CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 



53 



Substituting this expression into equation (51), we 

 obtain for the intensity / at the range R the expres- 

 sion 



/(fi.gp) _ cos Op 



F ~~ ^dR . ' (55) 



ado 



It is now necessary to obtain expressions for R and 

 for the partial derivative of R with respect to do. We 

 begin by calculating the range R. We have 



"'dx, 



(56) 



R 



—dy = I cot edy, 

 «o dy -J !/„ 



which, upon substituting cos 6 = {c/ca) cos 5o from 

 equation (26), becomes 



cdy 



(57) 



R = cos di. 



'"X.V^ 



We differentiate this expression for /2 as a product of 

 functions of Bo, assuming that the usual formulas for 

 differentiating under the integral sign are valid. When 

 the two resulting integrals are put over one denomi- 

 nator, the whole expression for the derivative simpli- 

 fies to 



dR , . . r* cdy 



R _ 2 . C 



ddo' '"'"'H.((^-c^cos^fl„)^ 



(58) 



Substituting the expressions for R and for dR/ddo, 

 equations (57) and (58), into equation (55), we find 

 for the intensity I the expression 



(59) 



sin 00 sin 



J y. sin 9 Jy, 



dy 

 !/„ sin'fl 



with 



and 



sin dh 



^^'-(-:) 

 ■/^ 



^ COS^ On 



COS^ Oo. 



(60) 



(60a) 



I 



For application, this formula suffers from two de- 

 fects. First, it is not sufficiently simple; second, it is 

 not sufficiently general. The second point will be 

 taken up later, where it will be seen that equation (60) 

 does not cover the important class of conditions 

 where the sound ray becomes horizontal anywhere 

 en route. As for the first point, we shall simplify 

 equation (59) for application to these cases where it 

 is valid. 



Under ordinary circumstances, c does not vary be- 

 tween the sea surface and operational depths by 



more than 5 per cent of the surface velocity. As a 

 result, those rays which leave the projector at a 

 moderate angle will not become so steep that the 

 sine of the angle cannot be replaced in good ap- 

 proximation by the angle itself. Thus we may re- 

 place the expression cj — c^ cos^ do, which appears 

 three times in equation (59), by the approximation 



^-e cos^ do « c?[i - (^^tA^)\i - eS)] 



« (^(^ - 2e), 



(61) 



in which e, as in Section 3.3.2, stands for the ex- 

 pression (c — Co)/co. As a result, we can replace equa- 

 tion (60) by the approximate relation 



-' « '- ^■. (62) 



dy 



while the range is given approximately by the expres- 



2ey 



sion 



R 





dy 



-„A/flg - 2e' 



(63) 



In most transmission work, the sound field inten- 

 sity is reported either as transmission loss or as 

 transmission anomaly. The transmission loss H is 

 defined as the ratio of the source strength F and the 

 soimd field intensity in decibels, 



ff=101og--- (64) 



The transmission anomaly A is defined as the ratio 

 of the intensity predicted by the inverse square law 

 and the sound field intensity I, also in decibels, 



A = 10 log 



F/R^- 



10 log 1^. 



(65) 



On the basis of this definition and equation (62) the 

 transmission anomaly will be given by the approxi- 

 mate relationship 



"" dy 

 ,e%y) 



A « -lOlogf^ + lOlog f 



JyJiy) •^v.i 



-f 10 log do + 10 log dH, 



(66) 



d{y) = VeS - 2«. 



where 



Where it applies, this formula is simple enough to 

 lead readily to results of practical significance, as 

 under "Layer Effect" in Section 3.4.2. 



From its mode of derivation the expression (59) for 



