Page:Philosophical magazine 21 series 4.djvu/19

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drops succeed each other with sufficient rapidity, in short, if they constitute a jet of water falling into the current, the successive waves will, by their intersection, give rise to a continuous series of points elevated above the general level of the liquid, and forming a ripple more visible than the waves which it envelopes. If we replace the jet by a solid cylinder, the effect will be essentially the same: the waves, it is true, may differ in form and even in the velocity with which they are propagated; but, as before, the ripple will be their envelope. It is in this manner that the pebbles and other partially immersed bodies on the banks of a stream give rise to ripples whose forms, as we shall see, indicate in every case the velocities of the adjacent parts of the current. It is scarcely necessary to add that bodies moving on the surface of still water produce precisely similar effects; the ripples caused by boats and water-fowl are examples familiar to all.

5.
The relation between the form of the ripple and the velocity of the stream may be easily determined without even knowing the forms of the waves of which the ripple is the envelope. Deferring this determination, however, to art. 10, let us first, for the sake of completeness, consider the following question: —

The initial form of a wave being known, through what variations will it pass as it floats down a stream where the velocity and direction of the current vary from point to point according to a given law? Let x and y be the coordinates of any point m on the surface of the stream, and let v and &alpha; be given functions of x and y, denoting respectively the velocity of the current at the point m, and the angle between its direction and the abscissa axis; the problem is to find the equation



y = f(x, t) $$. . . . . (1)

of the wave at the expiration of the time t, the equation



y = f(x, 0) $$. . . . . (2) of the wave at the origin of that time being known.

At the time t in question, the point m of the wave bmc will have two velocities; one, X, in the direction of the normal mn to the wave, and another, v, in the direction ma determined by the angleGmX = a. At the expiration of the element of time dt, there- fore, the point m will arrive at the opposite angle m' of a small pax'al- lelogram, whose sides mn=Xdt and jna=.vdt have the directions above defined. If we call x' and y' the coordinates of m', and <^ = nm the angle between the external normal B2