Lewis 



irregular sea (a), but it will be shown later that a short- crested sea and differ- 

 ent headings can also be taken into account. In determining the form of the re- 

 sponse amplitude operators, it must be recognized that the parameters describ- 

 ing ship performance should be non-dimensional. Pitch angle is a satisfactory 

 measure of angular motion, for it will be the same for ships of different size in 

 comparable situations as well as being related to wave slope. Fig. 2(b) shows 

 the pitching response amplitude operator in the form 



K g ^a 



with 6^ in radians. It is clear that if the response operator curves are ex- 

 pressed non-dimensionally — here pitch angle/wave slope — they will be identical 

 in shape for geometrically similar ships at the same Froude Number. However, 

 they are separated horizontally by an amount equal to log ^co^/co^. Furthermore, 

 points at corresponding values of VL will have the same ordinates, where L is 

 ship length. 



If, as in this case, one ship is twice the length of the other, we have Lj =2Lj, 

 and at equal values of ^A, A.2 = 2X.^. Hence, w^ = ^2 co^, and the separation of 

 corresponding points is log^Wj- log^a;2 = log^ co^/c^^^ log^v^ = 1/2 logg2 = 0.3468. 



Finally, we may multiply the wave slope spectrum (Fig. 2a) by the pitch re- 

 sponse operators (Fig. 2b) to give the non-dimensional response spectra (Fig. 

 2c). These non-dimensional response spectra are of direct quantitative signifi- 

 cance, since they represent (pitch amplitude)^ and the mean pitch amplitudes 

 will be a function of the areas under the curves. 



Similarly heaving acceleration — or vertical acceleration at any point along 

 the length of the ship --is properly referred to wave slope. For in long waves, 

 if we neglect forward speed, the vertical motion of the ship will approach that of 

 the surface wave particles, whose vertical acceleration is, when expressed non- 

 dimensionally, (a)Vg) ^g. Maximum wave slope at any particular frequency is 

 the same, for 



^ r ^ ^ r 

 X. '^'^^^ 277g g ^a • 



Hence, if vertical acceleration is referred to wave slope, this is equivalent to 

 relating it to the wave particle accelerations at the particular wave frequency. 

 The response amplitude operator for heaving acceleration (or vertical acceler- 

 ation at any point) can therefore be expressed as 



27tI.J\ 



o^'l.. 



Heaving motion is somewhat different. If one is concerned with the absolute 

 value of heaving, then the conventional wave amplitude spectrum is appropriate, 

 with a response amplitude operator in the form: 



192 



