13 



ourselves to this remark. 



Finally, the influence of viscosity must be mentioned. Except for 

 small Reynolds numbers, no information is available on the subject. One makes 

 the plausible assumption that because of the small displacement thickness of 

 the boundary layer, the influence of viscosity on tne magnitude of the added 

 masses is negligible. 



3 . SUMMARY 



The hydrodynamic masses of a body moving at the free surface are 

 mathematically formulated functionals of the body motion. In principle, 

 beside the form of the body, the history of its motion and thus the resulting 

 velocity field must be given. 



Only few solutions for some simple kinds of motion are known, from 

 which the dependence of the added masses from an acceleration parameter a/g, 

 the Froude number F, a frequency parameter F u , etc., can be established. 

 From a physical point of view we are just beginning to tackle the problem. 



From a point of view of technical application, conditions are 

 slightly more favorable, since one is able to make estimates which, however, 

 frequently are rather coarse. The most familiar estimates are: 



For oscillations in a vertical plane, one assumes 



k = k , k = k 



z z yy yy 



The values k , k are, at least up to moderate Froude numbers, 



z yy 



rather independent of the speed of advance. Because of the high values of 

 the frequency parameter involved, one can hope that the mentioned approximate 

 relations may even be accurate enough for calculating free vibration periods. 



But, again, we do not possess a satisfactory general solution and 

 serious endeavors will be needed in the field of theory and experiment to 

 improve the present state of knowledge. 



Acknowledgment is made to Mr. R.D. Cooper of the Taylor Model 

 Basin for kind help in editing this preliminary memorandum. 



