# Modeling of acoustic thermoviscous boundary losses

**Martin Berggren
**

Department of Computing Science

UMIT Research Lab

Umeå University

Umeå

### Abstract:

For acoustic wave propagation in air, losses due to viscous and thermal dissipation are often quite small. Therefore, numerical computations relying on equations lacking modeling of such losses, such as the linear wave equation or the Helmholtz equations, give usually very accurate results. The exception to this rule is applications dominated by boundary effects. Typical examples are sound propagation in narrow channels, appearing in musical instruments and measurements devices for instance, and propagation in small devices such as microphones, hearing aids, and micro-speakers. For such devices, the effects of the viscous and thermal boundary layers can be large, and a correct modeling of these losses can be crucial in order to obtain reliable numerical simulation results.

Such effects are well described by the boundary layers appearing in the linearized compressible Navier-Stokes equations, but the numerical treatment is complicated by the fact that such boundary layers are extremely thin, about 20-300 micrometer in the audio range, which means that numerical solutions will be computationally very costly in general. We propose instead to model the presence of the thermal and viscous boundary layers by a generalized impedance boundary condition, involving a reaction-diffusion problem on the boundary. This condition can be derived by boundary-layer analysis applied to the compressible Navier-Stokes equations, and they will in the end appear in the form of additional boundary terms in the standard Helmholtz equation, which are straightforward to implement in a finite element solver. Existence and uniqueness of the final equation can be shown by standard techniques such as the Fredholm alternative. In cases with special geometries, such as narrow straight pipes, the proposed damping model agrees with known expressions from the literature, but the proposed model has a much wider potential applicability.