }_j^0{)}^2+{\kappa }_n^2)]$. Therefore the requested real-branch frequencies satisfyThe limiting upper condition is effectively pressure release for these acoustic branches. No claim is made here about imaginary or other complex $\beta$ branches, which were excluded from the requested analysis.