Errors in array response calculations due to incorrectly folding the vertical acoustic arrival structure into beam pattern

Some ambient sea noise models assume for simplicity of computation that all sound arrives in the horizontal plane of a passive hydrophone array and thus use a two-dimensional as opposed to a three-dimensional beam pattern when determining array response. Using the properties of spherical harmonics, this paper gives an exact formula for the response of an equally spaced, N-element linear array in any physical orientation in the ocean and in any noise field. An exact expression is also given for the response predicted by a model that uses a two-dimensional beam pattern. Their difference, for the same noise field, gives the error due to the two-dimensional beam pattern assumption.