Numerical Simulation of Acoustic Waveguides for Webster-Lokshin Model Using Diffusive Representations
This paper deals with the numerical simulation of acoustic wave propagation in axisymmetric waveguides with varying cross-section using a Webster-Lokshin model. Splitting the pipe into pieces on which the model coefficients are nearly constant, analytical
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IRCAM, Analysis-Synthesis team, 1, place Igor Stravinsky, F-75004 Paris, France INRIA, Sosso project, domaine de Voluceau-Rocquencourt, B.P. 105, F-78153 Le Chesnay, France
1 Introduction This paper deals with the numerical simulation of acoustic wave propagation in axisymmetric waveguides with varying cross-section using a WebsterLokshin model. Splitting the pipe into pieces on which the model coefficients are nearly constant, analytical solutions are derived in the Laplace domain, enabling for the realization of the propagation by concatenating scattering matrices of transfer functions (§2). These functions involve standard differential and delay operators, as well as pseudo-differential operators of diffusive type, induced by both the viscothermallosses and the curvature. These operators are explicitly decomposed thanks to an asymptotic expansion, and the diffusive ones may be defined and classified (§3). Various equivalent diffusive realizations may be proposed, that are deeply linked to choices of cuts in the complex analysis of the transfer functions. Then, finite order approximations are given for their simulation (§4).
2 Deriving the model 2.1 Acoustic model A mono-dimensional model of the propagation of the acoustic pressure p in axisymmetric waveguides including viscothermallosses on the wall has been derived in [1, Chap 1], assuming the quasi-sphericity of isobars near the wall. Defining in the Laplace domain ;j;( Z, s) = R( z) fJ( z, s) where s is the Laplace variable, z is the curvilinear ordinate measuring the arc length of the wall, and R(z) is the radius of the guide, the Webster-Lokshin model may be written:
~ 8;'lj!z(s)
[(S)2 ~ ~ + 2E(Z) (S)~ ~ + Y(z) ] 'lj!z(s) = o.
(1)
* on sabbatical leave from ENST, TSI dept. & CNRS, URA 820. 46, rue Barrault
F-75634 Paris Cedex 13, FRANCE. G. C. Cohen et al. (eds.), Mathematical and Numerical Aspects of Wave Propagation WAVES 2003 © Springer-Verlag Berlin Heidelberg 2003
Diffusive Representations for Webster-Lokshin Simulation
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y'1-R'(z)2
is the sound speed, T(z) =R"(Z)/ R(z) the curvature, and c(z) = A R(z) quantifies the effect of the viscothermallosses. Note that contant curvatures correspond to cylinders or cones (T= 0), exponential or catenoidal shapes (T> 0), and sinusoidal shapes (T < 0), for the curvilinear ordinate z. For short pieces of guide on which T and c may be approximated by their constant mean value, Eq. (1) has the analytic solution ($z(s) = A(s)e Z r(sltB(s)e- Z res) C
where r(s) is a square root of
m+2c (~) 2+T. 2
;!
2.2 Scattering matrix The waves r(z, s) = (p(z, s) =f pcaz p)/2 defined in [2J are locally outwardly (P+) and inwardly (p-) directed. For a C1-regular profile R(z), their connection at z* is simply given by P;:(z*,s) = P;+1(z*,s) where nand n+1 index two concatenated pieces of guide. For this last reason, we are interested in the time-domain simulation of the scattering matrix defined for ~± ~ 1/Jz =R(z)p(z,s). For convenience, we consider the adimensional problem (c = ITI = 1, and z E [0,1]' f3 ex: c, an
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