SHIP MOTIONS 



169 



comparatively small maguitudej of various periial.s, rang- 

 ing through the whole gamut (so to speak) represented by 

 our diagrams [of behavior in regular waves] and more. 

 And the effect of such a compoimd wave series on the 

 model would be more or less a compound of the effects 

 proper to the individual units comprising it." 



This pronouncement of Froude, which antedates the 

 conclusion of oceanographers concerning a realistic model 

 of the sea surface, has become the basic hypothesis for 

 ship motions studies. Fuchs and MacCamy (1953) used 

 it in their deterministic approach, as has been discussed. 

 These, however, were not the first applications of the 

 idea of linear superposition. 



Legendre (1933) used this principle in investigating the 

 rolling characteristics of two cruisers by means of theory, 

 towing-tank tests and observations on ships at sea. 

 He summarized his findings as follows: 



1 "Rolling computed on the basis of the mean ob- 

 served .wa\'e does not lia\'e the character and is much 

 smaller than the observed rolling. 



2 "The summation of roll angles, computeil on the 

 basis of the observed wave, decomposed into its sinu- 

 soidal components, has for its period the natural period 

 of the ship. It is of the same order of magnitude as the 

 observed rolling. 



3 "In order to predict the rolling of a projected ship 

 it is necessary to investigate the general characteristics of 

 the actual wave." 



Actually the work of Legendre was ahead of his time 

 since neither mathematical nor physical tools were then 

 developed sufficiently to permit a thorough analysis. 



Section 2.1 of this chapter treated the ship response to 

 simple harmonic waves and such results in conjunction 

 with a given input (either an energy spectrum or a time 

 history) lead to an output which represents the ship mo- 

 tion. The transfer function, which dejiends on the geom- 

 etry and physical characteristics of the ship as well as 

 ship speed and relative heading, is determined either 

 theoretically or by model tests. It ^^■ill be worth while to 

 discuss the transfer function, before the statistical 

 method is demonstrated. 



It was shown in Section 2. 1 that the heaving and pitch- 

 ing amplitudes of a ship's motion in sinusoidal waves are 

 expressed as complex amplitudes 



Zoe" 



a nlf 



Zo (cos 5 + /' sin 5) 

 Bq (cos e -1- i sin e) 



(20) 



where Zo and da are the real amplitudes and 5 and e are 

 phase lags of a ship's motions with respect to wave form. 

 When a ship's response to waves is obtained by model 

 tests, the real amplitudes Zo and da and phase lags 5 and e 

 are measured directly from the test records. When the 

 amplitudes are computed by the methods outlined in 

 Section 2.1 and Appendix C, the result is first obtained 

 in the form, C + iQ, and is subsecjuently converted to 

 the form indicated by the left-hand sides of equations 

 (20). 



For use in connection with irregular wave inputs, the 

 foregoing ship-motion amplitudes are referred to the unit 



sinusoidal wave amplitude U> obtain 



7\(cc) = -° f'* = -' cos 5 + i - sin 5 = c. (co) + irL(o^) 

 a a a 



7 ^(co) = - e = - cos « + 7 - sm t = Cg(ic) + iqsicc) 

 a a a 



(21) 



antl in this form arc known as "frequency-response 

 functions" (Press and Tukey, 1-19.56). St. Denis and 

 Pierson (1-1953) used the term "response amplitude oper- 

 ator" for the square of the absolute value of the fre- 

 quency-response function. In what follows, one mode of 

 a ship's motion will be considered at a time, and the sub- 

 scripts z and 6 will be dropped. It should be empha- 

 sized, however, that the single mode response includes all 

 the effects of coupled motions. The symbol Ticc) was 

 used l)y Pi'ess and Tukey for the fr(;quency-response 

 function. In the present exposition it will be used only 

 for the absolute value: 



T(u:) = Zu/a or du/a 



(22) 



The symbol Ti(oi) will be used for the complex form 

 shown by equations (21); i.e., indicating both the 

 amplitude and the phase lag. 



3.4 Statistical Methods For Studying Motions in Ir- 

 regular Waves. St. Denis and Pierson (1-1953) treated 

 the relationship between the wav'e spectrum and a ship's 

 motion spectrum as given by 



$(a.J = EicoJ T-(o>„ 



(23) 



which states that the energy spectrum of a particular 

 seakeeping variable, $ (ship motion, strain, etc.), at a 

 given heading and speed, equals the product of the en- 

 countered wave spectrum and the response operator 

 (transfer function) for tho.se conditions. This presenta- 

 tion introduced, for the first time, probability concepts in 

 conjunction with the linear-superposition theory. 



A practical use of this expression is illustrated by Fig. 

 10. The upper part (A) of this figiu'e shows a wave spec- 

 trum, in this case a Nevmiann spectrum for a 40-knot 

 wind, corrected to the ship's speed of 12 knots (this will 

 be explained subsequently). The symbol We designates 

 the frequency of wave encounter. The middle part of the 

 figure (B) is a plot of T-{o3^, i.e., the square of the ab- 

 solute value of the frequency response function. The 

 lower part of the diagram (C) is the spectrum of the 

 ship's pitching, ^(co,). At each abscissa, corresponding 

 to a given frequency w, = 2-w/Te, the ordinate of the 

 lower section is the product of the ordinates of the upper 

 and middle sections. Once the spectrum ^(a),) is com- 

 puted, various average characteristics of the ship's mo- 

 tion are obtained from it, by means of relationships given 

 in Section 1-8.6. 



Care must be taken in the interpretation of the e\'ents 

 depicted in Fig. 10. The Neumann spectrum given there 

 is a scalar wave spectrum; that is, it embodies the di- 

 rectional properties of the waves it describes but does not 

 permit a quantitative evaluation of energy distribution 



