= Solution
At the change in <buoyancy frequency>, continuity of vertical <velocity> and <fluid pressure> gives
$$
[W]_{H^-}^{H^+}=0,\qquad [P]_{H^-}^{H^+}=0.
$$
The background <mass density> is continuous, so there is no interfacial density jump or separate surface restoring <force>. Since $P=i\omega W'/k^2$, these <internal-wave transmission across a stratification step> conditions are continuity of $W$ and $W'$. The <buoyancy perturbation> need not be continuous, because the background density gradient jumps.
Let $W_0=-i\omega\eta_0$ and $\ell=k\sqrt{N_2^2/\omega^2-1}$. The lower unstratified layer obeys $W''-k^2W=0$ and the upper <radiation condition> gives
$$
W(z)=C e^{-i\ell(z-H)}\quad(z>H),\qquad W'(H)=-i\ell C.
$$
Propagating these matching data downward gives
$$
W(z)=C\left[\cosh k(H-z)+i\frac{\ell}{k}\sinh k(H-z)\right]\quad(0\leq z\leq H).
$$
The <kinematic boundary condition> at the moving lower boundary therefore gives
$$
\boxed{C=\frac{-i\omega\eta_0}{\cosh(kH)+i(\ell/k)\sinh(kH)}}.
$$
The upper vertical-displacement <complex amplitude> at $H$ is
$$
\boxed{\eta_H=\frac{\eta_0}{\cosh(kH)+i(\ell/k)\sinh(kH)}},\qquad
(U,W)=\left(\frac\ell k,1\right)C e^{-i\ell(z-H)}.
$$
In particular, $|\eta_H/\eta_0|=[\cosh^2(kH)+(\ell/k)^2\sinh^2(kH)]^{-1/2}$. The unstratified layer attenuates and phase-shifts the transmitted <internal gravity wave>. The matching height is $H$ in the original PDF; the TeX transcription's $H'$ is a typo.
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