Deep Water due to Motion of Submerged Bodies. 57 



been dealt with by Prof. Lamb*, and we can avail our- 

 selves of his results for the. surface elevation f 2 - We" can 

 now write down the complete solution for the motion due to 

 u moving cylinder. With X = (vt — x) in the above equa- 

 tions, we refer the motion to an origin in the free surface, 

 vertically above the instantaneous position of the axis of the 

 moving cylinder. For values of X that are considerable 

 our approximate solution is given by 



$ =0i — <£/+$2 



a*vX 



a 2 i'X 



x- 2 + (^-/o 2 x*+(ztiiy 



(4#a 2 7r/t?) e 



-£(*+«) 



cos^ 2 X 



V 2 



(36) 



st=ei-ti'+& 



2a?h 



~ X 2 + A 2 



+ (tyaV/t> s ) e 



v ~ sin^X. 



(37) 



In the region in front of the moving cylinder the train of 

 regular waves represented by the periodic terms in these 

 equations is absent. 



§ 4. Fluid Motion due to Moving Submerged Sphere. 



The corresponding investigation of the wave-disturbance 

 due to a sphere moving with constant velocity v, at a uniform 

 depth h beneath the surface, proceeds exactly as in the case 

 of the cylinder. The velocity-potential cj) 1 of the fluid 

 motion due to a sphere moving steadily in an infinite liquid 

 in the positive direction of x is given by 



m 3 cos# va z x /00 . 



2r 6 



For the moving line pressure represented in (30) above 



Kll 



l 2 (X,Q,t)=±>ccnre sin 



kX — 2ea 2 J 



cos nih— Ks'm mh 



-??iX 



'0 m 2 -\-K* 



at distance X behind the moving line pressure, and 



dm, 



*a(X,0, *)=-2«a a f 



m cos mh — k sin mh 



■mX 



dm 



J m 2 +/c 2 



at distance X in front of the moving pressure ; with k= (#/?' 2 ) : i Hydro- 

 dynamics,' 3rd edition, p. 385. 



t This result differs from that given by Prof. Lamb (Annali di 

 Matematica, tome xxi,serie iii. p. 237) in the sign of the term representing 

 the system of regular waves, but gives the same value for the wave- 

 making resistance of the cylinder: E=47r 2 ^prtV« -2 ' cA , given in equation 

 (78) of his paper. 



