A semi-analytical solution for viscothermal wave propagation in narrow gaps with arbitrary boundary conditions.

摘要

Previous research has shown that viscothermal wave propagation in narrow gaps can efficiently be described by means of the low reduced frequency model. For simple geometries and boundary conditions, analytical solutions are available. For example, Beltman [4] gives the acoustic pressure in the gap between an oscillating, rigid, rectangular plate and a rigid surface. Assuming a pressure release boundary condition at the circumference of the plate, excellent agreement with experiments was obtained. In many engineering applications however, the boundary conditions may vary along the circumference of the plate. For instance, the vibrating membranes in hearing aid receivers are attached to complex structures and a simple pressure release (p = 0) or zero velocity boundary condition (dp=dn = 0) is only valid at some parts of the circumference of the vibrating structure. One can use numerical methods, like FEM or BEM, but often a large number of degrees of freedom is needed to obtain accurate results. Furthermore, a thorough understanding of the various phenomena can only be gained through a large number of calculations. In this paper a semi-analytical solution is presented for the viscothermal wave propagation in the gap between an oscillating, rigid, circular plate and a rigid surface for the boundary conditions just mentioned. The pressure in the gap is written as a series expansion of solutions satisfying the differential equations in the interior domain. Subsequently, either the pressure release or the zero velocity boundary condition is imposed on different parts of the circumference. The unknown constants in the series expansion are determined using a weak form of the boundary conditions. It is shown that only a limited number of terms is needed to accurately describe the total acoustic force on the plate. The solution is validated by means of a finite element calculation.