1179 
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and the corresponding velocity communicated to the plate Is from equation (27) of Report A given by 
areas eee (27) 
Vv — 
mn 
m 
Tne above equations (26) and (27) will hola as a good approximation for the present case 
of a Spherical wave provided (26) satisfies the conditions (23), i.e. provided 
Rc loge Brrc® ¢ woz €)? 
and —y—- | —— are small 
na €-1 na €e-1 
and these conditions can ve written in tne form 
2 2 
a ftoe€ and —_, € log € are small (28) 
Ba €-1 Bak, e-1 
Now in practice (see Appendix 1) 8 a will be large and thus unless € is large conditions (28) wil) 
hold and therefore equations (26) and (27) as In Report a, 
Since € can be large in practice, especially for small anjles of incidence, an approximation 
to cover this case is obtained in the Appendix wnere It is shown that under the conditlons B a and 
€ large, wnile 1- é R_ of order unity, then, correct to fractional errors of order z and Ba 
equation (16) becomes 
R ’ 
Ripi dco Bal ERE IE PN (a) eae erase (29) 
2 Py ct € nk 
and p vanishes when 
ne Log a, 1 (Log a)? 
€ 2 Bae 
(30) 
where ay st 
(1 - Sa 
0 
The velocity given to the plate if it leaves the water ‘s still given, however, by equation 
(27) correct to fractional errors of the first order. 
Thus although when € is large the conditions (23) do not hold up to the time when the plate 
leaves the water yet equation (27) is still a good approximation for the velocity given to the plate. 
The apparent anomaly is due of course to the fact that when € is large the pressure on the plate 
decreases very rapidly at first so that practically the whole of the velocity given to the plate is 
communicated in the very early stages during which the conditions (23) are valid. 
Now, as shown In the Appendix, the conditions (i) Ba large and, (ii) 1-£R_ positive of 
order unity, will usually nold in practice at distances from the explosion at which the explosion wave 
can be regarded as of small amplitude, We thus have finally that the velocity communicated by an 
underwater spherical wave to the plate if it leaves the water is given approximately, whether € be 
Yarge or not, by the same expression (27) as that obtained in Report A for an incident plane wave. 
Summary. 
A general solution is given for the problem of a spherical wave of smal) amplitude incident 
on an infinite plate which moves normally to itself and offers a purely inertia resistance to such 
motion, ; 
The case of an underwater explosion wave is specifically considered and approximate formulae 
are obtained which will usually hold in practice at distances from the explosion for which the 
explosion wave may be expected to behave as one of small amplitude. It is shown in particular that 
If NO eeoee 
