TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



RAY TRACING: 



Exact: 



Parabolic approximation; 



dr2 2 dz 



(12) 



(13) 



Both give ; 



n cosO = const 



(14) 



Rays are same except for range scale. 



NORMAL MODES EXPANSION: 



Exact ; 



i/k2 + u r 



= Y* A (f) (z)e o a 



a a 



(15) 



Parabolic approximation: 



i {k ^ ^) 



p =y!A (j) (z)e ^ o ■ 2k ' (16) 

 ^-' a a o 



a 



Normal modes are the same. 



1 ^a 



Phase error w r r ~ 2it when r « 10 nm 



8 k^ 



(17) 



Figure 3. VALIDITY OF PARABOLIC APPROXIMATION 

 (2-'D, Stratified Ocean) 



162 



