Communicated by Professor W. E. Ayrton, F.R.S. Received April 21; received in revised form May 26,—Read June 16, 1904
The object of the investigation was to determine how ripple-mark started in perfectly smooth sand, apart from accidental variations of surface friction; how the ripples grew; how the amplitude and depth of the water affected them; and whether there was any connection between ripple-mark and kindred phenomena, such as the dust figures in a Kundt's tube.
In the course of the investigation the causes, form, and mode of action of "ripple vortices" in the water were determined, and the fact that no vortices of this kind can form in a current flowing steadily in one direction over an obstacle was demonstrated, both theoretically and practically. Also many minor details came to light, such as the fact that each ripple is continually travelling, as a whole, and why it does so.
As regards the starting of ripple-mark, it was found that a single ridge forms, on perfectly smooth sand, wherever the water happens to have the same place of maximum longitudinal velocity during several oscillations. That as soon as this ridge is high enough (less than a millimetre is sufficient), the water, in flowing over it, forms a spiral vortex with horizontal axis, which starts a new furrow and ridge in the lee of the first-on one side during one swing and on the other side during the next. That the vortex that forms in the lee of each of these two ripples in turn originates a new ripple there, and that in this way, fresh ripples are begun with each succeeding swing of the water till the whole sand is ripple-marked.
It was shown that the "ripple vortices" only came into existence during the time when the water was rising above the mean level, and the causes and manner of formation of the vortices were given.
It was found that when the place of maximum horizontal velocity was constant, not merely for several oscillations, but for a considerable time, as at the loop, for horizontal motion, of a stationary wave, the ridge mentioned above grew into a ripple-marked heap which was highest at the loop and lowest at the nodes. By oscillating water in the simplest possible way, i.e., so that it alternately rose and fell at one end of the trough while it alternately fell and rose at the other, a stationary wave having its length twice the length of the trough was obtained, and in this case a ripple-marked mound arose of which the apex was at the middle of the trough—one loop of the wave—and the lowest parts were near the ends—the two nodes. When the stationary wave was of the same length as the trough two mounds arose, one near each end, and so on.
It is suggested that the tidal ridges in estuaries and the chains of sand banks under the sea are formed in this way, by stationary or nearly stationary water waves, and that the sand dunes of the sea-shore and of the desert, and the clouds in a "mackerel sky" may be similarly the products of stationary air waves.
[Note: Publication of the final paper was postponed by Ayrton's desire until June, 1910. It appeared in the Proceedings of the Royal Society of London, Series A, Vol. 84, No. 571 (October 21, 1910), 285-310. The entire paper is available online at the Internet Archive for the Contributions of Women in Physics site at UCLA.]