= Focusing power of internal-wave reflection
{title2=$\gamma$}
For a nonrotating <internal gravity wave> of fixed <angular frequency> in uniform <buoyancy frequency>, the <dispersion relation> fixes the ray angle $\theta$ to the horizontal. Consider a planar slope at angle $\alpha$ in the focusing orientation, with $0<\alpha<\theta<\pi/2$. Conservation of tangential <wavenumber> at reflection gives the ratio of total <wavenumber> magnitudes
$$
\gamma=\frac{K_{\mathrm{reflected}}}{K_{\mathrm{incident}}}=\frac{\sin(\theta+\alpha)}{\sin(\theta-\alpha)}>1.
$$
The reflected <wavelength> and beam width each shrink by $\gamma^{-1}$. At fixed <angular frequency>, the magnitude of <group velocity> also shrinks by $\gamma^{-1}$; conservation of beam-integrated <energy flux> therefore multiplies the wavelength-averaged <energy density> by $\gamma^2$.
The orientation matters. Reversing the ray itinerary gives the reciprocal, defocusing factor; a horizontal boundary gives $\gamma=1$. Thus subcritical slope magnitude alone does not imply amplification for every incident direction. The divergence as $\alpha\to\theta$ signals the failure of the ideal inviscid reflection model near <internal-wave slope criticality>.
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