■3 - 



223 



For a point source, at distance ) on the axis of the piston, emitting a wave of exponential 

 -nt 



form, P| " P, 2 at centre of piston, the kinetic energy (1. communicated finally to the pist 



on 



is then given by 



J_ZI_ 



mr- (") 



Q- 1 + sin0 



where tan 5 = | (ll) 



andHi, the energy* in the wave (neglecting afterflow contribution) directly incident 

 on the piston is given by 



77 X^ p ' (1 - sin 5) 



Oj = ^a_, (12) 



jocn 



Equation (lOj Is th<? spherical wave analogue of Butterworth's equation (2U) of report 

 "Note on the motion of a finite plate due to an underwater explosion' to which it reduces when 

 X->oo provided I in Butterworth's equation be correctly interpreted as twice the incident Impulse. 



A. 3. Surface effect . 



The effect of a free surface above the piston at right angles to the rigid wall which is 

 then semi-infinite can be taktn into account by the method of images. The problem then becomes 

 that of an infinite wall cont^iining two similar pistons subjected to the waves from positive and 

 negative sources symmetrically situ.'*ted with rvspi^ct to the pistons. For a rectangular piston 

 an equation similar to equation (7) but much more lengthy can be obtained in this way to allow 

 for effect of surfave, but in view of possible cavitation (see paragraph A. 4) exact calculation 

 of surface effect is of doubtful use. 



For incompressible flow the energy conmunicated finally to a circular piston of radius 

 a with centre at depth H below the surface can be expressed as a 0. where fL is given by equation 

 (lO) and a is given approxinetsly by 



a = , 'j;^'^'' (13) 



32 H (1 ♦ E^) 



where 1.an\p - — (lU) 



X 



In equation (l3) the denominator represents the effect of the "image piston' to the first 

 order in a/H, while the numerator gives the effect of the "Image source', the ratio of the tiean 

 incident pressures from the source and its imagf being taken approximately as the ratio of the 

 incident pressures at the centre. ^ 



A.I, Effect of cavitation . 



The preceding equations arc all based on the assumption that no cavitation occurs, i.e. 

 that tensions can be developed up to any required magnitude between the water and the piston and 

 in the water itself. A complete quantitative account of the effect of any cavitation seems 

 unobtaimble at prt^sent but two qualitative pjints seem worth menticnlng. 



Firstly, cavitation at or near the piston can be caused by the piston tending to move 

 faster than the water at certain stages of the motion. The point to be noted in this case, 

 neglecting surface affect considered later, is that after such cavitation the water will still 

 be following up the piston and can thus communicate further energy if and when it catches up the 

 piston. It is thus by no means certain that such cavitation will imply a drastic reduction In 

 the energy coimiun icated finally to the piston. 



Secondly 



