20 
WATER SURFACE 
1 E 15° 
09.9.0 
ai 7 Aes eas AAD ry - .: rs -"SE8 BED “(hAak) 
59™ A mathematical explanation of the "blind band" has not so far been 
given, no doubt because of the complexity of the problem. However, fast 
cine photographs of the motion caused by an electric spark (to similate 
an explosion) made at the interface of two liquids, the lower one having 
a slightly greater density than the upper, show a bubble of very peouliar 
form. The collapsing stages are indeed remarkable, but what is more 
important for the present considerations is that the bubble in the upper 
liquid even in the very early stages is slightly elliptical, with the 
vertical radius greater than the horizontal. The lower liquid throws 
a degenerate type of circular curtain up into the upper bubble and the 
inner regions of the interface move upwards. Now these peculiar bubble 
shapes must imply that the associated pressure pulses in the two liquids 
also are not hemi-spherically symmetrical. One would anticipate that 
the pressure pulse near the interface would be anomalous, as indeed is 
observed. The blind band and the oavity, or crater, in the lower mediun 
are in fact correlated manifestiations of the same mechanical phenomena, 
and both are generated in the very early stages of the expansion of the 
explosive gases. Accepting this view, one would not expect artificial 
obstructions on the sea bed to produce "screening", i.@. a blind band. 
This is found to be the case. Furthermore, one would not expect to be 
able to detect a "blind band” in the air blast from an explosion on the 
ground, except very close-in, because the non-linear terms in the hydro- 
dynamics of air blast are so very mich more important than they are for 
water blast, and the departure from “geometrical optics" is correspondingly 
far greater. 
60.* Two mathematical investigations, one supported by cognate experimental 
evidence relating to the reflection of pressure pulses at an interface, 
may be briefly mentioned here. The first is limited to weak pulses, so that 
the theory of sound is applicable, and is due largely to Arons and co-workers 
at Woods Hole. The second is purely mathematical(2) and relates to a 
finite step pulse in one medium meeting a second medium at a plane interface. 
According to the usual theory of wave motion, when an infinite train 
of plane harmonic waves moving through a “lighter” medium in which the wave 
velocity is c meets the plane interface with a "denser" medium in which the 
wave velocity is C, reflection and refraction occur, and the refractive 
index is 
pe= Wc 
Angles of inoidence greater than or equal to the "oritical angle" 
c = cosec™! , induce total reflection. Notice that depends only on 
the ratio of the two wave velocities and not on the frequency. 
By means of a Fourier transform, a pulse of any form may be expressed 
as a synthesis of various harmonic waves. The pulse shape of interest for 
underwater explosions has a sharp front and decays exponentially, anda 
manageable Fourier integral can be found for this case. Since the oritical 
angle of reflection is independent of requency, such a pulse should be 
reflected completely for angles greater than the critical angle. 
