CALCULATION OK SOUND INTENSITY FROM RAY PATTERN 



51 



When wt is small the terms in (we)^ may be neglected; The assumption for the case of constant sound 

 imder these same conditions Y — Yo is small com- velocity and its generalization for the case of variable 

 pared with I'o. Equation (47) thus provides a useful velocity are illustrated in Figure 13. In the left-hand 

 approximation when the depth error is relatively 

 small. By translating from t back to c this equation 

 becomes 



•I'o 



Y -Yo-' ;^\ (c- Co)dy. (48) - 



w 

 2co. 



The expression (48) has a simple interpretation in 

 terms of the velocity-depth diagram. The integrand 

 (co — c)dy is just the black area in Figure 12; thus 



Yo 



-^e 



C-Co 



Figure 12. Depth correction as area under bathyther- 

 mograph trace. 



the integral from to Fo is the shaded area between 

 the velocity-depth curve, the vertical line c = Co, and 

 the horizontal line y = Yo. Qualitatively, we may 

 conclude that the depth correction will be large for 

 steep gradients and larger if these steep gradients 

 are located at shallow depth. 



3.4 CALCULATION OF SOUND INTENSITY 

 FROM RAY PATTERN 



The foregoing sections were devoted exclusively to 

 tracing the paths of individual rays and stated 

 nothing about sound intensity in the ray pattern 

 except for the special case of spherical waves. For 

 that situation an assumption that the energy flows 

 out radially along the sound rays led to the same 

 inverse square law of intensity decay which was de- 

 rived rigorously under "Spherical Waves" in Section 

 2.4.2. It is a plausible generalization to assume that 

 energy always travels out along the rays even when 

 the soimd velocity is not constant and the rays are 

 curves. 



SOURCE 



SOUND VELOCITY CONSTANT WITH DEPTH 

 VELOCITY DISTANCE 



SOURCE 



SOUND VELOCITY CHANGING WITH DEPTH 



Figure 13. Effect of vertical velocity changes on ray 

 paths. 



drawing, the rays are straight lines; and the energy 

 radiated by the source into a small solid angle is con- 

 fined inside the indicated cone. Because of this as- 

 sumption, we get the exact inverse square law of in- 

 tensity loss. In the right-hand drawing, the rays are 

 curves; and the energy radiated by the source into 

 the same small solid angle is confined inside the horn- 

 shaped surface displayed. In this general situation 

 the energy flow through normal unit area depends 

 not only on the distance r from the source, but also 

 on the total cross-sectional area of the horn which, in 

 turn, depends on the way the rays are bent. Thus it 

 is clear that the inverse square law will not, in general, 

 be predicted even by this simplified ray treatment. 



3.4.1 General Formulas for Change 

 of Intensity along a Ray 



The prediction of shadow zones as described in the 

 preceding section is only one part of the description 

 of the expected intensity distribution. There remains 

 the problem of calculating the intensity in regions 

 traversed by the rays. We already know that this in- 

 tensity loss will not exactly obey the inverse square 

 law except in very special cases. 



