Formulation of Four Poles of Three-Dimensional Acoustic Cavities Using Pressure Response Functions with Special Attention to Source Modeling

Jay Kim Structural Dynamics Research Laboratory University of Cincirmati Cincirmati, OH 45221-0072 The procedure proposed by Kim and Soedel [1] for formulation of four pole parameters of threedimensional cavities is revised. In the procedure, four poles were formulated in terms of the pressure responses of the cavities to a point source. However, it is shown that using the point source model for such a purpose is not valid because the pressure response function becomes singular at the source point. In this work, the procedure is modified by employing a surface source. It is shown that the modified procedure can be applied to three-dimensional acoustic systems. Introduction A four pole matrix is a very convenient concept to analyze complex acoustic systems. It allows various acoustic elements of the system to be formulated independently and to be assembled to form the system equation. Also, using four poles reduces related analysis efforts substantially as the system equation remains a two-by-two matrix. Many applications are found for analysis of one-dimensional systems [2, 3] and lumped parameter systems [2]. It is very appealing to have four poles of three-dimensional cavities because they are easily integrated with those of one-dimensional acoustic cavities for the purpose of system analysis. Kim and Soedel [1] proposed a method to formulate four pole parameters of three-dimensional cavities in terms of the pressure response functions of the systems at the input and output points. Lai and Soedel [ 4] applied a similar concept to analyze shallow three-dimensional cavities by specializing the procedure for two-dimensional cases. In all these works [1, 4], pressure response functions were obtained by solving the wave equation of cavities and modeling flow input and output ports as point sources. Deriving a four pole matrix of a three-dimensional cavity implies that the cavity is cormected to onedimensional systems. Therefore, the size of its xpass flow source can generally be considered much smaller than other dimensions of the cavity. Hence, it appears to be logical to model an acoustic source as a point source, as it has been done in [1, 4, 5]. However, in this work, it is shown that the point source model cannot be used to derive four poles of threeor two-dimensional cavities because of the singularity at the source point. The alternative approach is to use the surface source model. It is shown that an extended model can be generated by cormecting one-dimensional pipes of proper lengths to three-dimensional cavities, which enables the four poles of such a sub-system to be derived. Numerical examples are shown, where the boundary element method is used for actual calculations, to illustrate how the concept is used for system analysis. Formulation of Four Poles using Pressure Response Functions A four pole matrix defmes the relationship between the input and output variables of an acoustic system in the frequency domain. For the acoustic system shown in Figure 1, the equation is defmed as: {~:} = [ ~ ~]{~:}. (1) where P and Q are the amplitudes of the acoustic pressure and volume flow rate, subscripts 1 and 2 indicate the input and output points, respectively, A, B, C and D are the four pole parameters. It was shown that four pole parameters of an acoustic system could be formulated from the pressure response functions of the system as follows [1]: A= fTl({j)) B = 1C =-f (W)+ };1 (W) f (w) D = };1 (w) (2) f..,(w), f..,(w), 21 f..,(w) 22 ' f..2(w), where w is the circular frequency and fij ( w) is defmed as the pressure response of the system at location i when the system is subjected to a single harmonic volume flow input with unit strength at location j.