Focusing power of internal-wave reflection ($\gamma$)

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 magnitudesThe 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.