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

62 3. Sound Wave Propagation and Characteristicsand for k 2 r 2 ≪ 1:u(r, t) ≈Ap(r, t)sin k(r − ct) ≈ ̸ 90 ◦ re p(r, t) (3.42)kρcr2ρckrThe difference between Equations (3.41) and (3.42) connotes, respectively, the farfield and near field effects of a spherical wave. As r approaches the center of thespherical source the sound pressure and particle velocity becomes progressivelymore out of phase, approaching 90 ◦ as the limit. In the near field the sound intensityis not simply related to the root-mean-square value of the sound pressure.A sound source is generally directional, and the sound intensity does not havethe same value at all points on the surface. In order to evaluate the integral ofEquation (3.37) it is necessary to execute an approximation by segmenting thesurface into a finite number of sub-elements, each subtending an area S i and toestablish the sound intensity on each sub-element in a direction normal to thatelement. A summation procedure over all of the surface sub-elements will yieldthe total sound powerW = ∑ iI i S i (3.43)whereI i = sound intensity averaged over the ith element of area S i , W/m 2S i = ith element of area, m 2Equation (3.43) can be expressed logarithmically as( ) ( )I SLw = 10 log + 10 logI0 S0( ) SLw = L I + 10 logS0whereL w = sound power level, dB re 10 −12 WL I = sound intensity level, dB re 10 −12 W/m 2S = area of surface, m 2S 0 = reference area = 1.0m 2I 0 = reference sound intensity, internationally set at 10 −12 W/m 23.18 The Monopole SourceA monopole can be described as an idealized point generating a spherical soundwave. A pulsating sphere can be considered a good approximation of a point sourcewhen its radius is small compared with the wavelength of the sound it generates.