The *Helmholtz wave equation* expresses constraints at every point in
in terms of partial derivatives of the pressure function. Its frequency-dependent form is

in which is the sound pressure, is the Laplacian operator from calculus, and is related to the frequency as .

Closed-form solutions to (11.8) do not exist, except in trivial cases. Therefore, numerical computations are performed by iteratively updating values over the space; a brief survey of methods in the context of auditory rendering appears in [211]. The wave equation is defined over the obstacle-free portion of the virtual world. The edge of this space becomes complicated, leading to *boundary conditions*. One or more parts of the boundary correspond to sound sources, which can be considered as vibrating objects or obstacles that force energy into the world. At these locations, the 0 in (11.8) is replaced by a *forcing function*. At the other boundaries, the wave may undergo some combination of absorption, reflection, scattering, and diffraction. These are extremely difficult to model; see [268] for details. In some rendering applications, these boundary interactions may simplified and handled with simple *Dirichlet boundary conditions* and *Neumann boundary conditions* [368]. If the virtual world is unbounded, then an additional *Sommerfield radiation condition* is needed. For detailed models and equations for sound propagation in a variety of settings, see [268]. An example of a numerically computed sound field is shown in Figure 11.14.

Steven M LaValle 2020-01-06