FUNDAMENTAL EQUATIONS 



45 



If n is a constant, that is, if the sound velocity 

 is independent of depth, the rays (17) are clearly 

 straight lines. 



3.2.2 Differential Equations of Rays 



It may be argued that the replacement of the wave 

 equation by the ray treatment as represented by the 

 differential equation (13) has little to recommend it- 

 self. It appears that one difficult partial differential 

 equation has merely been replaced by another, which 

 might resist attempts at solution as effectively as the 

 first one. 



djna) _ djnP) _ d(na) d(ny) d(nfi) d{ny) 



Figure 3. Specification of direction of ray element ds 

 by direction cosines. 



Further examination shows, however, that the new 

 equatiorf (13) has two properties which tend to sim- 

 plify its solution. First, equation (13) contains no 

 time derivatives. This means that it describes the 

 propagation of a disturbance in terms independent 

 of the frequencies which make up this disturbance. 

 Second, it is possible to set up ordinary differential 

 equations that describe the path of individual rays; 

 the latter equations will be derived in this section. 



We start with the equations (14), from which we 

 proceed to eliminate W. This can easily be done by 

 use of the formulas S^W/dxdy = d'^W/dydx, etc. By 

 differentiating the first equation of (14) with respect 

 to y, the second with respect to x, and by equating 

 the results, a relation between a, /3, and n is obtained. 

 Proceeding similarly with the other equations, we 

 obtain the following relationships which must hold 

 at any point of the ray pattern: 



dy 



dx 



dz 



dx 



dz 



dy 



(18) 

 These equations can be developed further to yield 

 the changes of a, /?, and y along the path of an indi- 

 vidual ray. If the arc length along the ray path from 

 a given starting point is denoted by s, we have 



d(na) 



d(na) dx 



— — h 



dx ds 



d(na) dy d(na) dz 

 ds dz ds 



dz 

 ds 



0; T = y 



ds dx ds ' dy 



We see from Figure 3 that 

 dx _ dy 



ds ' ds 



Thus equation (19) turns into 



d{na) d(na) , d(na) d(na) 



ds dx dy dz 



which, upon using the relations (18), becomes 



(19) 



(20) 



d(na) 

 ds 



d(na) d(na) 

 a—, h ;8^^ h r 



dx 



{a^ -H /32 + 7^) 



+ 



dx 

 dn 

 dx 



( da 

 \ dx 



d{na) 

 dx 



d^ dy 



dx dx 



>■ 



(21) 



The first parenthesis equals unity, because it is the 

 sum of squares of direction cosines; while the second 

 parenthesis, which is equal to one-half times the 

 derivative of the first one, vanishes. Thus equation 

 (21) simpUfies to 



d{na) dn 

 ds dx 



After similar calculations are carried out for d{n§)/ds 

 and d{ny)/ds, we get the following set of three ordi- 

 nary differential equations : 



d{na) dn d{n0) dn d(ny) dn 

 " ~ " ~ ' ~ dz' 



ds 



dx 



ds 



dy 



ds 



(22) 



It is understood that n, the index of refraction, is a 

 given function of x,y,z. 



We now deduce an important result for the special 

 case where the sound velocity is a function of the 

 vertical depth coordinate y alone. We shall show that 

 for this case the entire path of an individual ray lies 

 in a plane determined by the vertical line through 

 the projector and the initial direction of the ray. 



Let the origin of coordinates be taken at the pro- 

 jector, and let the direction cosines of a ray leaving 



