Index of content:
Volume 124, Issue 4, October 2008
- GENERAL LINEAR ACOUSTICS 
On the transient solutions of three acoustic wave equations: van Wijngaarden’s equation, Stokes’ equation and the time-dependent diffusion equation124(2008); http://dx.doi.org/10.1121/1.2973231View Description Hide Description
Acoustic wave propagation in a dispersive medium may be described by a wave equation containing one or more dissipation terms. Three such equations are examined in this article: van Wijngaarden’s equation (VWE) for sound propagating through a bubbly liquid; Stokes’ equation for acoustic waves in a viscous fluid; and the time-dependent diffusion equation (TDDE) for waves in the interstitial gas in a porous solid. The impulse-response solution for each of the three equations is developed and all are shown to be strictly causal, with no arrivals prior to the activation of the source. However, the VWE is nonphysical in that it predicts instantaneous arrivals, which are associated with infinitely fast, propagating Fourier components in the Green’s function. Stokes’ equation and the TDDE are well behaved in that they do not predict instantaneous arrivals. Two of the equations, the VWE and Stokes’ equation, satisfy the Kramers-Kronig dispersion relations, while the third, the TDDE, does not satisfy Kramers-Kronig, even though its impulse-response solution is causal and physically realizable. The Kramers-Kronig relations are predicated upon the (mathematical) existence of the complex compressibility, a condition which is not satisfied by the TDDE because the Fourier transform of the complex compressibility is not square-integrable.
124(2008); http://dx.doi.org/10.1121/1.2968671View Description Hide Description
The transmission of sound from a slanted side branch into an infinitely long rectangular duct is studied numerically using the method of finite element with absorptive domain exit boundaries. The sound transmission coefficients associated with various acoustic modes are investigated in details. The results show that the plane wave assumption is only valid at very low frequency. It is also found that the intensities of the higher modes are stronger than that of the plane wave once they are excited. Besides, a critical side-branch slant angle is found over which a significant change of sound propagation mode takes place. This affects substantially the energy distribution between various acoustic modes inside the main duct. A simplified model is proposed to explain the phenomenon and the relationship of this critical angle with the width ratio between the side branch and the main duct is established.
124(2008); http://dx.doi.org/10.1121/1.2967837View Description Hide Description
Sound propagation in an acoustic waveguide is examined using a hybrid numerical technique. Here, the waveguide is assumed to be infinite in length with an arbitrary but uniform cross section. Placed centrally within the guide is a short component section with an irregular nonuniform shape. The hybrid method utilizes a wave based modal solution for a uniform section of the guide and, using either a mode matching or point collocation approach, matches this to a standard finite element based solution for the component section. Thus, one needs only to generate a transverse finite element mesh in uniform sections of the waveguide and this significantly reduces the number of degrees of freedom required. Moreover, utilizing a wave based solution removes the need to numerically enforce a nonreflecting boundary condition at infinity using a necessarily finite mesh, which is often encountered in studies that use only the standard finite element method. Accordingly, the component transmission loss may readily be computed and predictions are presented here for three examples: an expansion chamber, a converging-diverging duct, and a circular cylinder. Good agreement with analytic models is observed, and transmission loss predictions are also presented for multimode incident and transmitted sound fields.