Testing Ship Models in Transient Waves 



where G( joj) is not properly the response of a physical system but the ratio of 

 the frequency response of two physical systems. 



With this philosophical restriction in mind, we will continue to call wave 

 height an input and ship motion an output, but at no time will the intervening 

 system be required to have the characteristics of a real physical system. 



Wave Transients 



The study of transient waves on a free surface is an advanced top in hydro- 

 dynamic theory, but it is amenable to the "systems" approach if linear wave be- 

 havior is assumed. Consider first that all waves are traveling in the same di- 

 rection on a surface of infinite extent and in a fluid of infinite depth. Suppose 

 further that a wave disturbance of finite energy per unit crest length has been 

 traveling for all time and is observed at a single stationary point Xj. The wave 

 height 77(Xj, t) may be expressed in terms of its Fourier transform by the re- 

 lation: 



-CO 



•^(Xi,t) = ^J da,N(x^,c^) e-'"^ (6) 



- 00 



Following the technique of Stoker [2], the complex quantity N(x^,oj) is vis- 

 ualized as the infinitesimal wave component with frequency + w. This wave at 

 any instant of time extends over the entire plane and at any one point persists 

 for all time. At another point Xj , which is a distance x along the direction of 

 wave travel from Xj, this same infinitesimal wave is observed but with a phase 

 lag of co^x/g radians, according to linearized wave theory. That is, the time 

 history at Xj is given by 



T7(x2,t) = 2^ J doj N(Xj,a)) e' 



ja)|a)|x/g jojt (7) 



where the absolute value of w is used in the phase operator to ensure that the 

 quantity 



N(x^,aj) = N(x,,o,) e-^"'""'/e (8) 



is conjugate with N(x2, - co) . This property is necessary in order that a Fourier 

 transform represent a real function of time. 



The operator e"^"''^'''/^ can thus be viewed as the "transfer function" of 

 water, or the frequency response function which relates wave heights measured 

 at two points separated by a distance x in the direction of travel. 



511 



