Frequency and Time Numerical Solutions of 3D Sound Propagation in Open and Closed Spaces

Abstract

Problems in acoustics can often be solved with the aid of formulas or expressions known as Green's functions. These functions, or fundamental solutions, relate the field variations (such as pressures) in an acoustic medium caused by sound sources placed elsewhere in the medium. The two fundamental solutions most often used are for harmonic point loads in a three-dimensional, infinite, homogeneous space, and for a harmonic line load acting within two-dimensional spaces. These are chosen because their solutions are known in closed-form and they are relatively simple in structure. This paper presents a set of three-dimensional solutions applied when the space domain is modeled with plane barriers placed together to reproduce spaces that vary from a simple half-space to a parallelepiped closed space. It is assumed that the homogeneous three-dimensional space is subjected to 3D point sources and spatially sinusoidal, harmonic line sound loads.

The resulting expressions are implemented to evaluate the field inside a rectangular space, whose walls allow different absorption coefficients. The time responses are obtained by means of Inverse Fourier Transforms. Complex frequencies are used to attenuate the response at the end of the time window. The effect of this attenuation is taken into account by re-scaling the time response.