Kotik and Lurye 



other. As one might have expected, the second approximation is somewhat bet- 

 ter; however both are very close to the exact distribution. 



In Fig. 2 we have plotted the normalized real amplitude of the time-harmonic 

 pressure on the surface of the spheroid vs normalized axial distance. (From 

 symmetry, the pressure is obviously a function of the axial coordinate only.) 

 The normalized real amplitude is defined as P/pcraV, where P is the real am- 

 plitude of the unnormalized pressure, and v is the real amplitude of the sphe- 

 roid velocity. The solid curve represents the exact pressure distribution, which 

 in the case of surge in an infinite fluid, is known to be a linear function of the 

 axial distance. As can be seen from their labels, two of the broken curves were 

 calculated from the approximate dipole distributions of Fig. 1. The third pres- 

 sure curve was obtained from a discrete distribution of 45 dipoles whose 

 strengths were computed by applying the mean square boundary condition to a 

 set of 200 points on the spheroid surface. Evidently it is only near the nose that 

 the approximate pressures depart sensibly from the exact one, and even there 

 the relative error is less than 15%. 



It is worth noting that neither the computation of the dipole strengths nor 

 the subsequent pressure calculations exceeded 0.01 hr of IBM 7094 machine 

 time for any one case. 



REFERENCES 



1. R. Timman and J. N. Newman, "The Coupled Damping Coefficients of a 

 Symmetric Ship," Journal of Ship Research 5, pp. 1-7, March 1962. 



2. H. Lamb, Hydrodynamics , 6th ed., Dover Publications, New York, pp. 10-11 

 (1945). 



3. W. E. Cummins, The Impulse Response Function and Ship Motions , David 

 Taylor Model Basin Report presented at the Symposium on Ship Theory, 

 Institut fiir Schiffbau der Universitat Hamburg, 25-27, January 1962. 



4. J. N. Newman, "The Damping of an Oscillating Ellipsoid Near a Free Sur- 

 face," Journal of Ship Research 5, pp. 44-58, December 1961. 



5. J. Kotik and V. Mangulis, "On the Kramer s-Kronig Relations for Ship Mo- 

 tions," International Shipbuilding Progress 9, pp. 361-367, September 1962. 



6. F. Ursell, "The Decay of the Free Motion of a Floating Body," Journal of 

 Fluid Mechanics 19, Part 2, pp. 305-319, June 1964. 



7. W. E. Cummins, "The Force and Moment on a Body in a Time- Varying 

 Potential Flow," Journal of Ship Research 1., pp. 7-18, April 1957. 



424 



