“Incompressible” Acoustic Near Field as a Spatial Distribution of Sources Generating the Far Field

Abstract

Principle: Suppose one calculates an “incompressible” flow (pressure field p(0)) by substituting Laplace's equation for the wave equation in a given acoustic problem. Then the far field p(1) is a solution of the wave equation for a spatial distribution of matter sources of strength −c0−2∂p(0)/∂t. (Boundary condition: zero normal velocity.) This is readily demonstrated. Physical significance: Since the fluid is actually compressible, isentropy gives −c0−2∂p(0)/∂t=−∂p(0)/∂t. This density fluctuation is precisely equivalent to a volume fluctuation of the fluid elements. Pulsating sphere: A pulsating sphere generating an “incompressible” field p(0) may be replaced by a rigid sphere surrounded by a cloud of sources of strength −c0−2∂p(0)/∂t; this will generate the far field p(1). Aeolian tones: The principle helps explain nonvanishing generation of sound by a stationary rod in an air jet. The sound energy flow at a stationary surface must be zero: the sound is actually emitted from the region of fluctuating compression surrounding the rod. Jet noise: [J. Acoust. Soc. Am. 31, 245 (1959)]. The “incompressible” approximation to the field within the jet is p(0). Source-like pulsations of the fluid elements in response to p(0) are considered to generate the far-field sound p(1). The source strength, modified for convection, is −c0−2Dp(0)/Dt. (This work has been supported partially by the Air Force Office of Scientific Research of the U. S. Air Research and Development Command.)