THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

64 3. Sound Wave Propagation and Characteristicswhere ρ is the mass density of the propagation medium. With the sound sourcebeing isotropic (i.e., omnidirectional, with no angular-dependent variations), andintegration of Equation (3.46) over 4π steradians, the sound power is given byW = Ir S = 4π r 2 Ir = 4πr 2 prms2 (3.49)ρcIf sound power is measured in half-space, i.e., if the source lies on a reflectivesurface, then integration occurs over 2π steradians andW = 2πr 2 prms2 (3.50)ρcHere, W denotes the sound power of the source in watts, I the sound intensity(W/m 2 ) in the direction of wave propagation, and r the distance from the centerof the source.For a spherical wave in full space, root-mean-square sound pressure and soundintensity in the direction of wave propagation are related to the sound power ofthe source byprms 2 = ρcW(3.51a)4π r 2I =W(3.51b)4π r 2The tendency of the sound intensity in Equation (3.51b) to decrease with increasingdistance from the source is called the inverse square law.3.19 The Spherical Wave: Sound Pressure Leveland Sound Intensity LevelCombining Equations (3.51a) and (3.51b) with the definition of sound pressurelevel given by Equation (3.22), we express the sound pressure Lp in terms of soundpower W of the source and distance r from the source:( p 2) ( )L p = 10 log rmsρcWpref2 = 10 log4πr 2 pref2= 10 log(ρcW) − 20 log r + 83 (3.52)where p ref = 20 μPa. The sound intensity level L I in the direction of sphericalpropagation is found from( ) ( )IWL I = 19 log = 10 logIref4πr 2 pref2= 10 log W − 20 log r + 109= L W − 20 log r − 11 (3.53)where I ref = 10 −12 W/m 2 and L W is the sound power level (dB re 1 pW).