This work has been carried out during sabbatical leave granted to the author Dr. Hamzeh Duwairi from the university of Jordan for the academic year 2007/2008 Forchheimer’s Sound Waves Propagation in a Cylindrical Tube Filled with a Porous Media H. M. Duwairi Mechatronics Engineering Department, School of Technological Sciences, The German-Jordanian University, 11180 Amman, Jordan Abstract: - A rigid frame, cylindrical capillary theory of sound propagation in porous media that includes the nonlinear effects of the Forchheimer type is laid out by using variational solutions. It is shown that the five main parameters governing the propagation of sound waves in a fluid contained in rigid cylindrical tubes filled with a saturated porous media are shear wave number, μ ω ρ / R s = , reduced frequency parameter, a wR k = , porosity, ε , Darcy number, K R Da = , and Forchheimer number, F s C C 2 * = . The manner in which the flow influences the attenuation and the phase velocities of the forward and backward propagating isentropic acoustic waves is deduced. It is found that the inclusion of the solid matrix increases wave’s attenuations and phase velocities for both forward and backward sound waves, while increasing the porosity and the reduced frequency number decreased attenuation and phase velocities. The effect of the steady flow is found to decrease the attenuation and phase velocities for forward sound waves and enhance them for the backward sound waves. Key-Words: - sound waves, porous medium, fluid flow 1 Introduction Acoustic problem covers a wide range of practical problems. If the acoustic improvements are restricted to interior spaces (buildings halls, theaters, dwellings, factories, vehicle cabins, etc.), usually mineral wools or open pore foams can be used to solve the problem. For outdoor problems, for instant, acoustic noise barriers against traffic noise, the absorption is provided by granular materials such as porous concrete or similar materials, as they behave better with bad weather and other atmospheric phenomena, as well as they can be cleaned (with a pressurized water) without loosing their acoustic properties. In porous materials such as fibrous and granular, the absorption process of the acoustic wave takes place through viscosity and thermal losses of the acoustic energy inside the micro tubes forming the material. The problem of a propagation of sound waves in fluids contained in a plain medium is a classical one, to which famous names are connected like Helmholtz [1], Kirchhoff [2] and Rayleigh [3]. Since then many papers have been written on the subject; often in relation to the studied dealing with the dynamics response of pressure transmission lines. A variational treatment of the problem of sound transmission in narrow tubes is described by Cummings [4] as an alternative to the more usual analytical procedure which is limited to mathematically tractable geometries. A first approximation to the effects of mean flow on sound propagation through cylindrical capillary tubes is achieved by Peat [5]. A sound transmission in narrow pipes with superimposed uniform mean flow and acoustic modeling of automobile catalytic converters is done by Dokumaci [6]. A numerical study on the propagation of sound through capillary tubes with mean flow is achieved also by Jeong and Ih [7] and finally an approximate dispersion equation for sound waves in a narrow pipe with ambient gradients is done by Dokumaci [8]. The problem of sound waves propagation in a stationary or flowing fluid in a porous medium is not addressed yet. An attempt is made in this article to develop a simplified nonlinear theory that predicts the propagation characteristics of a stationary or flowing fluid in saturated porous media. This theory is an extension of the classical plain medium theory, using a modification to Darcy’s law due to the Forchheimer effects. Analytical expressions for the propagation constant are obtained from variational solutions. Comparison with previous works in the limit of plain medium shows an excellent agreement. 2 Problem Formulation Consider a rigid tube filled with a saturated porous material, the fluid is assumed to be a stationary or movable inside the tube. The x- coordinate is measured along the tube and the r- coordinate is measured normal to the axial direction. 1st WSEAS Int. Conf. on COMPUTATIONAL CHEMISTRY, Cairo, Egypt, December 29-31, 2007 92