# The Boundary Element Method in Acoustics: A Survey
## 1 Introduction  
Acoustics is the science of sound and has particular applications in sound reproduction, noise and sensing [1; 2]. Acoustics may be interpreted as an area of applied mathematics [1; 3]. In special cases, the mathematical equations governing acoustic problems can be solved analytically, for example if the equations are linear and the geometry is separable. However, for realistic acoustic problems, numerical methods provide a much more flexible means of solution [4; 5]. Numerical methods are only useful in practice when they are implemented on computer. It is over the last sixty years or so that numerical methods have become increasingly developed and computers have become faster, with increasing data storage and more widespread. This brings us to the wide-ranging area of computational acoustics; the solution of acoustic problems on computer [6; 7; 8; 9; 10; 11], which has significantly developed in this timescale. In the context of this work, the acoustic field is assumed to be present within a fluid domain. If there is an obstacle within an existing acoustic field, then the disturbance it causes is termed _scattering_. If an object is vibrating and hence exciting an acoustic response, or contributing to an existing acoustic field, then this is termed _radiation_. If the fluid influences the vibration of the object or structure, and energy is exchanged in both directions between them, then this is termed coupled fluid-structure interaction [12] or, in the acoustic context, vibro-acoustics [13; 14]. Alternatively, if the acoustic field is an outcome of a background flow, for example the generation of noise by turbulence, then this is termed aero-acoustics [11; 15]. The determination of the properties of a vibrating or scattering object (for example, its shape, the surface impedance or the sound intensity) from acoustic measurements in the field is termed inverse acoustics [16]. Vibration analysis is not the focus of this work, but it clearly cannot be divorced from the study of acoustics. In vibration analysis, the domain of the structure oscillates, perhaps under variable loading or forcing. In the case of vibro-acoustics, the vibration can excite or be excited by the acoustic field and, in any analysis, both need to be modelled [17; 18; 19], and the equations coupled. The computational modelling of vibration is normally carried out by the finite element method (FEM) [20; 21], and this method is significantly established in this application area. Vibrational modelling may be carried out in the time domain. However, with the nature of structural vibration in that it is normally dominated by modes and their corresponding resonant frequencies, and hence vibration is often close to periodic. Similarly, the excitation forces on a structure, such as the driver on a loudspeaker of the explosions in an engine can similarly be close to periodic in short time scales. It is, therefore, found that vibratory problems are routinely analysed in the frequency domain, both in the practical work and computational modelling. An acoustic field is most straightforwardly interpreted as a sound pressure field, with the sound pressure varying over the extent of the domain, and with time. The transient acoustic field can be computationally modelled by the finite element method [22; 23], the finite difference--time domain method (a particular finite difference method that was originally developed for electromagnetic simulation [24; 25; 26]) can also be applied in acoustics [27; 28; 29; 30]) and the boundary element method [31; 32; 33; 34], but acoustics and vibration are more often analysed and modelled in the frequency domain. In acoustic problems, the most likely fluid domains are air or water and, in many cases in these fluids, the linear wave equation is an acceptable model. By observing one frequency at a time, the wave equation can be simplified as a sequence of Helmholtz equations.