title: Conditions under which a number of sinusoids may be instantaneously in phase.
abstract: The literature on the phase relationships between frequency components of a Fourier analysis is reviewed, with examples and theories from acoustics and neurophysiology. Given n sinusoids of different frequencies omega1, omega2, .., omegan and phase angles phi1, phi2, .., phin, it is shown that for n greater than or equal to 2 the set of initial phase angles allowing the n sinusoids to be in phase at some time t0 consists of one or more planes of constant dimension 2 and that for n = 2 such a time t0 always exists. The conditions under which the common phase of n sinusoids at one time t0 will be the same as the common phase at another time t0 are also investigated. The importance of incommensurately related frequency components is emphasized by proofs which do not depend on harmonic relationships. Proofs are formulated in a linear algebra format to demonstrate the versatility of the method for analysing long sequences of frequencies and phases.
mesh_terms: D000162:Acoustics; D006309:Hearing; D009424:Nervous System Physiological Phenomena; D010507:Periodicity