P = a exp (iu>T) . 

 The travel time T and amplitude a satisfy the eikonal equation 



(VT)2 = c-2 



and the transport equation 



respectively. 



y. a^vT = , 



The eikonal equation may be solved by using the method of characteristics 

 for first-order partial differential equations. Characteristic curves, better 

 known as rays, are orthogonal to surfaces of constant time. They satisfy the 

 ray tracing equations 



d /I dx\ a 1 



ds \c ds/ dx c 



_d_ /I dy\ ^_1_ 



ds \c ds/ dy c 



d /I dz\ d 1 



ds \c ds/ dz c 



dT 



ds 



0^ ^ (1) 



' . l^)" 



\ds/ 



Once the rays have been foimd, the divergence theorem applied to the trans- 

 port equation produces the geometrical spreading law for the pressure amplitude. 



±= (— Jl-\ 

 a^ \ ^ ^'^O / 



-1/2 



or for the equivalent plane wave intensity. 



96 



