= Solution
Let $p$ denote the <fluid pressure> perturbation divided by the reference <mass density>, and let $b=-g\rho'/\rho_0$ be the <buoyancy perturbation>. The <buoyancy frequency> satisfies $N^2=-g\hat\rho_z/\rho_0$. <Linearization> requires small boundary slope $k|\eta_0|\ll1$, small displacements compared with the vertical scales of the disturbance and background, and <advection> small compared with the oscillatory acceleration. For a propagating <internal gravity wave> this includes $|m\eta_0|\ll1$; boundary slope alone is insufficient when $|m|\gg k$. The <Boussinesq approximation> also requires small relative <mass density> differences over the region of interest. These conditions must hold for the resulting disturbance, including any amplification by <resonance>.
The inviscid <Linearized Boussinesq equations> and linear <kinematic boundary condition> are
$$
u_t=-p_x,\qquad w_t=-p_z+b,\qquad b_t+N^2w=0,\qquad u_x+w_z=0,\qquad W(0)=-i\omega\eta_0.
$$
Write the <velocity field> as $(u,w)=\operatorname{Re}\{(U(z),W(z))e^{i(kx-\omega t)}\}$. <Incompressibility> and horizontal <momentum> balance give
$$
U=\frac{i}{k}W',\qquad P=\frac{i\omega}{k^2}W',\qquad B=-\frac{iN^2}{\omega}W.
$$
The vertical <momentum> balance therefore yields
$$
W''+k^2\left(\frac{N^2}{\omega^2}-1\right)W=0.
$$
For a real vertical <wavenumber> $m$, this gives the <dispersion relation>
$$
\boxed{\omega^2=\frac{N^2k^2}{k^2+m^2}}.
$$
When $0<\omega<N$, define $\ell=k\sqrt{N^2/\omega^2-1}$. The <radiation condition> selects $m=-\ell$, because energy must leave the boundary upward. The <boundary-forced internal gravity wave> is
$$
\boxed{W=-i\omega\eta_0e^{-i\ell z},\qquad U=\frac{\ell}{k}W}.
$$
Its <wavevector> is $\mathbf K=(k,m)$, its <phase velocity> is $\mathbf c_p=\omega\mathbf K/(k^2+m^2)$, and its <group velocity> is
$$
\mathbf c_g=\left(\frac{Nm^2}{(k^2+m^2)^{3/2}},-\frac{Nkm}{(k^2+m^2)^{3/2}}\right).
$$
Thus the <phase velocity> points down and right, while the <group velocity> points up and right. They are perpendicular. <Internal-wave polarization> makes the particle motion an oscillation along the <group velocity> direction, with zero first-order mean transport. The <constant-phase lines of an internal gravity wave> are also parallel to the <group velocity>. Their inclination $\vartheta$ above the horizontal satisfies $\sin\vartheta=\omega/N$.
When $\omega>N$, set $\kappa=k\sqrt{1-N^2/\omega^2}$. Boundedness at infinity selects the <evanescent wave>
$$
\boxed{W=-i\omega\eta_0e^{-\kappa z},\qquad U=-\frac{i\kappa}{k}W}.
$$
The horizontal and vertical <velocity> components are in quadrature: fluid particles describe small <ellipses>, and the response decays over $\kappa^{-1}$. There is no upward time-averaged <energy flux>, because $P$ and $W$ are in quadrature. A real vertical <group velocity> is not defined for this <evanescent wave>. The pattern travels horizontally with <phase velocity> $\omega/k$.
For $N=0$ the response is the <evanescent wave> with $\kappa=k$; the ratios $\omega/N$ should not be used. At the cutoff $\omega=N>0$, the bounded harmonic solution has $W=-i\omega\eta_0$, $U=0$: it neither decays nor has nonzero upward <group velocity>. It is the limiting cutoff response, rather than a localized radiating disturbance. A uniform nonzero <buoyancy frequency> in an infinitely deep <Boussinesq approximation> is itself a local idealization of the background <mass density>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-345-wave-regimes.png]
{title=Propagating and evanescent responses}
The arrows distinguish the <phase velocity>, <group velocity>, and oscillatory particle motion of the <internal gravity wave>. The right panel shows the decay envelope and particle <ellipses> of the <evanescent wave>.
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