170 



THEORY OF SEAKEEPING 



600 



rJeiimonn Seo Energ;^ 

 Spec+rurn for 40 Knot 

 Wind- Corrected for 

 \Z Knot Ship Speed 



5: 



OZ 0.4 6 0.8 1.0 1.; 



Ship Frequency 2 ''^/Tg, 1/Sec. 



1.4 



Fig. 10 Energy spectra and response of a ship model in an ir- 

 regular sea. Series 60, 0.60-block-coefficient model 5 ft long, 

 expanded to 500-ft ship (from Korvin-Kroukovsky, 1956; ab- 

 stracted from Lewis and Numata, 1956) 



Pitch Spectra 

 Model ni3 

 Speed ,Z.53 Ft/Sec, 



6 8 2 4 6 6 



Frequency of Encounter, cc'g 



Fig. 1 1 Motion spectra predicted by three different methods 

 for destroyer model 1723 in irregular long-crested sea: (a) By 

 direct analysis of motions in irregular waves (solid lines). (V) 

 By calculation from wave spectra and experimentally measured 

 responses to regular waves (dash lines), (f) By calculation 

 from wave spectra and analytically computed responses to regu- 

 lar waves (+) (from Korvin-Kroukovsky and Jacobs, 1957) 



with direct ion. Tlii.s i.-: ('([uivaleiit to mea.'^urenipiit of the 

 wave .spectrum at a fixed point or to measurement of the 

 ilirectional spectrum and tlien integration with respect 

 to direction. In either event, the directional aspects 

 are not recoverable from the scahir spectrum. When the 

 wa\'e spectrum is transformed to a wave-encounter spec- 

 trum (for instance, for a ship speed of 12 knots in head 

 seas), the directional implications of the spectrum are 

 ignored so that the comiiuted response presupposes that 

 all the waves trax'eled in the same direction. If that is 

 the case (a spectrum of swell, for example), the work reji- 

 rcsented in Fig. 10 is valid; if not, the wave .spectrum 

 is inadequate and the transfer function is likewise in- 

 adeciuate, since it was only made for the head-seas condi- 

 tion. Fig. 10 applies to towing-tank tests in irregular, 

 long-crested waves so the procedure is here valid. 



The most significant feature of the St. Denis-Pierson 

 work is that it permits evaluation of ship motions in ir- 

 regular short-crested seas by embodying the product of a 

 ilirectional spectrum with a "directional transfer func- 

 tion." In so doing, it was necessary to de\'elop a com- 

 plex freciuencv mapping based on 



., V cos X 



y 



g 



(24) IS 



in ordei' for all terms of (2;^) to be in the w, domain. The 

 difficulty in such a mapping arises in the inverse trans- 

 formation which is not uni(|ue but results in a cjuadratic 

 expression which forces the authors to make their tran.s- 

 formation in three separate regions to avoid confusion. 

 A further contribution to the elegant treatment of the 

 frequency mapping is the inclusion of the Jacobian in 

 the transformation in order to I'etain the total energy in 

 the spectrum being mapped. 



The true significance of eciuation (23) is accomplished 

 by a decomposition and resynthesis of its elements to ar- 

 rive at the response spectrum. 



The directional spectrum (either measured or as- 

 sumed) is divided into three nonoverlapping parts ap- 

 propriate to the regional divisions just mentioned, so that 

 uni(|ue inverses olitain when mapping into the oj, — x, 

 plane by eciuation (24) 



A'(uj,x) = /ti(co,x) + Ki (co,x) + ^'3 (^,x) (25) 



I'.ach term in eciuation (25) is multiplied by the square 

 of the respon.se-amplitude operator (transfer function), 

 such that three expressions of the form 



/^,(a),x) r*,(^.X,"). {i = 1,2,:3) (26) 



result. Each of the terms in equation (26) is then 

 mapped into the oj^ — Xf ilomain to give three terms of the 

 f ( )i'm 



i?,(c.„ xJ 7'*, (co... X.), (i = I. n, III) (27) 



The spectrum of the ship response is th(>n obtained by in- 

 tegration of the terms in (27) 



•'' The symbol x rppresent.s the relative lieiiding of ship to wave 

 ;is given in St. Deni.s and Pierson (1953). This is equivalent to 

 9 in Chapter 1 and is retained in this account of their work. 



