408 
Proceedings of Royal Society of Edinburgh. [sess. 
water in the space between x = -1 and x — +: 1 becomes more and 
more nearly an increasing number of inward travelling waves, 
with lengths slowly diminishing to zero ; and, as we see by the 
exponential factor in (144), with amplitudes and with slopes also 
slowly diminishing to zero : as time advances to infinity. 
§ 111. The semi-period of one of these quasi standing waves is, 
9 2 
as we find from (139), approximately equal to when the 
time is so far advanced that J- gt 2 is very great in comparison with 
p. Thus we see that the period is infinite at the origin. This 
agrees with the history of the whole motion at the origin, which, 
as we see by putting x = 0 in (139), with 2=1 and p = 4,, is 
expressed by the formula 
The motion of the water in the space between x = — 1 and x = + 1 
is of a very peculiar and interesting character. Towards a full 
understanding of it, it may be convenient to study the simplified 
approximate solution 
which the realised part of (139) gives when \ gt 2 is very large in 
comparison with p. 
§ 112. The outward travelling zeros on the two sides, beyond 
the distances ± 1 from the origin, divide the water into con- 
secutive parts, in each of which it is wholly elevated or 
depressed. These parts we may call half-waves. They travel 
outwards with ever-increasing length and propagational velocity. 
Each of the half-waves developed after t = Jtt , as it travels 
outward,, increases at first to a maximum elevation or maximum 
depression, and after that diminishes to zero as time advances 
to infinity. 
§ 113. It is interesting to trace the progress of each of the zeros 
in the intervals between the times of our six diagrams. This is 
facilitated by the numbers marked on several of the zeros in the 
different diagrams. Thus, confining our attention to the left-hand 
(147). 
(148), 
