1911] HYDKODYNAMICAL NOTES 11
giving
f+"co8««.<fa = ./(£); f+Mcosa^a=~ra) ....... (13)
J -00 \ \A/ J -00 VO
The former is employed in the derivation of (3).
The occurrence of stationary values of U is determined from (7) by means of a quadratic. There is but one such value ( Z70), easily seen to be a minimum, and it occurs when
...................... (14)
On the other hand, the minimum of V occurs when 7c2 == gp/T simply.
When t is great, there is no important effect so long as x (positive) is less than U0t. For this value of x the Kelvin formula requires the modification expressed by (11). When x is decidedly greater than U^t, there arise two terms of the Kelvin form, indicating that there are now two systems of waves of different wave-lengths, effective at the same place.
It will be seen that the introduction of capillarity greatly alters the character of the solution. The quiescent region inside the annular waves is easily recognized a few seconds after a very small stone is dropped into smooth water*, but I have not observed the duplicity of the annular waves themselves. Probably the capillary waves of short wave-length are rapidly damped, especially when the water-surface is not quite clean. It would be interesting to experiment upon truly linear waves, such as might be generated by the sudden electrical charge or discharge of a wire stretched just above the surface. But the full development of the peculiar features to be expected on the inside of the wave-system seems to require a space larger than is conveniently available in a laboratory.
Periodic Waves in Deep Water advancing without change of Type.
The solution of this problem when the height of the waves is infinitesimal has been familiar for more than a century, and the pursuance of the approximation to cover the case of moderate height is to be found in a well-known paper by Stokesf. In a supplement published in 1880J the same author treated the problem by another method in which the space coordinates x, y are regarded as functions of $, i/r the velocity and stream functions, and carried the approximation a stage further.
In an early publication § I showed that some of the results of Stokes' first memoir could be very simply derived from the expression for the
* A checkered background, e.g. the sky seen through foliage, shows the waves best. t Camb. Phil Soc. Trans. Vol. vm. p. 441 (1847) ; Math, and Plnjs. Papers, Vol. i. p. 197. £ Loe. cit. Vol. i. p. 314.
§ Phil. Mag. Vol. i. p. 257 (1876) ; Scientific Papers, Vol. i. p. 262. See also Lamb's Hydrodynamics, § 230.