- 3 - 285 



Assumption (a) is equivalent to the statement that diffraction effects due to the arrival 

 of the pulse at the edge of the plate die away very quickly. The motion of the platt then becomes 

 quasi-steady, every element of the plate having the same equation of irotion, counting as zero the 

 time at which the incident pulse arrives at the element. Since the incident pulse sweeps over 

 the plate with a velocity C/sinv/j, where <// is the angle of incidence {defined as the acute angle 

 between wave-front and plate), thTs means that once we have calculated the motion of any element 

 we can calculate tnat of any other element by "post dating" the time-scale by the appropriate 

 values of ' ^i" r , where the z-axis lies in the plate and in a plane parallel to the direction 

 of propagation of the wave-front. 



Assumptron (b) is similar to that introduced in II in support of which both theoretical 

 and experimental evidence was given. It is slightly less drastic in that in II it was applied to 

 the diffraction wave, moving with precisely the velocity C, whereas here the pulse is sweeping 

 over each element of the plate in turn with a velocity effectively greater than C (due to the tilt 

 of the wave-front) and therefore the plastic and elastic forces will have still less chance of 

 propagating themselves ahead of the incident wave-front than they have of propagating ahead of the 

 diffraction wave. 



If we make these two assumptions, the two boundary conditions at the plate can be 



satisfied, as Taylor (3) shows, by assuming that the effect of the plate can be represented as a 



simple reflected wave, the angle of reflection being equal to the angle of incidence. Besides 



the "post-dating" of the time-scale for elements of the plate with different z co-ordinates, the 



only other changes that have to be made in order to convert the theory for direct incidence into 



the theory for oblique incidence are, first that we must replace the quantity p by p cos and 



p (the density of the plate) by p cos fp. When we come to consider the propagation of the 



cavitation front, the theory proceeds exactly as in I, except that we have x cos in all the 



equations instead of x, (x being the co-ordinate whose axis is perpendicular to the plate). 



The fact that we have replaced p by p cos i// necessitates our also Veplacing a by a cos v//. The 



discussion of the various energies involved can also be taken over if we replace a i,y a cos i// 



and fm _ hy ''m _ cos v//. This latter change is the same as replacing p by p cos\p because 



^0 ^0 



E^ contair, 5 the factor pa. For E , E and the tjtal energy one of the factors cos \p is removed 



when we integrate over the whole pulse. 



It is a little more difficult to see what modifications have to be made to the theory of 

 the "follow-up" stage of the motion. Since the angles of incidence and reflection are equal, it 

 follows that the particles will have no velocity in the z-Oirection at the instant when the pressure 

 drops to zero, ano therefore that the velocity of the cavitated water will also be perpendicular 

 to the plate. we can take over the argument of I, but here we leave x unchanged and do not replace 

 it by X cosy. We thus obtain, instead of equations (18) and (8) of I, the following relations 

 between t , x, y, and vc, n/c being the veloc i ty of the water at cavi tat ion ) , 



SlnJk . t!L.i°ljL f -i. - log - /s-n R fi<.£2JJ^ ) 

 C 1 - a cos 1// ou9c J \ ^'- / 



X 

 vc 



l^p^sj/.- 



^0 = 



* a cos lA cosh (^1^)) 



g cos >A cosh /" "s>//\ ^ sin R |^lifi^\] " ^°= "P 

 \ ec / \ 0C Jj T-a 



(1) 



xo - "' '^°^ ^ {2 - 0- cos </') 

 ec (1 - a cos i/;) 



cos 1/; 



(2) 



Since equation (19) in I is derived purely by momentum considerations, it cannot involve the 

 quantity cos i/i di rectly. Also the thickness of the "reconstituted" layer of water on the plate 

 is just X for when particles that were originally at a distance x from the origin are just 

 arriving at the layer. In other words, equation (l9) is exactly the same as in I, that is:- 



dt L 



^0"' fA-Pof^ ""^ * /^"^y - ° (3) 



Examination 



