Natural Boundary Conditions

The surface integrals that appear in the weak form of the wave equation when applying integration-by-parts to the stress term lead to the natural boundary condition of the wave equation. Specifically, dropping these terms from the weak form corresponds to the following physical conditions in the strong form.

Using the general parameterization this gives

in acoustic media, and

in elastic media, respectively. The latter condition is the so-called free-surface condition, which describes that traction pointing in normal direction out of the domain vanishes.

A physically more common condition in acoustic media is that pressure or the displacement potential, respectively, is zero at the boundary, see Dirichlet conditions.