DEEP WATER WAVES, PROGRESSIVE OR STATIONARY, TO THE THIRD ORDER OF APPROXIMATION.
[Proceedings of the Royal Society, A, Vol. xci. pp. 345—353, 1915.]
As is well known, the form of periodic waves progressing over deep water without change of type was determined by Stokes* to a high degree of approximation. The wave-length (A.) in the direction of x being 2?r and the velocity of propagation unity, the form of the surface is given by
y= a cos (x — t) — | a.2 cos 2 (as — £) + fa3cos3 (x — t), .........(1)
and the corresponding gravity necessary to maintain the motion by
<? = l-<*2..................................(2)
The generalisation to other wave-lengths and velocities follows by " dimensions."
These and further results for progressive waves of permanent type are most easily arrived at by use of the stream-function on the supposition that the waves are reduced to rest by an opposite motion of the water as a whole, when the problem becomes one of steady motionf. My object at present is to extend the scope of the investigation by abandoning the initial restriction to progressive waves of permanent type. The more general equations may then be applied to progressive waves as a particular case, or to stationary waves in which the principal motion is proportional to a simple circular function of the time, and further to ascertain what occurs when the conditions necessary for the particular cases are not satisfied. Under these circumstances the use of the stream-function loses much of its advantage, and the method followed is akin to that originally adopted by Stokes.
* Gamb. Phil. Trans. Vol. vm. p. 441 (1847); Math, and Phys. Papers, Vol. I. p. 197. t Phil. Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. i. p. 262. Also PMl. Mag. Vol. xxi. p. 183 (1911) >• [This volume, p. 11]. * representing the passage of heat per unit area and per unit time.