In the simplest ocean models the ocean surface is modeled as a pressure release surface and the ocean bottom is assumed perfectly rigid. This leads to Dirichlet and Neumann boundary conditions respectively and the modal problem is a conventional Sturm-Liouville eigenvalue problem. Considering the bottom boundary we note that there really is no well-defined bottom depth--- below sediment lies basalt and one may continue from mantle to core, ... The truncation of the interval is justified when including additional depth no longer results in a significant change in the result. This somewhat nebulous transition occurs when the ocean subbottom is thick enough that material absorption eliminates significant energy return from deeper depths by refraction or reflection.
Mitigating against a conservative policy in carrying depth varying properties to great depths is the increased cost of solving the modal equation on a large domain. Thus it is desirable to truncate the problem at the shallowest possible depth. The rigid bottom model makes sense at a sediment/basalt interface where there is a strong impedance contrast. Basalts, however, are typically characterized by a strong elastic wave speed gradient which refracts ray paths back into the ocean. At mid- to high-frequencies, say above 50 Hz, this refracted energy will be severely attenuated and may be safely ignored. A more realistic bottom boundary condition is obtained with an acoustic half-space.
The boundary conditions corresponding to these various cases are provided below. The results are presented in three forms: 1) as a Robin condition on the pressure, 2) as boundary conditions on the stress-displacement vector and 3) as boundary conditions on the solvability vector . The first form is used for problems where all internal media are acoustic; the second form is used for problems where some internal media are elastic; the third form is used when elastic displacements are required ( KRAKEL ).