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

3.17 Sound Intensity 61Here the surface integral in Equation (3.37) is the integral of the sound intensity Inormal to the element dS of the surface area. The integration can be executed over aspherical or hemispherical surface enclosing the source. If the source of power W ismounted on an acoustically hard surface (i.e., a surface which is totally reflective),the sound waves expand within a hemisphere. Other surfaces, such as those of aparallelopiped (representing, for instance, the walls of a room), are often used inpractical applications. When the integration is performed over a spherical surfaceof radius r for a nondirectional source, sound intensity is related to sound powerbyI (r) = W S = W(3.38)4π r 2where S denotes the area of a sphere having radius r. Equation (3.38) constitutesthe inverse-square law of sound propagation, which accounts for the fact that soundbecomes weaker as it travels in open space away from the source, even if viscouseffects of the medium are disregarded. For sound radiation within a hemisphere,with the sound source mounted at the origin above a totally reflective surface,Equation (3.38) becomesI =W2π r 2Intensity, which represents the transfer of sound wave energy, equals the productof sound pressure and particle velocity,I = p · u (3.39)and for a simple cosine spherical wave, the pressure p(r, t) is given as a solutionto the spherical coordinate form of Equation (2.25)p(r, t) = A rcos k(r − ct)A is a constant amplitude with its physical units in N/m. From Equation (2.22) thevelocity isu(r, t) =− 1 ∫ [ Aρ r k sin k(r − ct) + A ]r cos k(r − ct) dt2=− 1 [− kA cos k(r − ct) −A]ρ kcr r 2 kc sin k(r − ct)oru(r, t) =A[ρcr cos k(r − ct) 1 + 1]kr tan k(r − ct)At large values of kr Equation (3.40) becomesu(r, t) ≈p(r, t)ρc(3.40)k 2 r 2 ≫ 1 (3.41)