# 3. Problem statement¶

The problem of sound field synthesis can be formulated after as follows. Assume
a volume \(V \subset \mathbb{R}^n\) which is free of any sources and sinks,
surrounded by a distribution of monopole sources on its surface \(\partial
V\). The pressure \(P(\x,\w)\) at a point \(\x\in V\) is then given
by the *single-layer potential* (compare p. 39 in [CK98])

where \(G(\x-\x_0,\w)\) denotes the sound propagation of the source at
location \(\x_0 \in \partial V\), and \(D(\x_0,\w)\) its weight, usually
referred to as *driving function*. The sources on the surface are called
*secondary sources* in sound field synthesis, analogue to the case of acoustical
scattering problems. The single-layer potential can be derived from the
Kirchhoff-Helmholtz integral [Wil99]. The challenge in sound field
synthesis is to solve the integral with respect to \(D(\x_0,\w)\) for a
desired sound field \(P = S\) in \(V\). It has unique solutions which
[ZS13] explicitly showed for the spherical case and [Faz10]
(Chap.4.3) for the planar case.

In the following the single-layer potential for different dimensions is discussed. An approach to formulate the desired sound field \(S\) is described and finally it is shown how to derive the driving function \(D\).