Deep Water due to Motion of Submerged Bodies, 53 



(b) Take next the velocity- potential of the image of the 

 given solid in the free surface from that already obtained. 

 This term, — <£/, involves an equal and opposite impulsive 

 pressure applied to the free surface at the commencement of 

 motion, and an equal surface elevation subsequently. The 

 total elevation of the surface at each point is now double that 

 due to the velocity-potential (a), and in addition the term 

 added in [b) violates (but to a much smaller degree) the 

 conditions required at the surface of the solid (22). 



(c) Find by means of (2) the pressure acting at the free 

 surface required by the fluid motions referred to in (a) 

 and (b). This is the pressure applied by the fluid above the 

 surface which is to be the free surface ultimately to the 

 fluid below it. This pressure must be applied to the surface 

 to maintain the two motions represented by the terms in (a) 

 and (b) when the infinite mass of fluid until now assumed to 

 be above this surface is removed. Further, this system of 

 pressure must move along the surface so as to accommodate 

 itself to the motion of the solid, that is, it moves along the 

 surface with the same velocity as the solid. The terms 

 <£i~~<£i' of the velocity potential imply that this pressure 

 system acts on the surface. We must therefore apply 

 to the surface a system of pressure equal and opposite 

 to that required by the terms introduced in (a) and (b) ; 

 and the surface then becomes a free surface. The motion 

 due to this system oE pressure can readily be expressed 

 by means of the results obtained in § 1. The resultant 

 fluid motion given by the three terms which have been 

 indicated above fulfils all the required conditions except 

 that contained in equation (22) ; which is however satisfied 

 to a first approximation, since the solid is assumed to be at 

 a considerable depth beneath the free surface. 



(d) To proceed to a higher order of approximation we 

 must add a motion giving a normal velocity at the surface of 

 the solid equal and opposite to that given by the resultant 

 of the terms introduced in (b) and (c), with zero velocity at 

 infinite distance. This term in turn violates the condition 

 for a free surface (21): the next term introduced to fulfil 

 the requirements of a free surface violates conditions at the 

 surface of the solid; and so on. The whole process amounts 

 to finding a series of reflected motions founded on the motion 

 due to the solid and its negative image in the free surface; 

 and it may of course be applied to all cases where the solu- 

 tion for translational or rotational motion of a solid in an 

 infinite liquid has been obtained. The case of viscous liquid 

 can be treated in the same way. 



