Ogilvie 

 i indicates that a Cauchy principal value is to be used. Similarly, we find: 







Equation (17a) is useful if we know mju = MjkC'^)? but it is easily shown that 

 knowledge of mjuC'^) fo^ any single frequency is sufficient. The same comment 

 applies to (17b) with respect to b*;^(w). The values at co = oj are most likely to 

 be amenable to calculation, although in some cases it may be easier to calculate 

 the values at w = o. In either of these two limiting cases, the free surface prob- 

 lem degenerates into a much simpler problem, and in fact numerical solutions 

 for quite complicated geometries are possible by methods such as that of Hess 

 and Smith (1962). 



Equations (17a) and (17b) have been proven by Kotik and Mangulis (1962) for 

 the special case of heave disturbances at zero forward speed, and these authors 

 surmised that a similar result would be valid for all modes, with or without for- 

 ward speed. Their argument was based on the observation in other fields of sci- 

 ence* that such formulas are obtainable whenever the system response obeys a 

 linear law and there is a clear causality relation between input and output. In 

 the ship motions problem, linearity has been demonstrated for certain types of 

 motion, as described in Chapter II, and Cummins' analysis was based on a line- 

 arity hypothesis. Therefore it is not surprising that the formulas can be derived 

 from Cummins' results and the experiments indicate that we should expect the 

 formulas to be valid. Also, there can hardly be any question about the validity 

 of the causality assumption, t 



An alternative derivation of these relations is presented in Appendix C, 

 wherein we avoid the double transform operations which were used to derive 

 (17a) and (17b). It is seen in the Appendix that the formulas are really just 

 corollaries of Cauchy's Integral Formula. 



Two points should be made with respect to use of these formulas: 



1. Contrary to statements by Kotik and Mangulis, it does not follow that an 

 approximate formula tor, say, a damping coefficient can be used in (17a) to ob- 

 tain an approximate formula for the corresponding added mass coefficient. The 

 reason for this is that an approximate formula for damping coefficient may give 

 good results in the range of interest for damping coefficients (especially near 

 resonance) but the asymptotically wrong at extreme values of the frequency. 



*Such relations are known as the "Kramers-Kronig relations" in statistical me- 

 chanics. They may be interpreted as Hilbert transforms. 



'Davis and Zarnick (1964) have questioned this, because in their experiments 

 they observed a response before t = o when an impulse occurred at t = 0. How- 

 ever, I consider their paradox a result of their choice of time coordinates and 

 their definition of an impulse. Certainly there can be no ship disturbance until 

 the ship encounters a free surface disturbance, and so a causality hypothesis 

 is valid. 



42 



