exact theory. 
Waves in the Presence of An Inclined Barrier," Fritz John, 
November 1947. 
This paper deals with the motion of water of infinite depth 
in the presence of a fixed barrier of length A which extends down- 
ward from the surface and forms an angle 7/@2n with the horizontal 
direction where n is the positive integer. Jt is based on the work 
of J. J. Stoker and H. Lewy for determining waves on sloping 
beaches, "Waves over Beaches of Small Slope, Under a Dock, Under 
an Overhanging Cliff, and Past Plane Barriers," (IMM,NYU Lice 
By the use of sonatas functions the determination of the motion 
is reduced to the solution of an ordinary differential equation 
with constant coefficients instead of an integral equation. The 
role of the various physical assumptions in uniquely determining 
the solution is brought out clearly, and assumptions that have to 
be made to make Fourier integral transformations valia. are 
avoided. The paper treats the following: (1) the method for find- 
ing the general solution, (2) the case of a vertical barrier, 
(3) the submerged infinite barrier, (4) asymptotic behavior of ex- 
ponential integrals, (5) crivation of the complete system of con-— 
ditions on the general solutions for the inclined barrier, (6) 
the complete finite system of conditions for the solution, and 
(7) a uniqueness theorem. 
"The Solitary Wave and Feriodic Waves in Shallow Water," Joseph 
B. Keller, December 1947. 
The paper states that the object of the present investigation 
is to discuss waves of permanent type in shallow water by a 
method in which the character of the approximation is quite clear, 
and which is capable of being carried out to include terms of 
any desired order. Actually stationary solutions are found; 
progressive waves may be obtained from them by adding a constant 
velocity to the fluid. The method consists in expanding the 
solution of the exact hydrodynamic problem systematically in 
powers of a dimensionless parameter @=wh where h is the 
depth of the undisturbed fluid andes is the curvature at some 
point on the surface. The expansions are inserted into the 
equation of motion and the boundary conditions, and coefficients 
of like powers of @& are equated. The variables are chosen in 
such a way that the terms of zero order in the expansion in 
powers of @ satisfy the well-known equations of the nonlinear 
shallow water theory, which are analogous to the equations of gas 
dynamics. The derivation of these equations were attributed to 
the work of K. 0. Friedrichs. 
It is stated that it is easily shown that the only solutions 
of these equations for the first approximation (satisfied by the 
terms indenendent of @& ) which are of permanent form are the 
constant or piecewise constant (shock type) solutions. However, 
