1914] MOMENTUM OF PROGRESSIVE WAVES 235
from which we deduce almost exactly as before that the momentum (13) is positive if /3 (being positive) is less than 1 and negative if /3 is greater than 1. If /3=1, the momentum vanishes. The conclusions formerly obtained on the supposition of small disturbances are thus extended.
We will now discuss the momentum in certain cases of fluid motion under gravity. The simplest is that of long waves in a uniform canal. If 77 be the (small) elevation at any point x measured in the direction of the length of the canal and u the corresponding fluid velocity parallel to x, which is uniform over the section, the dynamical equation is*
As is well known, long waves of small elevation are propagated without change of form. If c be the velocity of propagation, a positive wave may be represented by
xl .............................. (18)
where F denotes an arbitrary function, and c is related to the depth according to
c* = ghQ .................................. (10)
From (17), (18)
' ........................... (20)
is the relation obtaining between the velocity and elevation at any place in a positive progressive wave of small elevation.
Equation (20), however, does not suffice for our present purpose. We may extend it by the consideration that in a long wave of finite disturbance the elevation and velocity may be taken as relative to the neighbouring parts of the wave. Thus, writing du for u and h for A0, so that ij = dh, we have
and on integration
The arbitrary constant of integration is determined by the fact that outside the wave u = 0 when h = 7i0, whence and replacing h by h0 + 77, we get
as the generalized form of (20). It is equivalent to a relation given first in another notation by De Morgan f, and it may be regarded as the condition
* Lamb's Hydrodynamics, § 168.
t Airy, Phil. Mag. Vol. xxxiv. p. 402 (1849).imited to the supposition of small disturbances.