= Solution
Retain the rigid-wall cosine in $x$ and the lower-wall cosine in $y$:
$$
\phi=A\cos(n\pi x/a)\cos(\beta y/b)e^{-i\omega t}.
$$
The <wave equation> gives $\omega^2=c_0^2[(n\pi/a)^2+(\beta/b)^2]$. The upper-wall condition gives $-k\beta\sin\beta/b-\omega^2\cos\beta=0$, hence
$$
\boxed{\beta\tan\beta=-\omega^2b/k.}
$$
For 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.
\Image[/past-exam-of-the-theoretical-mechanics-course-of-the-university-of-oxford/2009-compliant-wall-spectrum.png]
{title=Real vertical-mode intersections for three horizontal mode numbers}
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:
$$
\beta_j=(j+\tfrac12)\pi+O(M^{-1}).
$$
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 satisfy
$$
\boxed{\omega_{n,j}\sim\pi c_0\sqrt{\frac{n^2}{a^2}+\frac{(j+1/2)^2}{b^2}},\qquad n,j=0,1,2,\ldots.}
$$
The 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.
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