Acoustical background

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


The mathematical background for functions defined on the unit sphere was presented in Chap. 1. Spherical harmonics played an important role in presenting and manipulating these functions. In this chapter, functions on the sphere are defined through the formulations of fields in three dimensions. Although sound fields are of primary concern in this book, which is oriented towards microphone arrays, the material presented here can be applied to scalar fields in general. This chapter begins by presenting the acoustic wave equation in Cartesian and spherical coordinates, with possible solutions. Solutions to the wave equation in spherical coordinates are shown to involve spherical harmonics and spherical Bessel and Hankel functions. Having formulated the fundamental solutions, sound fields due to a plane wave and a point source are presented, including an analysis of the effect of a rigid sphere introduced into the sound field. The latter is useful for describing the sound field around a microphone array configured over a rigid sphere, for example. The chapter concludes with a formulation of the three-dimensional translation of sound fields.

Original languageEnglish
Title of host publicationSpringer Topics in Signal Processing
PublisherSpringer Science and Business Media B.V.
Number of pages25
StatePublished - 1 Jan 2015

Publication series

NameSpringer Topics in Signal Processing
ISSN (Print)1866-2609
ISSN (Electronic)1866-2617


  • Microphone Array
  • Rigid Sphere
  • Sound Field
  • Sound Pressure
  • Spherical Harmonic

ASJC Scopus subject areas

  • Signal Processing
  • Computer Networks and Communications
  • Electrical and Electronic Engineering


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