Topographic sideband of an internal gravity wave

Reflection from periodic topography couples an incident horizontal wavenumber $k$ to sidebands $k+j{k}_T$. Expanding the impermeability condition on a boundary of small amplitude $h_0$ produces the $j=\pm 1$ sidebands at order $h_0$ and the $j=0,\pm 2$ corrections at order $h_0^2$.