Let \(D_t\) be the time derivative.

Let \(D_x\) and \(D_y\) be the spatial derivatives in the \(x\) and \(y\) directions.

The Laplacian is \(\nabla^2 = D_x^2 + D_y^2\).

Let \(c\) be the propagation speed.

The d'Alembertian is \(\square^2 = \frac{1}{c^2} D_t^2 - \nabla^2\).

Let \(u\) be the displacement field.

The wave equation is \(\square^2 u = 0\).

With a position-dependent speed field: \((D_t^2 - \nabla \cdot c^2 \nabla) u = 0\).

The Klein–Gordon equation is \((\square^2 + m^2) u = 0\), where \(m\) is the mass.

The relativistic heat equation is \((\square^2 + \alpha D_t) u = 0\), where \(\alpha\) is the diffusivity.

Boundary conditions:

• Fixed (Dirichlet): \(u = 0\)
• Free (Neumann): \(u_x = 0\)
• Absorbing (1st approx): \(u_x - \frac{1}{c} u_t = 0\)
• Absorbing (2nd approx): \(\frac{1}{c} u_{xt} - \frac{1}{c^2} u_{tt} + \frac{1}{2} u_{yy} = 0\)