TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



Wigner distribution 

 f (z,e,r) 



= ^ ydz' e^'^o^^' r{^ + ^ z'.r) i>{z - I z',r) (33) 



where 



y f (z,0,r) de = |ijj(z,r) |2 (34) 



/ f (z,e,r) dz = l^e.r) |2 (35) 



F(z,e,r) = <f(z,e,r)> (36) 



8F ^ 8F ^ 1 ^ (^ - l^l ^ 

 gr ''az 2 8z \c2 / 38 



'/ 



where 



and 



de'w(z,e,e") [F(z,e',r) -F(z,9,r)] (37) 



dz ^ de ^ 1 8 /fo _ ^^ (38) 



dr ' dr 2 9z \c2 



w(z,e,e') = 2TT k^ f(z,e - e-, I e2 - I e-2j (39) 



Figure 24. WAVE KINETIC EQUATION 



191 



