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

3.18 The Monopole Source 63The three-dimensional Equation (2.25) is expressed in spherical coordinates, withoutangular dependence, as∂ 2 (p ∂ 2∂ t = p2 c2∂ r + 2 )∂p(3.44)2 r ∂rThe solution to Equation (3.44) must be of the formp = 1 r [F1 (ct − r) + F2(ct + r)]with the term F 1 (ct − r) describing waves moving away from the source. Wediscard the term F 2 (ct + r) which describes waves traveling toward the source.With time t = 0 in order to eliminate a phase angle, a harmonic solution to thespherical wave equation (3.44) is given byp = A rcos[k(ct − r)]where A is a constant and k the wave number equal to ω/c. The root-mean-squaresound pressure prms 2 at a distance r from the source is given by∫ Tp 2 rms = 1 p 2 dt =A2cos[k(ct − t)]dt = A2(3.45)T 0 r 2 T 02 r 2Here T = 1/f = 2π/ω or the period needed to complete one cycle. If many frequenciesare present, prms 2 and the root-mean-square pressure can be measuredfairly accurately if the integration time is sufficiently large compared with theperiod of the lowest frequency. The sound power of the spherical source can befound by the use of Equation (3.37) as follows:∫∫W = I · n dS (3.46)∫ TEquation (3.46) represents the sound power of a source wheres∫ TI = 1 ρu dt (3.47)T 0constitutes the vector sound intensity; u represents the particle velocity vector, Sany closed surface about the source, n the unit normal to surface S, and T theaveraging time. The surface S for a spherical wave is defined by a sphere of radiusr about the source, so that Equation (3.46) becomes∫∫W = Ir dS (3.48)The magnitude of the sound intensity I r is directed radially, i.e., it runs parallelto unit normal n. When sufficiently far from the source, the sound pressure andparticle velocity are in-phase. Applying Equation (3.39) yieldsIr = p rms u rms = p2 rmsρcS