TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC- EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



. d^ 1 a^ll; o 



dr ' 2k 9z2 

 o 



Let A = 



1 3^ 



n^(r,z,e)T-l + ia(r,z,i 

 k 



\li = 



2k 9z2 

 o 



B = 



o 

 2 



n^^l + ia 



(18) 



(19) 



j^ = lA^ + iBt 



(20) 



, . , iAAr/z iBAr iAAr/2 , , 

 ii(z,r + Ar) = e e e <|^(z,r) 



(21) 



iAAr =Tr-l I -ik2Ar/2k [ cr- , , ) 



(22] 



FEATURES ; 



1. Exponential accuracy in z 



2. Second order accuracy in r 



3. Exactly energy conserving (when a = 0) 



4. Unconditionally stable 



5. Computationally efficient 



6. Readily implemented. 



Figure 4. SPLIT-STEP FOURIER ALGORITHM 



164 



