14 
stipulating that there should be no progressing wave coming to- 
ward the barrier from. the left hand infinity. However, in the 
case of a barrier extending from the surface dowmward a finite 
distance the situation is quite different in this respect. No 
uniquely defined reflection coefficient can be determined 
because of the fact that the strength of the singularity at the 
intersection of the barrier with the water surface cannot be 
determined a priori. This fact was observed earlier in dealing 
with a similar problem for waves on sloping beaches; the fact is 
that one would probably have to utilize knowledge of some sort 
about the phenomena of breaking of waves--which are essentially 
non-linear phenomena and hence out of the scope of the theory in 
question here--in order to determine reflection coefficients in 
these cases without recourse to experimental results. The results 
of observations (cf. the paper of Miche)on sloping beaches seem 
to indicate the: following to be likely: If a barrier slopes at a 
small angle then practically all the energy of a progressing wave 
moving in the direction of decreasing depth over the barrier will 
be converted either into heat or into the energy of a flow (the 
undertow) through the occurrance of oreakers. If however, the 
wave amplitude is very small or the slope angle is large enough 
(greater than 40°, according to Miche) a standing wave, denoting 
perfect reflection, will be observed over the barrier in practice. 
MYaves Against a Cliff Overhanging at an Angle of 135°, Eugene 
Isaacson, September 1947. = 
Lewy's method is applied to treat numerically waves pro- 
eressing against a cliff overhanging at 135°. The nature of the 
waves ars analyzed by computing the shape of the two fundamental 
waves and @ , from whicn all progressive wave solutions 
can be constructed. The principal results are: At shore the 
amplitude damping effect of the cliff is about 3 per cent shorter, 
and beyond one wave length from the shore the effect of the cliff 
is negligible. 
"Theory of Underwater ixplosion Bubbles," Bernard Friedman, 
September 1947. 
This paper supplements and greatly extends previous work 
on underwater explosions, in particular AMP Report 37.1R, "Studies 
on the Gas Rubble Resulting from Underwater Explosions: On the 
Best Location of a Mine Near the Sea Bed," by the author and 
Max Shiffman. Here is shown that the effects of surface, bottoms, 
walls, target, etc., can be approximated by the addition of a 
suitable term to the kinetic energy. The evaluation depends upon 
the solution of an "electrostatic problem." Section IV develops 
in detail the case of a bubble between a free surface anda 
bottom. Section I presents a collection of formulas and a summary 
of methods which can be used to predict the period of oscillation 
of the bubble, the distance its center moves, during the first 
oscillation, the maximum and minimum radius of the bubble, and 
finally, the peak pressure emitted by the bubbie. The formulas 
