Wave Analysis Techniques to Achieve Bow-Wave Reduction 



method is also best suited to application on a full-size ship for investigating 

 scale effects. 



Although the longitudinal-cut methods had been proposed with a certain 

 reservation regarding their feasibility, see Ref. 4b, the numerical and experi- 

 mental experience gained so far and reported in this paper seems to confirm 

 their validity and practicability. 



DEFINITION OF AN AMPLITUDE SPECTRUM 



The surface disturbance produced by a moving ship cannot in general be 

 described in terms of a one -dimensional wave spectrum. However, if one ne- 

 glects certain local effects and considers the asymptotic behavior of the far 

 field, it turns out that only one wavelength, the so-called free wave, predomi- 

 nates in each direction of propagation. The remaining terms decay so rapidly 

 that the wave-making resistance in an ideal fluid is determined entirely by 

 these free waves. For most practical purposes, therefore, the wave-making 

 characteristics of a ship form may be considered as established if the free 

 wave spectrum, that is, the pair of functions relating the amplitudes and phases 

 of the component plane waves to their wave numbers, is known. Since each 

 component wave has a different direction of propagation and wavenumber, the 

 spectrum can be defined in various ways depending upon whether the total free 

 wave system is expressed as an integral in the direction of propagation or in 

 one of the possible wavenumbers. It will be found convenient for the present 

 work to first define an unambiguous amplitude spectrum. 



Consider a right-handed Cartesian coordinate system Oxyz moving steadily 

 with the ship at speed V in the direction of Ox. Let Oxy be the undisturbed free 

 surface, Ozx the center plane, and Oyz the midship section of the ship. Take Oz 

 vertically upward, i.e., against the acceleration of gravity g. The steady sur- 

 face deformation produced by the moving ship can then be expressed by the 

 equation 



z = ^(x.y), (1) 



where i is an arbitrary function of (x,y). 



From now on it will be advantageous to use only nondimensional quantities. 

 This can be easily achieved by multiplying or dividing all lengths and wavenum- 

 bers by the quantity kg = g/v^, which is a fundamental unit of the problem. The 

 only relevant dimensional quantity other than a length or a wavenumber needed 

 here is the wave resistance R^, which will be rendered nondimensional by the 

 relation 



Rw=RwkoVpv2, (2) 



where p denotes the density of water. (The term "wave resistance" will be used 

 in this paper to designate the dynamic reaction of any given free wave system 

 as predicted by linearized potential theory.) When dimensional quantities are 

 used in the following pages for special reasons they will be identified by under- 

 lining. 



733 



