TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



ELLIPTIC WAVE EQUATION; 



-f (^1^) +ff + 4l|f + k2 [n2(r,z,e) + ia(r,z,e)] 

 r 8r \ 3r / dz-^ r^ dQ'^ o 



P = (1) 



k = — = const, n = c /c(r,z,e) (2) 



o c o 



o 



Let p (r,z,e,aj) = i|j(r ,z, 6 ,03) H (k r) (3) 



o o 



Approximations: 1. k r >> 1 (far field) (4) 



2. l|^| « k \il)\ (small angles) (5) 



PARABOLIC WAVE EQUATION: 



2ik 1^ + |-T^ + ^ 1^ + k2 [n2(r,z,e)^l + ia(r,z,e)] i|; =^ (6) 

 o 3r 9z^ r^ 80^ o 



Neglect coupling between azimuths: 



i|^ + ^ f-l + ^ [n2(r,z,e)-l + ia(r,z,e)] i|; = (7) 



3r 2k 9z^ 2 

 o 



Figure 1. PARABOLIC EQUATION METHOD 



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