Waves of Finite Amplitude 
Two theories have been developed for deep water 
waves of finite amplitude: the Stokes theory deal- 
ing with irrotational waves, and the Gerstner 
theory dealing with a specific type of rotational 
waves. For the Stokes’ waves the velocity of 
progress depends also upon the steepness of the 
wave, as expressed by the ratio =H/L (Lamb, 
1932, p. 420) 
C= LL] 14 at Gat Pets eer et a | (9a) 
27 a 
For moderate values of 6 the wave form is very 
nearly trochoidal, byt for larger values the troughs 
become wider and flatter, and the crests steeper. 
According to Mitchell (Lamb, 1932, p. 418), the 
greatest possible value of 6 is equal to 1/7: 
Ones iG (9b) 
When this value is reached the wave profile 
becomes unstable and the wave breaks. The 
velocity of progress of a wave for which 6 = 1/7 is 
1.12 times the velocity of a low wave of the same 
length. 
Generally waves in the ocean are much less 
steep, and equations 1 to 8 are sufficiently accurate 
for waves of finite height. The latter differ, how- 
ever, from waves of very small amplitude in one 
important respect: the particle velocity is not 
uniform but is at a maximum when the particles 
are at the highest point in their orbit and moving 
in the direction of the wave. Upon the completion 
of each nearly circular motion the particles have 
advanced a short distance in the direction of 
progress of the wave and have brought about a 
small transport of mass (figure 1). The average 
velocity of this forward motion during one wave 
period, the mass transport velocity, is denoted 
by u’. In deep water, at a depth z (Lamb, 1932, 
p. 419), 
w =F Cetttlt (10) 
ne 
Figure 1.—Orbital motion during two wave periods of a water 
particle in a deep water wave of finite height. 
Rossby (1945) has shown the Stokes wave to 
be a special case of an infinite number of possible 
irrotational waves. His very general treatment 
has no bearing upon the following discussion of 
the growth of waves, but may modify some of the 
conclusions regarding the propagation of swell. 
The Gerstner waves are exactly trochoidal and 
equations 1 to 8 are valid for them without ap- 
proximation. The Gerstner waves are without 
mass transport velocity. 
PROPAGATION OF A DISTURBANCE THROUGH A REGION PREVIOUSLY 
UNDISTURBED 
General Considerations and Observations 
It has long been a controversial question 
whether a disturbance of the sea surface in @ 
storm area advances into an area of calm with 
the velocity of the individual waves, or at half the 
wave velocity (group velocity). A theoretical in- 
vestigation, which is presented in this section, 
leads to the conclusion that for practical purposes 
the disturbance shall be taken to advance at half 
the wave velocity. Some waves travel faster and 
arrive earlier, but their height is probably small. 
As a simple form of wave motion, consider a 
train of waves represented by sin k(a—Ct) where x 
is positive in the direction of propagation and C 
is the velocity of transmission of phase, termed 
wave velocity in this report. If the medium in 
which the motion occurs is such that Cis the same 
for all wave lengths the velocity C has additional 
physical significance, it equals V, the rate of 
transmission of energy. If waves are emitted 
from a source in such a manner that full intensity 
is reached with the emission of the very first wave, 
any subsequent position of this first wave defines 
a wave front. As an effect of viscosity the height 
of the wave decreases with increasing distance 
from the source, but at any given distance from 
