^.y» 2 + ((f) 2 + (f) 2 )^-(f) 2 -(|f) 2 -o (6o, 



Because of the exponential decay in the z direction, the wave motion is 



-1 2 



significant only in a surface layer of thickness 0(y q ). Therefore, z' is 



0(Fn ) and the second term of the above equation is to be omitted. Then 



we have 



«*» - /(I) 2+ (f) 2 



that means the local wave number. Substituting Equation (59) in Equation 

 (58) and taking terms of the lowest order, we obtain 



Ki + *oH) - k<x - y) (62) 



Kf- f) 2 V(f) 2 + (|f) 2 



This is identical with the dispersion relation which has been indicated by 



19 

 Keller. The solution of the above differential equation determines the 



phase function S(x,y). Keller has proposed a kinematical theory of waves 



superimposed on the nonuniform flow around the ship by the analogy with 



the geometrical optics. 



Now let us show the possibility of giving an analytical expression for 



the wave function by a coordinate transformation. To find the solution, 



we employ the curvilinear coordinates along streamlines and equipotential 



lines of the double body flow in the plane z = 0. Designate the velocity 



potential of the double body flow by $ and the stream function on the 



plane z = by f , the latter of which is slightly different from the stream 



28 



