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APPENDIX B. 
NOTE ON 1'HE EFFECT Of A THIN STEEL PLATE IN THE 
PATH OF AN UNDERWATER EXPLOSION WAVE es 
S, Butterworth 
\f a small amplitude wave of any type passes normally through the surface of separation of two 
media having densitles p, P2 and characteristic velocities of propagation Cy, C, then at the surface 
of separation there will be produced a reflected and transmitted wave whose pressure amplitudes are 
obtained from that of the incident wave by multiplying by co-efficients r, S such that 
[Pil Pagal eS Oe 
P22 Wey) 
ands = i1¢+rfr 
r = 
In these formulae the incident wave is suppose to be travelling in the medium of density Py If the 
direction of trave} be reversed the reflection and transmission coefficients are ry S; where 
qe Gp S; s i+ ry In the case where the media are water and stee) we find r = 0.932, 
If this result be applied to the case of a thin steel plate placed normally in the path of an 
explosion wave we find that tne emerging wave is built up of a succession of waves each of the same 
form as the incident wave but delayed in time by internal to and fro motion between the surfaces of the 
plate. Thus the first wave emerging is of amplitude ss, (the incident wave being assumed unity) and is 
followed by similar waves of amplitudes ss, My, SS} eo SSy Ler see at intervals 27, UT, 67 where T is 
the time taken to pass across the plate with the velocity appropriate to steel. 
If, as is approximately the case for an explosion wave, we assume the incident wave to be 
represented by € (when the time t*) and zero when t negative the emerging wave is given by 
t or ur or 1 Pa et x2 
Sa Ker Wee le opie teres te ieee) be cog eT 
1 1 1 1 1 re 
1- tte 
when t lies between 2(n-1) T and 2n7. Thus the merging wave varies by jerks at intervals 2 7. 
!f these intervals are short as is the case with thin plates we can obtain the general course of the 
wave with time by putting 
ane = ty mae = ev 
and the emerging wave is then 0 (Gus ef) 
ss 
where p = a Be t lode t 
| 
The maximum pressure in the emergent wave occurs when t = log 8 and has the value 
Bee tte (G - 1)/ B/R- 1, 1m illustration suppose the incident’ wave to fall to 1/€ in 107? second 
(as is approximately the case for a wave developed by a 500 Ib, charge of T.N.T.) and let the steel 
Plate considered be 1 cm. in ERIGKHGSS: Then since the unit of time is 10°” second and the velocity 
of propagation in steel is 5 x 10° cm. per second, T= 0.002 and with ry = 0.932, B= 36, 6 = 1 giving 
Prax = 9-877 and t = 0.102 x 107? second. 
The theory given above is of course only accurately applicable to waves of smal) amplitude. it 
has, however, been used In the text to estimate the effect of the mine-case in altering the form of the 
pressure wave. Since in these applications the alteration is small for a small amplitude wave, it may 
be fairly safely concluded that the effects for large amplitudes waves will also be small. 
For simplicity the unit of time is chosen to be the time taken for the wave to 
fall to i/e. 
