3D Sound

The mathematics of 3D sound is quite elementary, as we will see below. The hard part of the theory of practical systems typically lies in the mathematical approximation to the ideal case. Examples include Ambisonics [159] and wave field synthesis [49].

Consider a point source at position $\underline{x}_s\in\mathbb{R}^3$ . Then the acoustic complex amplitude at position $\underline{x}_l\in\mathbb{R}^3$ is given by

$$p(\underline{x}_l;\underline{x}_s) = p_1(\underline{x}_s) \frac{e^{jk\vert\underline{x}_l-\underline{x}_s\vert}}{\vert\underline{x}_l-\underline{x}_s\vert}$$

where $p_1(\underline{x}_s)$ denotes the complex amplitude one meter from the point source in any direction, and $k=2\pi/\lambda$ denotes the wavenumber (spatial radian frequency). Distributed acoustic sources are handled as a superposition of point sources, so the point source is a completely general building block for all types of sources in linear acoustics.

The fundamental approximation problem in 3D sound is to approximate the complex acoustic field at one or more listening points using a finite set of $M$ loudspeakers, which are often modeled as a point source for each speaker.