In this section, the integral equations of problems associated with Helmholtz equation are established. These integral equations constitute the basis of the boundary element method (BEM).
Consider a tri-dimensional closed volume Ω− with boundary ∂Ω (see Figure 7.1). The boundary ∂Ω separates an interior region (volume Ω−) from an unbounded exterior region Ω+. This exterior region can be seen as externally bounded by a fictitious outer sphere ΓR whose radius R tends toward infinity (Γ∞). n and n° denote the outward normal to Ω− and the outward normal to Ω+ on Γ∞, respectively. It is common to introduce the concepts of interior and exterior problems (see Figure 7.2). For the interior problem, one is interested in the sound pressure field inside Ω−* whereas for the exterior problem, one is interested in the sound pressure field in the exterior region Ω +† (outside Ω−).