Frequency and Time Numerical Solutions of 3D Sound Propagation in Open and Closed Spaces António Tadeu, Julieta António, and Luís Godinho Department of Civil Engineering at Faculty of Sciences and Technology, University of Coimbra, 3030-290 Coimbra, Portugal (Received 6 March and accepted 13 July) 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-dimen- sional, infinite, homogeneous space, and for a harmonic line load acting within two-dimen- sional 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 parallelepipedclosed space. It is assumed that the homoge- neous three-dimensional space is subjected to 3D point sources and spatially sinusoidal, har- monic 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 frequenciesare 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. 1. INTRODUCTION The frequencies involved in studying the acoustics of open or closed spaces tend to restrict the use of most of the techniques used to study wave propagation, such as the finite element and finite difference methods, because of the calculation effort required. The study of sound propagation in closed spaces often resorts to the ray tracing and image model techniques (Allen [1], Hammad [2,3], Kulowski [4]). With ray tracing, only a finite number of acoustic rays is traced between the sound source and the receiver. This means that the resulting simulation can be calculated in a reasonable time, even in more complicated geometries. There is a problem, however, in deciding on the number of rays to study, since the same geometric model may yield different results. Furthermore, certain rays that would be useful to the final outcome may not be included in the analysis at the outset (Kulowski [5]). BUILDING ACOUSTICS, VOL. 7, NO. 4 (2000) 247