12 
THEORETICAL STUDIES ON SURFACE 
GRAVITY WAVES 
A group working in the Institute for Math- 
ematics and Mechanics, New York University, 
under Navy contract Néori-201 Task Order No. 1 
has made theoretical studies of phenomena con-— 
cerning surface gravity waves. Some of the 
results are reported in this series of papers. 
‘Waves over Beaches of Small Slope, Under a Dock, Under an Qver- 
hanging Cliff, and Past Plane Barriers," September 1947. 
"The problem of obtaining two dimansional progressing waves 
over beaches with slope angle w=7/2n , nan integer, has been 
discussed recently by Bondi, Miche, Lewy, and Stoker. All of the 
solutions given by these authors become more complicated and 
cumbersome as n becomes larger, that is, as the beach slope 
becomes smaller. +n fact, the solutions consist of finite sums 
of complex exponentials and exponential integrals, and the number 
of the terms in these sums increases with n. Actual ocean beaches 
usually slope rather gently, so that many of the interesting cases 
are just those in which the slope angle is small--of the order 
of a few degrees, say. K. 0. Friedrichs has obtained a representa- 
tion of thesolution of the problem for all n in the form of a 
single complex integral, which can in turn be treated by the 
saddle point method to yield asymptotic solutions valid for large 
n, that is for beaches with small slopes. The resulting 
asymptotic representation turns out to be rather simple-— even 
for numerical computations--and it appears to be very accurate. 
A comparison with the exact numerical solution for W=6° shows 
the asymptotic solution to be practically identical with the ex- 
act solution all the way from infinity to within a distance of 
less than a wave length from the shore line. 
H. Lewy has obtained progressing wave solutions over beaches 
with slope angles w=p™%/2n, with p an odd integer such that p< 
2n. Thus the theory is available for cases in which @ is 
greater than T/2 , so that the "beach" becomes an overhanging 
cliff. The solution for a special case of this kind, i.e. for 
@ = 135° or p = 3, n = 2, has been carried out numerically by 
E. Isaacson in a report now being reproduced. It turns out that 
there is at least one interesting contrast with the solutions 
for waves over beaches in whichw<m/2 . In the latter case it 
has been found that as a progressing wave moves in toward shore 
the amplitude first decreases to a value below the value at oo, 
before it increases and becomes very large at the shore line. 
The same thing holds for standing waves; At a certain distance 
from shore there exists always a crest which is lower than the 
crests atco . In the case of the overhanging cliff with w= 135°, 
however, the reverse is found to be true: The first maximum 
going outward from the shore line is about 1% higher than the 
