The use of (bi-)normal modal expansions for modeling sound propagation in the atmosphere

Abstract

One method for solving the equations of sound propagation in the atmosphere is to expand the propagating acoustic pressure with respect to the eigenfunctions (or modes) of the vertical part of the equations in the frequency domain. In the approximation that the atmosphere is vertically stratified, using an expansion in vertical modes is an implementation of seperation of variables. In the simplest case, the propagation is described by a wave equation with the sound speed depending on altitude. Finding the vertical modes requires that one solve the eigenvalue problem for the resulting vertical operator. When attenuation is included, this operator is not self-adjoint, increasing the complexity of the problem and leading to modes which are not orthogonal, but rather bi-orthogonal: that is, orthogonal to the corresponding modes of the adjoint operator. The use of and methods for obtaining modal expansions for sound propagation in the atmosphere will be discussed, ranging from low frequency audible sound propagating in the nocturnal boundary layer to infrasound propagating in the global atmosphere. If the number of modes required is small, the modes provide a compelling physical picture complimentary to that provided by geometrical acoustics.