Testing Ship Models in Transient Waves 



w(t) Ri v'g/TTx cos f^ - ^j 



(16) 



as ar becomes large. This function can be heuristically interpreted as a linear 

 frequency sweep which looks back into the past of the wave height signal being 

 processed and detects those frequency components that have gone by at a past 

 time which would influence present wave height at a distance x in the direction 

 of the wave. This is motivated by the convolution integral 



v(- 



x^,t) - At vj(t) t^Cxj, t -t) 



- 00 



which is the time domain equivalent of 



N(x2,w) = N(Xj,a)) e 



j wl OJl x/g 



(17) 



(18) 



Fig. 4 - Weighting function of unidirectional waves in water 



Suppose we ask the physically ridiculous but mathematically interesting 

 question: "What signal would be observed at Xj if a unit "impulse" of wave 

 height were recorded at X2 ?" A unit impulse is described as a signal which is 

 zero except at an instant in time where it has infinite amplitude, such that the 

 integral over this point has a value of unity. The Fourier transform of a unit 

 impulse is 1.0. 



Solving Eq. (18), we find that 



N(Xj,aj) 



jcol 0)1 x/g 



(19) 



This can be readily shown to have an inverse transform which is equivalent to 

 that shown in Fig. 4 except for a reversal of the time variable. 



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