A linear rigid frame, cylindrical capillary theory of sound propagation in porous media is extended to include nonlinear effects of the Forchheimer type, by making a particle velocity-dependent correction to the complex density. This type of nonlinearity becomes important at incident sound-pressure levels above approximately 120 dB re: 20 μPa for highly porous fibrous materials. Data from three experiments on air-saturated porous media and foams, at levels up to 170 dB, are compared to the rigid frame theory with Forchheimer-type extension. One of the experiments measured a low-frequency approximation for the complex density directly; the others measured internal attenuation and surface admittance (inverse impedance). The generally good agreement found between predictions based on the Forchheimer-type nonlinear theory and each of the experiments suggests that this type of nonlinearity is the dominant one for propagation in many types of air-saturated fibrous porous media. Some interesting surface absorption characteristics are predicted at incident sound-pressure levels above 140 dB, especially at lower frequencies.
It is well known that the flow resistance, as defined by Darcy's law, becomes a function of the fluid velocity when the fluid velocity becomes sufficiently large. For porous materials in air, there appears to be two separate nonlinear flow regimes: For low velocities, the resistance coefficient increases with the square of the fluid velocity; and, for high velocities, the resistance coefficient increases linearly with the magnitude of the fluid velocity. Similar regimes also exist for the mass coefficient; however, the mass coefficient is observed to decrease with increasing fluid velocity. (For pure-tone excitation, it is shown that these relationships then hold for the complex magnitude of the particle velocity. ) In this paper, the two nonlinear flow regimes are parametrized for several porous materials. Nonlinear corrections are made to linear wave theory, and numerical solutions for pure tones are obtained that are then used to predict surface admittance and internal pressures of finite length sample. A simple surface pressure criterion is put fourth based upon the materials nonlinear parameters that gives an approximate maximum surface pressure that a porous material could be exposed to before nonlinear phenomena become important in describing the material's characteristics. PACS numbers: 43.25.Ba, 43.25.Ed, 43.55.Ev
A standing wave apparatus employing multiple microphones for measuring fundamental properties of acoustic materials is analyzed. The apparatus is basically a tube with a compression driver at one end with a rigid plug at the other and a finite length sample of acoustic material strategically located in between. A microphone array is then used in obtaining the pressure and velocity boundary conditions of the sample from which basic acoustic properties can be calculated. Such properties as internal propagation constant, characteristic admittance (inverse impedance), and bulk modulus are measured and data for a 100-pore-per-inch (ppi) open-cell acoustic foam is presented. A relatively new parameter called the complex flow impedance is measured under low-and high-intensity levels and is shown to exhibit finite amplitude properties. Of particular concern is how finite sample lengths effect the accuracy of the measurements. It is shown that in general a long sample length (on the order of a wavelength or more) is required for the accurate measurement of propagation constant, characteristic admittance, and bulk modulus measurements while a short length (much less than a wavelength) is required for good finite amplitude flow impedance measurements. Previous studies of flow impedance have been carried out using a similar but not identical apparatus [W. E. Zorumski and T. L. Parrott, "Nonlinear AcousticTheory for Rigid Porous Materials," U.S. NASA TN-6196, 1971; K. U. Ingard and T. A. Dear, "Measurements of the Acoustic Flow Resistance," J. Sound Vib. 103, 567-572 (1985) ]. However, the previous studies have lacked an analysis of the possible errors in the measurement methods that can be very significant for these types of flow impedance measurements. PACS numbers' 43.25.Ed, 43.25.Zx, 43.85.Bh LIST OF SYMBOLS a• effective pore radius in rigid frame model Zz. d sample thickness Zs, , kb complex propagation constant of wave in porous material ]•b K complex bulk modulus of fluid in porous material M mass loading part of flow impedance, = I'm(Zr)/cø Po static pressure of fluid Pb P complex excess (acoustic) pressure amplitude (taken to be the peak value, not rms) Po Npr Prandtl number of fluid rrø R resistive part of flow impedance, = Re(Z•.) v U complex particle velocity (the peak value, co not rms) COT
A wave perturbation method of analyzing the internal propagation and surface admittance characteristic of highly porous acoustic materials with rigid frames at high particle velocities is presented. It is an iterative method whose implementation is based on experimental observations. Numerical results for an open-cell foam material are compared with both experiments and exact numerical solutions of the fundamental equations of nonlinear motion. Most of the salient features of the surface admittance of a bulk material are predicted analytically with the accuracy increasing as the frequency of excitation is increased. The method is restricted to pure-tone excitation. PACS numbers' 43.25.Ba, 43.25.Ed, 43.55.Ev
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