416 WAVE not matter, though in some cases the wave permanently displaces, usually to a small amount only, the medium through which it has passed. Currents, on the other hand, imply the passage of matter associated with energy. The subject is one which, except in a few very simple or very special cases, has as yet been treated only by approximation even when the most formidable processes of modern mathematics have been employed, so that this sketch, in which it is desired that as little as possible of higher mathematics should be employed, must be confined mainly to the statement of results. And the effects of viscosity, though very important, cannot be treated. There are few branches of physics which do not present us with some forms of wave, so that the subject is a very extensive one: tides, rollers, ripples, bores, breakers, sounds, radiations (whether luminous or obscure), tele graphic and telephonic signalling, earthquakes, the pro pagation of changes of surface-temperature into the earth s crust all are forms of wave-motion. Several of these phenomena have been treated in other parts of this work, and will now be but briefly referred to ; others require more detailed notice. When a medium is in stable equilibrium, it has no kinetic energy, and its potential energy is a minimum. Any local disturbance, therefore, in general involves a communication of energy to part of the medium, and it is usually by some form of wave-motion that this energy spreads to other parts of the medium. The mere with drawal of a quantity of matter (as by lifting a floating body out of still water), local condensation of vapour in the air, the crushing of a hollow shell by external pressure, the change of volume resulting from an explosion, or from the sudden vaporization of a liquid are known to all as common sources of violent wave-disturbance. Waves may be free or forced. In the former class the disturbance is produced once for all, and is then propagated according to the nature of the medium and the form of the disturbance. Or the disturbance may be continued, provided the waves travel faster than does the centre of disturbance. In forced waves, on the other hand, the disturbing force continues to act so as to modify the pro pagation of the waves already produced. Thus, while a gale is blowing, the character of the water-waves is con tinually being modified ; when it subsides, we have regular oscillatory waves, or rollers, for the longer ones not only outstrip the shorter but are less speedily worn down by fluid friction. The huge mass of water which some steamers raise, especially when running at a high speed, is an excellent example of a forced wave. The ocean-tide is mainly a forced wave, depending on the continued action of the moon and sun ; but the tide- wave in an estuary or a tidal river is practically free, being almost independent of moon and sun, and depending mainly upon the con figuration of the channel, the rate of the current, and the tidal disturbance at the mouth. In what follows we commence with a special case of extreme simplicity, where an exact solution is possible. This will be treated fully, partly on account of its own interest, partly because its results will be of material assistance in some of the less simple, and sometimes apparently quite different, cases which will afterwards come up for consideration. (1) Transverse Waves on a Stretched Wire. In the article MECHANICS, 265, it has been proved by the most elementary considerations that an inextensible but flexible rope, under uniform tension, when moving at a certain definite rate through a smooth tube of any form, exerts no pressure on the interior of the tube. In fact, the rope must press with a force T/r (where T is the tension and r the radius of curvature) on the unit of length in con sequence of its tension, and with a force /x,v 2 /r (where <i- is the speed, and ^ the mass of unit length) in consequence of its inertia. That there may be no pressure on the tube, i.e., that it may be dispensed with, we must therefore have T - /j.v- = 0, or v = /T/ft.. From this it follows that a disturbance of any form (of course with continuous curva ture) runs along a stretched rope at this definite rate, and is unchanged during its progress. In the proof, the influ ence of gravity was left out of consideration, and this result may therefore be applied to the motion of a transverse disturbance along a stretched wire, such as that of a pianoforte, where the tension is very great in comparison with the weight of the wire. But the italicized word any, above, gives an excellent example of one of the most difficult parts of the whole subject, viz., the possibility of a solitary wave. This is a question upon which we cannot here enter. If we restrict ourselves to slight disturbances only, theory points out and trial verifies that they are super- posable. In fact, in the great majority of investigations which have been made with regard to waves, the disturb ances have been assumed to be slight, so that we can avail ourselves of the principle of superposition of small motions (MECHANICS, 73), which is merely an application of the mathematical principle of "neglecting the second order of small quantities." The verification by trial is given at once by watching how the ring-ripples produced by two stones thrown into a pool pass through one another with out any alteration ; that by observation is evident to any one who sees an object in sunlight, when the whole inter vening space is full of intense wave-motion. Returning to our wire, let us confine ourselves to a small transverse disturbance, in one plane, and try to dis cover what happens when the disturbance reaches one of the fixed ends of the wire. Whether a point of the wire be fixed or not does not matter, provided it do not move. In the figure below, two disturbances are shown, moving in opposite directions (and of course with equal speed). Of these, either is the perversion, as well as the inversion, of the other. When any part of the one reaches O, the point halfway between them, so as to displace it upwards, the other contributes an exactly equal displace ment downwards. Thus O remains permanently at rest, while the two waves pass through it without affecting one another ; and we may therefore assert that the wave A when it reaches the end of the wire is reflected as B, or rather that each part of A when it reaches O goes back as the corresponding part of B. B, in the same way, is seen to be reflected from O as A. Now we can see what happens with a pianoforte wire. Any disturbance A is reflected from one end O as B, and at the other end is reflected as A again. Hence the state of the wire, whatever it may be, recurs exactly after such an interval as is required for the disturbance to travel, twice over, the length of the string. Remembering that the displacements are supposed to be very small, our fundamental result may now be ex pressed by saying that the force acting on unit length of the disturbed wire, to restore it to its undisturbed position, is T/r or fj.v 2 jr. Thus the ratio of the acceleration of each element to its curvature is the square of the rate of pro pagation of the wave. It will be shown below that this is the immediate interpretation of the differential equation
of the wave-motion.