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Let $W$ be the complex vertical velocity amplitude in $w=\mathrm{Re}\{W(z)e^{ikx}\}$. It satisfies $W^{\prime \prime }+m^{2}W=0$, with $W(0)=ikU{\eta }_0$ and $W(H)=0$. Away from resonance, solve these two boundary conditions:The resulting field is a standing vertical superposition of equal-amplitude upward and downward internal gravity waves. Write $W=A_{+}{e}^{imz}+A_{-}{e}^{-imz}$. The rigid lid gives $A_{-}=-A_{+}{e}^{2imH}$, and the bed givesThe upward vertical-velocity amplitude is $|kU{\eta }_0|/[2|\sin (mH)|]$. For real positive ${\eta }_0$, its phase relative to the bed-height coefficient is $-mH+\pi$ if $\sin (mH)>0$ and $-mH$ if $\sin (mH)<0$, modulo $2\pi$. Relative to the imposed bed-velocity coefficient $ikU{\eta }_0$, the phase is instead $-mH+\pi /2$ in the positive-sine case, with an additional $\pi$ when the sine changes sign. The complex coefficient gives the answer without convention ambiguity. The upward displacement coefficient, if desired, is $A_{+}/(ikU)$, since $w=U{\zeta }_x$ for the stationary wave.

A reflecting lid reverses the vertical energy direction while retaining horizontal wavenumber and laboratory frequency. In the terrain frame the incident and reflected energy rays therefore travel downstream while zigzagging between the boundaries. The velocity field has nodes at the lid and a vertically standing pattern; the sketch marks this ray geometry, rather than falsely drawing the reflected ray as a specular optical ray.

The intrinsic group velocity makes an angle $\theta$ with vertical satisfying $\tan \theta =|c_{gx}^{\mathrm{fluid}}|/c_{gz}=m/k$. Consequently the denominator vanishes whenThese are the rigid-lid resonance of stationary topographic internal waves. At such a nonzero vertical mode the homogeneous rigid-lid eigenfunction is also zero at the bed, so it cannot satisfy the nonzero prescribed bed velocity; a bounded stationary inviscid forced solution does not exist. Time-dependent forcing produces secular modal growth until dissipation or nonlinearity intervenes. The formal $n=0$, $m=0$ limit is not this resonance: $\sin [m(H-z)]/\sin (mH)\to (H-z)/H$ is finite. Propagating resonance also requires $k<N/U$; an evanescent field has no real standing-wave resonances of this kind.