15 
are given in the form of certain integrals which were evaluated 
by the method given in AMP Report 37.1R. Graphs of these 
integrals for the most frequently occurring values of the para- 
meters are included. Section III contains a discussion of these 
formulas and a short indication of their proof. A careful 
analysis is made of the dependence of the parameters in the 
bubble motion upon the properties of the explosive in Section 
II the theory is applied to the analysis of some experimental 
data obtained at Woods Hole by Arons and his co-workers. It is 
stated that the agreement between theory and experiment as regards 
period is excellent; as regards pressure and distance the bubble 
moves, the agreement is only fair. The problem is idealized by 
assuming that (1) water is an ideal incompressible fluid, (2) 
the bubble remains spherical in shape, and (3) the gas inside 
the bubble expands adiabatically. 
"~The Dock Problem," K. OQ. Friedrichs and Hans Lewy, October 
1947. 
"Suppose one half-plane of an infinite water surface is 
covered with a rigid plate, the "dock". How does the dock 
influence waves standing or traveling perpendicularly to the 
edge of. the dock on a free water surface?" A solution is pre- 
sented based on a method related to the La Place transformation. 
It is stated that the problem permits a simple explicit solution 
which offers the possibility of discussing the nature of the 
solution and of determining it numerically. The behavior of the 
wave motion near the edge of the dock, i.e. near the line along 
which the water surface and the dock meet is discussed. The 
amplitudes of the waves on a free surface and the pressure under 
the dock are shown graphically. It is said that aside from its 
interest as a flow problem, the dock problem is of significance 
as a new example of a potential problem with different linear 
boundary conditions on different parts of the boundary. 
"Water Waves on a Shallow Sloping Beach," K. 0. Friedrichs, 
October 1947. 
It is stated that the problem of standing and progressive 
waves on a beach with a plane bottom has been solved by Miche, 
Lewy and Stoker for slope angles @ which are integral fractions 
of a right angle,w="/2n ,n=1, 2,-... In this report, in 
which various suggestions by J. J. Stoker and H. hewy are in- 
corporated, a simple asymptotic representation for small angles 
@ is presented. lt is shown that for small slope angles the 
flow can be described as consisting of waves with a varying wave 
length, which depend on the depth in the same way as Airy's waves 
in channels with constant depth. Formulas are developed which 
give the dependence of the amplitude and the phase on the depth 
for small slope angles. It is said that the wave shape calculated 
on the basis of developed formulas for a beach with a 6° slope 
agree almost perfectly with the calculated on the basis of the 
