An efficient computation of sound fields due to a monopole source placed above a porous layer is presented. This paper examines an improved scheme whereby the steepest descent path is selected for the numerical evaluation of the Sommerfeld integral. Along the steepest descent path, a standard Gaussian-Hermite quadrature can be used to calculate the sound fields effectively. The suggested numerical scheme is accurate at all frequencies except in the very near field. The proposed method is more numerically efficient than other computational schemes, especially at long ranges and high source frequencies.

The current study examines the propagation of sound in street canyons with geometrically reflecting surfaces. An image source method is a popular numerical model to estimate the propagation of sound energy in a street canyon. This numerical model calculates the total sound energy received at a field point by summing the contributions from individual image sources incoherently. The discrete image source model is generalized by replacing rows of point sources with their respective line sources. An integral formulation is derived, which can be evaluated exactly to give a simple analytical solution. The expression permits rapid computations of the sound energy due to a point source placed in a street canyon. The transient sound energy at a receiver point is also examined. It has been demonstrated that the transient sound energy can be expressed in terms of a standard exponential integral. The Schroeder integration method is then used to calculate the reverberation times, which allow a straightforward assessment of the acoustic environment in street canyons. Indoor and outdoor experiments were conducted to validate the proposed integral formulation. The analytical formulas were also compared with numerical results based on the standard image source method and with published experimental data.

An approximate analytical formula has been derived for the prediction of sound fields in a semi-infinite rigid-porous ground due to an airborne source. The method starts by expressing the sound fields in an integral form, which can subsequently be evaluated by the method of steepest descents. The concept of effective impedance has been introduced by using a physically plausible assumption. The integral can then be simplified and evaluated analytically. The analytical solution can be expressed in a closed form analogous to the classical Weyl-Van der Pol formula that has been used for predicting sound fields above a rigid-porous ground. Extensive comparisons with the wave-based numerical solutions according to the fast field formulation and the direct evaluation of the integral have been conducted. It has been demonstrated that the analytical formula is sufficiently accurate to predict the penetration of sound into a wide range of outdoor ground surfaces.

This study examines the sound field within a hard-backed rigid porous medium due to an airborne source. The total sound field can be approximated by two terms: A transmitted wave component arriving at the receiver directly through the porous interface, and a second transmitted wave component reflected from the rigid backing plane before reaching the receiver. These two components can be expressed in an integral form that is amenable to further analyses. A standard saddle path method is applied to evaluate the integral analytically, leading to a uniform asymptotic solution that allows the prediction of the sound field within the rigid porous medium. The validity of the asymptotic formula is verified by comparison with the numerical results computed by a more accurate wave-based numerical scheme. The asymptotic solution is shown to provide a convenient means of rapid and accurate computations of sound field within the rigid porous medium. The accuracy of the numerical solutions is further confirmed by comparison with indoor experimental results. The measurement data and theoretical predictions suggest that when the receiver is located near the bottom of the hard-backed layer, the reflection of the refracted wave gives rise to a significant contribution to the total sound field.