Equivalent acoustic impedance model. Part 2: analytical approximation

In the context of the prediction of noise levels in vibroacoustic systems, numerical models or analytical models can be developed. Generally, numerical models are adapted to the low and medium frequency ranges and analytical models to the medium and high frequency ranges. For analytical models, a classical approximation consists of modelling the multilayer system by an equivalent acoustic impedance. This paper deals with a multilayer system consisting of a porous medium inserted between two thin plates. Part 1 of this paper is devoted to the experiments performed and to the development of a probabilistic algebraic model for the equivalent acoustic impedance. In the present Part 2, an analytical method is constructed for this multilayer system. This method consists of introducing the unbounded medium in the plane directions x1 and x2 while the medium is bounded in the x3-direction. A two-dimensional space Fourier transform introducing the wave vector co-ordinates k1 and k2 is used. For a given frequency and for k1 and k2 fixed, the boundary value problem in x3 consists of 12 differential equations in x3 whose coefficients depend on k1 and k2, with boundary conditions. This system of equations is solved by using adapted algebraic calculations. By inverse Fourier transform with respect to k1 and k2, the equivalent acoustic impedance is deduced. The method which is proposed is not usual. Finally, a comparison of this analytical approach is compared with the experimental results.