54 



RAY ACOUSTICS 



the intensity at a point P on a ray is not valid if at 

 any place between the projector and P the ray has 

 become horizontal. For, at a point where the ray is 

 horizontal, 6 = 0, which implies that Co — c cos 6q 

 = 0, by equation (26). This means that the inte- 

 gral in equation (57) becomes infinite at points where 

 the ray is horizontal and cannot be differentiated 

 under the integral sign. We therefore conclude that 

 the expression (59) is valid only for rays that are 

 always climbing or always dropping, but cannot be 

 used, for example, to examine the intensity near 

 places where the ray diagram predicts shadow zones. 



We now derive an expression for I{9o,R) which will 

 be valid even at points on a ray beyond where the ray 

 has become horizontal. This will be done by deriving 

 an expression similar to equation (58) in which the 

 variable of integration is 8 instead of y. 



In all cases, we have, because of equation (56), 



R = I cot edy 



r* dy dc , 

 I cote- -^ 



dcde 



Since 



dc dV cos n Co . 



— = V Co = sm d, 



do ddL cos ^0-1 cos do 



it follows that R = — 



Co 



cos doJ e, 



cos So 



dy, 

 cos d—dO. 

 dc 



(67) 



This expression for R has the advantage over equa- 

 tion (56) in that the integrand does not become 

 infinite for 8 = 0. This expression can, therefore, be 

 differentiated with respect to 8o even when the ray 

 passes through points at which it is horizontal. 



The variable 8o occurs explicitly in the factor in 

 front of the integral and as the lower limit of inte- 

 gration and implicitly in the terms 8^ and dy/dc. 

 Taking this into consideration, it follows that 



dR 



ddo 



Co sin 8, 

 cos^ do 

 c? sin 6t 



cos^ 8i 



J $0 



do n d^y 

 'o J So dc^ 



dy 



— cos 8d8 



dc 



cos^ dd8 



Co sin Sofcos' Ok 

 cos^ 9o L sin 8h 



s^ 8j dy\ cos^ 8o /dy\ 

 n8h\ dc/h sin 8o \dc/o. 



(68) 



Though the expression (68) is much lengthier than 

 the expression (58), it has the advantage of being 

 valid at points on the ray where 8 = 0. The resulting 

 intensity, calculated by using equation (55), will also 

 be valid at all points on the ray. The quantities R and 

 dR/ddo must be substituted from equations (67) and 



(68). These expressions can be simplified by means 

 of the assumption that all angles are small. With this 

 assumption all cosines of angles can be replaced by 

 one; the sines may be replaced by the angles them- 

 selves; and among a number of terms those multi- 

 plied by higher powers of the angles may be disre- 

 garded. The simplified expressions for R and for 

 dR/d8o then take the form 



■<,] - 



J So C 



and 



^o"tWo~^wJ' ^^^^ 



From equations (55) and (65) we have, as a general 

 expression for the transmission anomaly A, 



R 



-Co I 



d8 



(69) 



10 log 



/dR . \ 



(71) 



\ R cos da I 



If we assume that 8o and 8h are both so small that 

 cos 8o can be replaced by one, and sin 8h by 8^, 

 formula (71) becomes 



IdR 



A « lOlogl 



(71a) 



By putting in the approximate values of R and 

 dR/d8o from equations (69) and (70), an explicit ex- 

 pression can be obtained for the transmission 

 anomaly. 



In the appUcation of equations (67) to (70) one 

 precaution must be taken. While the integrands of 

 the integrals that occur remain finite when the angle 

 of inclination becomes zero, these expressions ap- 

 proach infinity as the gradient approaches zero. They 

 are, therefore, not suitable for the treatment of 

 propagation through isovelocity layers. 



Another method of computing the transmission 

 anomaly that may be used whether or not a ray has 

 become horizontal and is in a more convenient form 

 for numerical computation is given under "Combina- 

 tion of Linear Gradients" in Section 3.4.2. 



3.4.2 



Applications 



Section 3.4. 1 was devoted to deriving formulas for 

 the intensity out along a ray as a function of the hori- 

 zontal range and the velocity-depth variation. These 

 formulas involve line integrals and are too compli- 

 cated to use for practical intensity computations. 



