K = 0(e ) and the singular perturbation in the near field suggests 

 3/8y = 0(e~ ), 3/3z = 0(e~ ), we obtain for the lowest order, the 

 Helmholtz equation 



l!| + l!| = K 2 ? (218) 



3y 3z 



or, by the Fourier inversion, 



^f + ^f=K 2 * (219) 



3y 3z 



The omitted terms are of the order higher by e. If we' employ the cylindri- 

 cal coordinates r, 0, 



z = - r cos 0, y = r sin (220) 



particular solutions of the differential equation are 



I (Kr) \ i cos n0 

 n (221) 



K (Kr) ) I sin n0 



n 



where I and K are Bessel functions of imaginary arguments, 

 n n 



Since the forward velocity is absent in the boundary value problem, the 

 boundary condition on the free surface for the diffraction potential 

 applies to the function lp too. 



|^-K^=0atz=0 (222) 



dZ 



One can construct a solution which satisfies the above equation and 

 vanishes at infinity by combining the basic solutions of Equation (221) 

 as follows. 



79 



