When the depth is not infinitely great compared with the length of a wave, theory and experiment agree in showing that the motion of each particle takes place in an ellipse whose major axis is horizontal. These ellipses diminish rapidly in length as we descend in the liquid, but still more rapidly in breadth; so that, as was to be expected, the particles at the bottom oscillate in horizontal straight lines. The expression for the velocity of propagation is now by no means so simple as in the previous cases—but is easily shown to include the values already given.
So far, the first approximation. A section of the surface made by a vertical plane in the direction of the wave's motion, is shown to he bounded by the harmonic curve, or curve of sines, the form assumed by a vibrating string (see SOUND); from which it fol lows that the crests are similar to the troughs. The second approximation makes the troughs flatter, and the crests steeper, and also shows that the particles are, on the whole, carded forward by each successive wave. The amount of this progression diminishes rapidly with the depth below the surface. A third approximation shows that the veloc ity is, ceterix paribus, greater the greater is the height of the waves.
When waves advance toward the shore, their circumstances change, in general gradu ally, from those of oscillatory waves to those of waves of translation, as the depth of the water becomes less and less considerable in comparison with the length of the wave; and it is found by experiment that they " break," as it is called, when the depth of the water is about equal to the height of the crest above the undisturbed level. All thu
curious phenomena of breakers are thus easily explained by the results we have already given, when they are considered with referende to the gradual alteration of the depth of the water.
Finally, we must notice a singular phenomenon often observed, viz., that of a series of waves breaking on the coast, every eighth, or ninth, or tenth, etc., is seen to be higher than its predecessors or successors. The explanation is simple enough, and points to the simultaneous existence of two or more sets of oscillatory waves of different lengths, due in general to quite distinct causes, which reach the shore together.—For further infor mation on this subject the reader is referred to papers by Stokes in the Cambridge and Dublin Math. Jtiurnal, vol. iv., and the Cambridge Phil. Trans., vol. viii., and to Airy's " Tides and Waves " in the Encyclopadia Metrop.
This might lead us to consider the very interesting case of " Co-existence of Small Motions" presented by the interference (q.v.) of such waves; but we have already in Various articles (see POLARIZATION. SOUND, UNDULATORY TnEonv) given sufficient examples to illustrate the great principle.
There remains the consideration of the propagation of waves in elastic solids, among which, at least so far as lmniniferous vibrations are concerned, it appears that the ether (q.v.) must be ranked. This is a subject of a higher order of difficulty than any of those before mentioned, and, in the case of light at least, has not yet been treated in a thor oughly satisfactory manner, though such men as Cauchy, Neumann, Maccullagb, Green, and Stokes have written profound memoirs upon it.