Deep Water due to Motion of Submerged Bodies. 59" 



To fulfil the condition of a tree surface, we must apply a 

 pressure equal and opposite to that given by (44) — that is, 

 a pressure symmetrical about a point in the surface at each 

 instant vertically above the instantaneous position of the 

 centre of the moving sphere. Referred to an origin ver- 

 tically above the initial position of the centre of the sphere, 

 the expressions representing the fluid motion due to the 

 moving pressure system are — 



fcfe* *, = - ^ £'«Jr f" e " fa J.(W) 



i I °° J o( feat) or. da , . ~\ 



*«*fo*(«-T)»}*<ttj o (ft °; + ; tf)t • . . ■ («) 



JK*. y, «, = ^J o ' <**£ e _& J„(te') 



H-BW ,_ T) ^"'ifl^, . . (46) 



with -cj-' 2 =(«/• — t?T) 2 + lf . 



These become, on integration with respect to a, 

 qa 6 f ? 7 J' 00 -*(*+*) 

 *" Jo */o 



/.•cos {gk(t — T)*ftdk, .... (47) 



tfsin{0Jfe(*— r) 1 }*^ . . . (48) 



which correspond to (29) and (30) above and are open to an 

 interpretation similar to that given for the motion reflected 

 from the free surface in the case of the moving cylinder. In 

 this case the reflected motion is the same as would be pro- 

 duced by a point pressure, acting on a free surface at a 

 height h above the free surface in our present problem, and 

 coinciding at each instant with the centre of the image 

 sphere. An important part of the motion can again be 

 obtained by applying the principle of stationary phase to thf> 

 integrals contained in (47) and (48)*. Corresponding to 



* T. H. Havelock, Proc. E.S. 1910. 



