Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



DISCUSSION 



Edwin C. James 



California Institute of Technology 

 Pasadena, California 



I would like to direct a question to Dr. Lee concerning the 

 pure steady force. Apparently this type of force can arise in free 

 surface problems and is attributed to a mean drift of mass in the 

 direction of wave propagation. The action of such a force applied 

 to an unrestrained body results in a sinkage or a lift. The question 

 is then, how does one physically explain the steady force when the 

 symmetry of the problem dictates that the mass transport at the 

 station x = should be zero? 



REPLY TO DISCUSSION 



Choung Mook Lee 



Naval Ship Research and Development Center 

 Washington, B.C. 



A mass transport phenomenon arises in the higher-order 

 theory of surface waves (see, e.g. Wehausen and Laitone [I960, 

 pp. 660-661]). Since the present work deals with a second-order 

 problem of free-surface waves, it may be expected that mass- 

 transport will occur in the present problem also. Although I have 

 not touched upon this subject in the text, I discussed it in some 

 detail in my previous work (Lee [ 1968 , pp. 317-318] ) . 



As the discusser pointed out, there is no mass flux across 

 the y-axis. Then, the question arises as to the origin of the mass to 

 supply mass transport. I answered this question in this previous 

 work by showing that the role of the steady potential ^^(x.y) is to 

 counteract the mass transport phenomenon. This means that <py 

 should behave like a steady sink whose strength is equal to the total 

 nnass drift through two vertical control planes encompassing the 

 cylinder, divided by Ztt. The lowest-order contribution from <Pj 

 to the steady force is fourth order, as is proved by Bernoulli's 

 equation. Thus, the second-order steady force still exists while 

 the mass transport phenomenon is nullified by the pure steady 

 potenticd tpj . 



951 



