% Absorption profile for an open boundary, in three dimensions.
%
%   s = sponge3(x, y, z, Lx, Ly, Lz, w, smax)
%
% All in SI: positions and widths in metres, the result in inverse seconds.
% Zero in the interior, ramping up quadratically over a layer of width w
% inside each face of the box [-Lx/2, Lx/2] x [-Ly/2, Ly/2] x [-Lz/2, Lz/2]
% and reaching smax at the wall. Added to a scene's absorption, this is what
% makes the finite grid stand in for open air: a wave that leaves the
% instrument is attenuated before it reaches the outer boundary, and whatever
% reflects off that boundary is attenuated again on the way back.
%
% The ramp is gradual on purpose. An absorbing layer is itself an impedance
% mismatch, so a sudden one reflects; spreading it over a couple of
% wavelengths keeps that reflection small. It is not a perfectly matched
% layer, and at grazing incidence it does leak.
function s = sponge3(x, y, z, Lx, Ly, Lz, w, smax)
  dx = max(0, w - (Lx/2 - abs(x))) / w;
  dy = max(0, w - (Ly/2 - abs(y))) / w;
  dz = max(0, w - (Lz/2 - abs(z))) / w;
  d = max(max(dx, dy), dz);
  s = smax * d.^2;
end