= Solution
The applied pressure changes the dynamic condition to $\phi_t+g\eta=-p/\rho$. For its specified spatial mode, the two amplitude equations become
$$
\dot A=\lambda B,\qquad
\dot B+gA=-\varepsilon gh\sin\Omega t,\qquad
\ddot A+\omega^2A=-\varepsilon h\omega^2\sin\Omega t.
$$
When $\Omega\ne\omega$, the corresponding particular oscillation is
$$
\boxed{\eta=-\frac{\varepsilon h\omega^2}{\omega^2-\Omega^2}\sin\Omega t
\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{b}.}
$$
Free oscillations with amplitudes $C\cos\omega t+D\sin\omega t$ may be added. If the mode starts from rest at $t=0$, its total amplitude instead is
$$
A(t)=-\frac{\varepsilon h\omega^2}{\omega^2-\Omega^2}
\left(\sin\Omega t-\frac{\Omega}{\omega}\sin\omega t\right).
$$
Near <resonance>, the response grows too large for the <linearization>. At exact resonance the rest-start limit is $A=(\varepsilon h/2)(\omega t\cos\omega t-\sin\omega t)$, with secular growth. Dissipation or nonlinear effects would be required to limit it.
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