3) 
height of the crest at o@ . Still another fact regarding the 
behavior of the solutions near the shore line is interesting. 
In all cases there exists just one standing wave solution which 
has a finite amplitude at the shore line; H. Lewy has shown 
that the ratio of the amplitude there to the amplitude at co 
is given in terms of the angle by the formula Yw/aq - Thus 
for angles w less than 7/2 the amplitude of the standing wave 
with finite amplitude is greater on shore than it is at infinity 
(becoming very large as w becomes small) while for angles w 
greater than 7/2 the amplitude on shore is less than ato . 
This result has been verified by BH. Isaacson for the special 
case W= 3%/, . Since the observations indicate that the 
standing wave of finite amplitude is likely to be the wave which 
actually occurs in nature for angles w greater than about 40°, 
the above results can be used to give a rational explanation for 
what might be called the "wine glass" eficct: Wine is much more 
apt to spill over the edge of a glass with an edge which is 
flared outward than from a glass with an edge turned over slight- 
ly toward the inside of the glass. 
A limit case of the problem of the overhanging cliff has a 
special interest, namely the case in which w annroaches the value 
7 and the problem becomes what might be called the "dock 
problem"; The water surface is free up to a certain point but 
from there on it is covered by a rigid horizontal plane. The 
solutions given by Lewy are so complicated as p and n become 
large that it seems hopeless to consider the limit of his 
solutions as @—->7r. Friedrichs and Lewy have, however, attacked 
and solved the dock problem directly. Their results will appear 
shortly. 
A fourth report by F. John deals with the effect of a plane 
rigid barrier on the surface waves in water which is, apart from 
the barrier, everywhere infinite in depth. The barrier is assumed 
to be inclined at an angle W/on , n an integer, to the 
horizontal and to extend in the one case from the surface down 
into the water for a finite distance and in the other case to ex— 
tend from infinity up to a point a certain finite distance below 
the water surface. The special case of vertical barrier has 
been treated by F. Ursell by a method different from that used by 
F. John, but which does not yield all possible solutions behaving 
like the classical steady progressing wave solutions at infinity. 
One of the interesting results obtained by F. John is the follow- 
ing: Consider first the case of a plane barrier extending from a 
point below the water surface down to infinity. If it is pre- 
scribed that a progressing wave of fixed amplitude and freyuency 
comes in from the right hand infinity, say, and that a progressing 
wave of the same frequency but unknown amplitude goes past the 
barrier to the left hand infinity while another is reflected back 
to the right hand infinity, it turns out that the problem has a 
uniquely determined solution provided that quite reasonable 
assumptions are made about the singularity at the tip of the 
barrier. In other words, a uniquely determined “reflection coef-— 
ficient" for the barrier is obtained in this case simply by 
