We consider first an acoustic bottom half-space characterized by a single wave speed, and a density, . The general solution in the half-space is given by,
and the Pekeris branch of the square root is used to expose the leaky modes. In order to have a bounded solution at infinity, we require B to vanish. At the interface, we require continuity of pressure and normal displacement which implies,
Thus, we obtain the bottom impedance condition,
A similar procedure yields the result for a top homogeneous half-space,
which differs by a sign change. Note that by letting we obtain the free-surface boundary condition and gives the perfectly rigid boundary condition.