= Solution
The central stratified layer supports upward and downward <internal gravity waves>, whose counterpropagating components form a <standing wave>. The unstratified regions support <evanescent waves>, and the upper disturbance decays as $e^{-k(z-2H)}$. Consequently the central layer is a <stratified internal-wave guide>, rather than a source of propagating energy at infinity.
The largest response occurs at its trapped <normal mode> <frequencies>. They can be specified without solving the forced problem. Set the boundary forcing to zero to find a free <normal mode>. The lower solution is proportional to $\sinh(kz)$, so continuity of <fluid pressure> and vertical <velocity> gives
$$
W'(H)=aW(H),\qquad a=k\coth(kH),\qquad W'(2H)=-kW(2H).
$$
In the central layer write
$$
W\propto\sin\bigl(\ell(z-H)+\delta\bigr),\qquad
\delta=\arctan(\ell/a),\qquad
\ell=k\sqrt{N_2^2/\omega^2-1}.
$$
The upper <Robin boundary condition> then gives the exact trapped-mode condition
$$
\boxed{\ell_nH+\arctan(\ell_n/a)+\arctan(\ell_n/k)=n\pi,\qquad
\omega_n=\frac{N_2k}{\sqrt{k^2+\ell_n^2}},\quad n=1,2,\ldots}.
$$
The left side increases strictly from zero to infinity, so there is one positive root for each $n$; the <frequencies> accumulate at zero. This is constructive phase matching after reflection at both ends. The phase shifts from the <evanescent waves> matter: simply imposing integer half-wavelengths across the stratified layer is generally incorrect.
\b[In the ideal inviscid model, exact resonant forcing has no bounded steady harmonic solution.] The undamped <normal mode> grows secularly under sustained forcing. Weak <viscosity> or other losses would produce large finite peaks near the displayed <frequencies>; the linear approximation eventually fails if the disturbance becomes too large.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-345-trapped-wave.png]
{title=A trapped internal-wave mode}
The illustrated free <normal mode> has an oscillatory central region and <evanescent wave> tails. Its lower tail reaches the fixed zero-displacement boundary; its upper tail decays to infinity.
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