Wave-based numerical methods for acoustics

Recent developments in wave-based numerical methods are reviewed in application to problems in acoustics where small perturbations of pressure and velocity propagate on a steady compressible mean flow. In the most general case, such wave fields are represented by the solution of the linearized Euler equations. In the case of steady time-harmonic solutions in a homogeneous medium and in the absence of mean flow, these reduce to the solution of the Helmholtz equation. When irrotational mean flow is present, they can be represented by a convected Helmholtz-like equation. When rotational mean flow effects are significant, the time harmonic, coupled, first order linearized Euler equations must be solved. Wave-based numerical methods have been applied to all three categories of problem. The solution of the time-harmonic acoustic equations, particularly for large three-dimensional domains, is computationally challenging. When traditional finite element methods are used with polynomial basis functions within each element, approximability arguments alone require the use of many nodes per wavelength to resolve the resulting spatially harmonic solutions. It is moreover well known that these requirements are exacerbated when the computational domain spans many wavelengths of the solution. In such cases the ’pollution ’ effect [1] means that the global error increases with frequency, irrespectiv