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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

m1∇ϕ⋅n⃗=0,on Γ,m_1 \nabla \phi \cdot \vec{n} = 0,\quad\text{on~}\Gamma,

in acoustic media, and

(C:ε(u))⋅n⃗=0,on Γ,\left(C : \varepsilon(u)\right) \cdot \vec{n} = 0,\quad\text{on~}\Gamma,

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.

Practical Advice

Keep in mind that the natural boundary conditions will be applied on all boundaries, where you do not explicitly specify other conditions.

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Boundary Conditions
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