where Ai is the Airy function of the first kind, and T, p , g, and h are to be 

 found. Upon substituting this ansatz into the reduced wave equation and compar- 

 ing similar powers of frequency, one obtains 



T = (T+ + T_)/2 , 



2/3 p^/^ = (T+ - T_)/2 , 



g = p 1/4 (a^ + a_)/2 ^ 



-1/4 

 h = p (a^ + a_)/2 . 



As illustrated in figure 8, subscripts + and - refer to the two rays that touch 

 and do not touch the caustic before reaching the field point, respectively. There- 

 fore, all the quantities appearing in Ludwig's representation may be expressed 

 in terms of the travel times T+ and amplitudes a+ of classical ray theory. 

 Brekhovskikh's solution lacks the term involving the Airy function derivative, a 

 term that is important away from the caustic. As a result, Ludwig's solution 

 has a larger domain of validity. 



CAUSTIC 



Figure 8. Classical Rays Used to Compute a 

 Uniform Asymptotic Solution at a Caustic 



105 



