does not asymptotically approach the Lewis value at high frequencies as might be expected. 

 It is seen to be 20 percent higher than the Lewis value at the highest frequency shown and 

 is still rising. 



The two-dimensional solutions obtained by Grim'* can be used to obtain the frequency 

 dependence of the added mass. Grim's results show the added-mass coefficient dropping 

 sharply from infinity to a minimum value as the frequency is increased and then rising asymp- 

 totically toward the high frequency solution for the section reflected in the waterplane. This 

 general form of the curve was also obtained in Ursell's exact solution for the circular cylin- 

 der. Observing the test results in Figure 8, we see that the experimental curve for a three- 

 dimensional form has this same general shape. Similarly, the experimental results of Haskind 

 and Riman^ show this same trend. 



Grim has indicated that at low frequencies the added mass of any section can be approxi- 

 mated by - —pb"^ In^ where the frequency parameter f is .°£tL^ j is the half-breadth of the 



rr ^ . . . 



section, p is the mass density, and co is the circular frequency. Comparison of this approxi- 

 mate formula with Grim's exact solution (Figures 1 and 2 of Reference 3) would indicate that 

 the approximation is good up to values of about ^ = 0.4. 



The Grim approximation has been applied to the model under test and the two-dimensional 

 results integrated over the length of the body. The resulting added-mass values are shown in 

 Figure 8. The small three-dimensional end correction has not been applied to the curve. 



It can be seen that the sharp drop of added mass with increasing frequency is ade- 

 quately represented by this approximation up to a frequency a yB/g = 0.8 or ^ = 1/3. This 

 confirms the utility of the Grim method. 



The two-dimensional theoretical solutions do not reveal any dependence of added mass 

 on forward speed. The experimental results indicate the speed effects are negligible except 

 at very low frequencies where the added mass at 4 knots is lower than that at slower speeds. 

 However, the data at these low frequencies is not considered accurate enough to warrant any 

 definitive statements on speed dependence. 



COUPLING OF HEAVE AND PITCH 



Havelock^ has recently investigated the coupling of the heaving and pitching motions 

 for a body with forward velocity. A long spheroid half immersed in a uniform stream executes 

 small heaving and pitching oscillations. A rigid wall surface condition is satisfied and damp- 

 ing is neglected. He finds a heaving force equal to - pMUd and a pitching moment equal to 

 qMUz where M is the displaced mass, U is the forward velocity,^ is the pitching velocity, 3 

 is the heaving velocity, and p, q are positive coefficients. Calculations show that p and q 

 are approximately 1 and 1/2 respectively. 



21 



