236 CALCULATIONS CONCERNING THE MOMENTUM OF PROGRESSIVE WAVES [3£
which must be satisfied if the emergence of a negative wave is to 1 obviated.
We are now prepared to calculate the momentum. For a wave in whic the mean elevation is zero, the momentum corresponding to unit horizont breadth is
f f p u (ha + 97) das = fp */(gjho) \tfdx,...................(22)
.when we- omit cubes and higher powers of f]. "We may write (22) also in tt form
,.. . 3 Total Energy /0_N
Momentum = -T -- si., ..................... (23)
...... - TJ C
G being the velocity of propagation of waves of small elevation.
As in (14), with <y equal to 2, we may prove that the momentum positive without restriction upon the value of 77.
As another example, periodic waves moving on the surface of deep wat' may also be referred to. The momentum of such waves has been calculate by Lamb*, on the basis of Stokes' second approximation. It appears th; the momentum per wave-length -and per unit width perpendicular to tl plane of motion is
...... .............................. (24)
where c is the velocity of propagation of the waves in question and the wa1 form is approximately
77 = a cos (ct — as) ............................ (25)
A.
The forward velocity of the surface layers was remarked by Stokes. F< a simple view of the matter reference may be made also to Phil. Mag. Vol. p. 257 (1876); Scientific Papers, Vol. I. p. 263.
* Hydrodynamics, § 246. by the fact that outside the wave u = 0 when h = 7i0, whence and replacing h by h0 + 77, we get