A finite element for viscothermal wave propagation

The well known wave equation describes isentropic wave propagation. In this equation, non-isentropic boundary layer effects are neglected. This is allowed if the characteristic dimensions of the acoustic domain are large with respect to the thickness of the boundary layers. However, in small acoustic devices such as hearing aid loudspeakers, the boundary layer effects are significant and can not be neglected. A model that describes viscothermal wave propagation is needed to model such devices. For viscothermal wave propagation, the compressibility of air depends on the thermal behavior that can range from adiabatic to isothermal. Moreover, the propagation behavior can range from propagation with negligible viscosity to propagation with negligible inertia (Stokes flow). This complete range is accurately described by the low reduced frequency model. This model’s major drawback is that it is only defined for simple geometries such as thin layers and narrow tubes. It is not valid for arbitrary geometries. To overcome this drawback, a three dimensional viscothermal finite element has been developed. Like the LRF model, it covers the complete range from isothermal Stokes flow to isentropic acoustics. As opposed to the LRF model, the viscothermal finite element can be used to analyze complicated geometries. This paper presents the weak formulation of the finite element. Furthermore, two examples are presented in which the results of the finite element models are compared to measurements.