168 



THEORY OF SEAKEEPING 



0.5 



4 5 

 OJ 



1,0 2.0 3,0 4,0 5,0 g,0 7.0 

 OJ " 



9.0 lao 



Fig. 8 Empirically determined response factors for heaving of Fig. 9 Empirically determined response factors for pitching of 

 ship model (from Fuchs and MacCamy, 195 3) model (from Fuchs and MacCamy, 1953) 



floating rectangular block using experimental values of 

 added masses and damping coefticients, as determined 

 from oscillations in still water. These kernel functions 

 are applied to recorded irregular wave motions and the 

 predictions are compared with \'alues read from o5-mm 

 motion picture records of the motion. The experimental 

 work has been reported separately hy Sibul (1953). 

 Similar investigations have been made with a model 

 ship. In order to avoid the laborious numerical inte- 

 grations mvolved in computing the response functions, 

 these were determined experimentally." 



The theory of Fuchs (1952) is applied to a ship motion 

 in long-crested irregular waves so that the heaving and 

 pitching motions are expressed by the aforementioned 

 convolution integral 



E{t) = -^ 



nit) K^{t - T)dT 



(15) 



(16) 



where the kernel function is given by 



K,{t) = -^ r 7'.(oj)c-'"'fZ.o 

 V2;r J -" 



T,(oj) is the response of the block to a unit amplitude 

 sinusoid of frecjuency co. Since the kernel function is a 

 form of transfer function by virtue of Ti(ci)), it is seen 

 that the heave or pitch function, E{t), is a deterministic 

 re.sponse which results from linear superposition of a 

 number of simple responses to regular \\-a\'e trains. 



Fig. 7 shows the results of computations made from 

 equation (15) where K^ir) was determined for a tow- 

 ing-tank model. The upper curve is the record of the 

 water surface elevation, the middle one shows heaving 

 of the model, and the lower one pitching of the model. 

 Solid lines show the measured model motions and the 

 broken lines show the motions computed by equation 

 (15) using the responses Ti(u) measured in a series of 

 sinusoidal waves. Figs. 8 and 9 show the real and imag- 

 inary parts of these responses for heaving and pitching 

 respectively. 



In equation (15), the kernel is independent of the par- 

 ticular wave system encountered and conseciuently needs 



to be computed or measured only once for each sea- 

 keeping event of each ship. This is the case in the sta- 

 tistical treatment as well, as will be shown. The appeal 

 of equation (15) lies in the fact that in one operation it 

 separates r]{t) into its Fourier components, multiplies the 

 Fourier components by their respective unit responses 

 and synthesizes the results into the predicted motion 

 E{t). 



The solution of the inverse problem, that of evaluating 

 the model's frequency-response function T'i(co) from tests 

 in irregular long-crested waves is accomplished by not- 

 ing that the recorded wave surface elevation can be rep- 

 resented by 



'7(0 



a((iLi)e 



(17) 



while the dependent function E(t), shown by equation 

 (15) can, in view of (IG) and (17), be written as 



Eit) 



= _L r 



V-IttJ- 



r,{o))a{i^)c'"'du 



(18) 



Designating the Fourier transform of equation (18) by 

 ^^(to), the frequency-response function is obtained as 



rxco) 



(19) 



a(a)) 



The application of Fourier transform pairs here is rem- 

 iniscent of the statistical treatment of ocean waves in 

 Section 1-8 and indeed it will be .shown that an expression 

 similar to (19) is obtained from the wave and motion 

 spectra, both of which may be considered as Fourier 

 transforms of particular realizations of their respective 

 time histories. In fact, it will be seen that this mathe- 

 matical model of ship response to waves is essentially 

 equivalent to the statistical model. It fails, however, to 

 achieve the generality of the statistical model in that it 

 treats only the case of long-crested irregular waves. 



3.3 Ship Motion Studies Based on Linear Superposi- 

 tion. R. E. Froude (1905) wrote: "Irregular waves such 

 as those commonly met when at sea... are only a com- 

 pound of a number of regular systems (individually of 



