Wavemaking Resistance of Ships with Transom Stern 



II. A SINK ON THE TWO-DIMENSIONAL FREE SURFACE 



Lamb [ 5] showed a formula for the free-surface shape due to a 

 point sink with the strength M located at x = 0, y = 0, moving with 

 constant speed U to -x direction on the free surface, where +x 

 is on the mean free surface pointing right, and +y points vertically 

 upward. The wave height is 



= -77 — < - 2 COS k-x + — " \ — 9 5 dm > , m x > 



(1) 



'0 m . ..Q 

 where 



k = -^ 



^0 \ --5 — ^ = "!(•?- Si kgx) COS k.x + Ci k x sin koX> 



r.- U , U^ ,_. 



Si u = u - .^-r^rr + 5-7-5T • . • (3) 



2 4 



Ci u = Y + log u - ^^ J. + XT-Zl - • • • (4) 



in X > (2) 



3-3! 5-5 



2-2! 4 • 4! 

 Y = 0.577215665 (5) 



and g is the acceleration of gravity. If this is compared with the 



wave height due to the distribution of constant pressure p^ on x < 0, 

 it can be easily seen that 



2SL = ^^1^1 (6) 



Pg U ^^' 



holds. Thus, from the Bernoulli equation the form of the free surface 

 in X < can be given by 



, p«> mx 



(1 + i^ \ __| dm) , in X < 



^ ^ Jq m^ + k^ ^ 



The wave height for x $ is plotted in Fig. 1 and it clearly shows 



575 



