254 



THEORY OF REVERBERATION INTENSITY 



volume in which the effective scatterers lie is about 

 half the volume illuminated by the ping at time i/2. 

 We shall now determine the intensity of the rever- 

 beration which reaches at time t from the volume 

 element dV, located at X in Figure 2. We use the 



h = 



(7) 



Figure 2. Diagrams used in developing volume rever- 

 beration' formulas. 



system of coordinate axes indicated in Figure 2; the 

 origin is at 0, and the ray from to X leaves with 

 spherical coordinates {d,<i>) defined by the tangent to 

 the ray at the origin. As drawn in Figure 2, 8 is the 

 angle made by the ray OT with the horizontal plane; 

 thus 6 is the complement of the polar angle made by 

 OT with the vertical direction OH. The amount of 

 energy which the pro j ector radiates per second into the 

 solid angle rfO in the direction {6,4>) is just Fb{9,<t>)dQ, 

 where F is the emission per unit solid angle in the 

 direction of maximum emission and 6(S,<^) is the 

 pattern function of the projector defined in Part I, 

 Section 12.4.4. The sound intensity at unit distance 

 (1 yd) from 0, along this ray, is therefore just Fh{d,4>). 

 If /i is the intensity at the point X, the "intensity 

 diminution" between the point one yard from the 

 projector and the point X may be denoted by h and 

 defined by 



The quantity h is simply related to the (positive) 

 decibel transmission loss H between the point 1 yd 

 from the projector and the point X by the formula 



H = -IQ log h. (8) 



The small tube of rays emitted by the projector 

 into the solid angle dO will have, at the point X, a 

 cross-sectional area, perpendicular to the sound rays, 

 which may be denoted by dS. Let the volume element 

 dV at X be defined as a cylindrical element whose 

 base area is dS and whose height is ds, an infinitesimal 

 extension along the direction of the wave propaga- 

 tion (see Figure 2). The volume included by dF is 

 therefore given by 



dV = aSds. (9) 



On the average, the sound which returns to from 

 X traverses the same ray traced out by the sound 

 which was incident on dV; this assertion, a conse- 

 quence of Fermat's principle, will be defended in 

 Section 12.5.5. Therefore, the scattered sound giving 

 rise to reverberation has been scattered directly 

 "backward." By the definition of the backward 

 scattering coefficient m, the intensity at a point, 1 yd 

 from dV, of the sound which returns to from dV is 

 just m/4x times the incident sound intensity at X, 

 times the volume of dV, or 



m 

 —hFb(e,(t>)dSds. 



If we now define h' as the intensity diminution be- 

 tween a point 1 yd from dV and the receiver at 0, 

 the intensity of the sound reaching from dF is 



m 

 4ir 



hh'Fbie,<l>)dSds. 



(10) 



The expression (10) gives the intensity of the sound 

 scattered backward from dF in the water at the re- 

 ceiving hydrophone. Let F' be the output of the re- 

 ceiver in watts when a plane wave of unit intensity 

 is incident on the receiver in the direction of its 

 maximum response. A plane wave of unit intensity 

 from some other direction (5,</)) will stimulate the re- 

 ceiver to an output of F'h'{d,i>) where h'{d,<f} is called 

 the pattern function of the receiver. Finally, a plane 

 wave of intensity J incident on the hydrophone from 

 the direction d,(l> will cause a watts output at the 

 terminals of the receiver of 



J-F'b'{e,4>). 



(11) 



