Recent Research on Ship Waves 



where A is the wave amplitude, k is the wavenumber (Ztt/X., if 

 \ is the wavelength), 9 the direction of propagation of the wave with 

 respect to variations of time t. The kinematic and dynamic properties 

 of the wave motion can be readily determined from the velocity poten- 

 tial, which differs from the above only by a factor 



(-ig/co)e 



if the fluid depth is large, the vertical z^-axis is positive upwards 

 with Zq = the plane of the undisturbed free surface, and the 

 (xQ,yQ,z-) coordinates are fixed with respect to the bulk of the fluid 

 volume. Finally, with the above restrictions, the frequency oj and 

 wavenumber k obey the dispersion relation 



k = coVg. 



The most general distribution of these elementary plane waves 

 is obtained by integrating over all scalar wavenumbers k (or fre- 

 quencies w) and wave directions 6, so that 



Uxo,yo)=J dk^ deA(k,e)e ° 



However, steady-state ship waves are independent of time, when 

 viewed from a moving coordinate system which translates with the 

 ship, say with velocity V in the +x direction, and this condition 

 restricts the frequency, or wavenumber, of the contributions to tlie 

 integrand in (2). If (x,y) denote moving coordinates with 



Xg = X + Vt 



then by direct substitution 



'°° ^^^ ik(xcos0 + y sJn&) + it(KV cos^-w) 



;(x,y) =^ dk j de A{k, 



e)e . (3) 



This integral will depend on time, for arbitrary amplitude functions 

 A(k,0) , unless 



cu = kV cos e (4) 



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