3 - 



489 



ever a larje erwujn ranj? .-.f the two oarameters tc enaOle intcroolation to D9 carriea out. 

 Hovovcr a luCKy guess snow.'O that tne vari'jt/ cf oressure - time curves resultinj from (6) and 

 (8) coulO oe rcouceO with fair dcoroximation to a single universal curve oy suitable aojustments 

 c' oressure and time scales. Tne crocess Oy wnich this is achisved is OsscriDco Delow. 



A v?,1u9 c' c = 0.10 corresconaing to a charje of 100 ID. was selected as a useful starting 

 joint. Equ-^tl-n (6) was then integrated numerically, as described in Aooendix 2, for the 

 foil, wing v-lues of m, viz., lO^m = 0, 25, 6U, 100, 225 an: 400. Fcr each -f these values ;f m 

 the pressure function given Oy tne excression on the right of (8) was comouteu from the values of 

 a and olcttsd against time. Insc^ctlon of these pressure time curves showed that whereas for 

 small values of m the Curves were narrow with large values of the oeal< pressure, for large values 

 of m tne curves were broao with low values cf oeah oressurs. Nevertheless the curves all had a 

 certain similarity. The excerlment was therefore tried of adjusting the pressure scales in eacn 

 to jive the same num:ricsl value f?r the caal< cressure, wnlle the time scales were adjusted tc make 

 the family of curves coincident at tne point where the pressure had fallen to half its maxitnjm 

 value. 



Thus fcr each of the values of the mo'nentum constant m tne quantity o/o was olotted against 

 the quantity t/t. (the origin cf the time scale being taken at tne instant of minimum radius) where 

 t. Is the time taken oy thv pressure tj fall from its maximum to half its csak value. It may be 

 called conveniently the "hal f-oeri od". Two sjcn curves for the extreme cases ra = 0, and m = 0. Oi» 

 have been plotted in Figure 2. It was found that in the region where p/p is greater than about 

 0.2 these curves fitted each other very closely. In fact the closeness of fit is probably better 

 than it is reasonable tc cxoect as b.;t*3en pressure-time curve; obtained from exact solution of 

 equations (i) and (2), and from actual exoerlrents. Accoroingly it seems quite adequate for all 

 practical purposes tc take any one of these pressure-time curves, expressed in this manner, as a 

 universal :ressure time curve - 3S rsgards shape - for the sulse produced by the collapse and 

 re-expansion :f the sx:l:si.n bubble*. 



So far we have only considered one specific value of the parameter c depending on the 

 charge weight. In Appendix l it Is shown that tncse solutions may oe converted Into solutions 

 for any other desired vilue of tne charge weight ay a simple linear transformation cf the length 

 ind time scales and for a corresponding altered valu? of the momentum constant m. Since, as 

 stated above, and demonstrated in Figure 2, change in m m^kes an almost negligible Tifferencs 

 to the- shape of the pressure time curve when expressed in terms of tne oe?k pressure ano tho half 

 perl:: it follows that all solutions for ther values cf c an: m (within reasonable limits) will 

 lie very cl :si. t the tw: curves in Figure 2< 



Also Plotted on Figure 2 are points obtained from the full integration of Taylor's -equations 

 for four cases ranging f-om i gram 3 feet deep to 460 lb. at a depth of 60 feet. It will be 

 oBserved that tne aoproximations used- in this report n^.ve not resulted in errors of more than about 

 five per cent of tne peok pressure. Tne agreement will become progressively worse at times longer 

 than tw; ano one half times the half period due t -. the neglect of the term in z in equation (1) 

 which becomes more important the larger the radius. Nevertheless the differences of the order of 

 five per cent arising due to the approximations made are a small price to pay to be able to 

 delineate tne shape of the main part of the pressure time curve for 3 very wlae range of charge 

 weights and depths without recourse to the very laborious integr'.tion of the full equations. 



In order that Figure 2 may yield a oressure-t ime curve in real units it is necessary tc 

 be able to calculate the peak oressure : and the half period t, - preferably in real units. 

 8y using Figure 1 to get a^ ^nd inserting in equation (9) the peak oressure may be easily obtained. 

 The values of t, obtained frjn the numerical integrations mentione': ibove h^.ve been collected ana 

 plotted as a function of tne momentum constant m. As explained in Appendix 1 this curve may now 

 be made apol icaple to all values of the parameter c (and hence to all values of the charge weight) 

 by relabelling the abscissa m/lOOc^ end by relabelling the ordinates t,/(jOc) i^. However, this 

 latter quantity can be converted to real times by multiplying t, by the time scale factor/(iO M^J j) . 

 Thus the ordinates of this graph nay also be labelled 4,67 T./m"^, Thes?" values of thf half period 

 nave been plotted in Figure 3 (after a further change of scale length to remove the numerical 

 factor 4,67) using open circles. Also plotted are 'our values obtained from the full integration 

 of Taylor's equations. ( It will be seen that except for large values of m the agreement between 

 the more exact values and those obtained oy the approxinet ions of this report are very close 1?. 

 Accordingly the curve drawn in Figure 3 may be regarded as a universal curve for obtaining the half 

 period of the bubble pressure pulsa. 



3ince the equation of radial motion (e) is symetrical witn regard to the time of minimum 

 radius the resulting pressure time curve is also symetrical and only one half of it is 

 given in Figure 2. 



Large values of m only arise in general for large shallow charges when large vertical 

 migration of the bubble occurs, witn considerable deoarture of tne bubble from scn.ricity, 

 and the pressure puis-' orcduced are relatively feeble .'.nd unimpcrtmt. 



