FUNDAMENTAL EQUATIONS 



43 



wave fronts, we must also generalize the definition of 

 a ray. We can no longer assume that the rays are 

 straight lines since we concede the possibility of re- 

 fraction and reflection. We shall, however, retain the 

 property that the rays are everywhere perpendicular 

 to the wave fronts. It is, of course, by no means 

 obvious that the results of this new approach will 

 agree with results from a direct solution of the wave 

 equation plus initial and boundary conditions. A 

 comparison between the results from the ray pattern 

 approach and the results from a rigorous treatment 

 of the wave equation will be carried out in Section 3.6 

 once the ray method has been fully described. It will 

 be found that in many practical situations these two 

 approaches lead to similar results. 



Geometrically, the rays and successive wave fronts 

 can be constructed as in Figure 1. The wave front at 



Figure 1. Huyghens' method for constructing suc- 

 cessive wave fronts. 



time t = (whose equation is given by T^ = — Coio) 

 is first drawn. In order to determine the wave front 

 at the time dt, the small ray elements are drawn as 

 straight-line segments perpendicular to the initial 

 wave front, as at {xi,yi,Zi). In the time dt, the end 

 point of the ray starting at {xi,yi,Zi) will have pro- 

 gressed to a point a distance c dt from the initial 

 wave front, where c is the velocity at the point 

 (.Xi,yi,zi). If this process is carried through for all the 

 points on the initial wave surface, the end points of 

 all the small ray elements will determine a second 

 surface, which may be regarded as the wave front at 



the time dt. By performing this process many times, 

 the wave front can be obtained at any time t. This 

 method of determining wave fronts by gradually 

 widening an initial wave front was first suggested by 

 the Dutch physicist, Huyghens, in the seventeenth 

 century, for the solution of problems in optics. 



3.2 



FUNDAMENTAL EQUATIONS 



3.2.1 Diiferential Equation of the 

 Wave Fronts 



Since the construction of wave fronts described in 

 the preceding section is purely geometrical, it must 

 be reformulated in mathematical terms for use in an 

 algebraic analysis of the sort we are carrying out. 



cdt 



m~P' 



Figure 2. Differential ray path. 



Let P in Figure 2 be any point on the wave front at 

 time t. The equation of the wave front is given by 

 equation (7). Let the coordinates of P be {x,y,z); let 

 PP' be the ray element emanating from P at the end 

 of a time interval dt; and let a,p,y be the direction 

 cosines of PP'. Then the coordinates of P' are 

 (x + acdt, y + ficdt, z -\- ycdt). Further, the wave 

 front at the time t -\- dt'is given by the equation 



W{x + acdt, y + ficdt, z + ycdt) = Co(t-ta + dt). (8) 



If cdt is assumed to be very small, the left-hand side 

 of equation (3) is very nearly equal to 



Wix,y,z) + [a- + /3- 



If we substitute this expression into equation (8), and 

 use equation (7), equation (8) reduces to 



dW , dW , dW Co ,„, 



dx dy az c 



The direction cosines a,P,y will next be eliminated 

 from equation (9). It is a well-known theorem' of 

 analytical geometry that the direction cosines of the 

 normal to the surface W = constant at the point 

 {x,y,z) satisfy the proportion 



dW dW dW 

 dx dy dz 



+ /3^— + 7 



jcdt. 



dz / 



