Sharma 



The nondimensional free surface deformation due to the moving ship is now 

 given by 



z = ^x,y). (3) 



Following Havelock (7) this function may be represented for large distances 

 (x^ + y2) by a general superposition of elementary plane waves 



/- 



C(x,y) = J[f(6') sin (sx+ uy) + g(0) cos (sx+ uy)] d0 , (4) 



s - sec 6 , u = sec 6 tan 6 , 



where 9 is the direction of propagation, s and u are the kinematically linked 

 longitudinal and transverse (circular) wavenumbers respectively, f(0) and g(d) 

 are two arbitrary functions, and the limits of integration may depend on the lo- 

 cation of the point (x,y). In the relevant literature i(d) and g(d) are sometimes 

 called the amplitude functions of the sine and cosine wave systems respectively. 

 However, this terminology can be misleading, since i(0) and g(9) do not actu- 

 ally have the physical significance of wave amplitudes. Thus Takahei (3a) and 

 Inui (2a) introduce a "wave slope" as the ratio of amplitude to wavelength in 

 direction 6: 



f(0)A((9) = i(d)/2rT cos^ 6 (5) 



and then question the validity of linearized wave theory because this ratio for a 

 simple cosine ship (Inuid C-201) approaches infinity as d tends to 7t/2. 



In the author's opinion such fallacious conclusions can be avoided by repre- 

 senting ^(x,y) as an integral in the transverse wavenumber u: 



^(x,y) = JlF(u) sin (sx + uy) + G(u) cos (sx+uy)] du , (6) 



u = s ^/7^^ , s2 = (^1+ (2u)2 + l)/2 . 



The redefined amplitudes F(u) and G(u) contain an additional factor 



d^/du = cos^e/(l + sin 2^) 



as compared to f(^) and gid); therefore the ratios 



F(u)/2-n- cos^ e and G(u)/277- cos^^ 



always remain finite, except when the linearized theory is genuinely over- 

 strained (for example by locating a singularity on the free surface). 



Even otherwise, the functions F(u) andG(u) come quite close to what may 

 be considered as physical amplitudes except for a constant factor of proportion- 

 ality. For instance, the average rate of flow of total energy across a transverse 

 vertical plane for large negative x is given by 



734 



