Wave velocity

“ The whole theory of the trail of waves made by a canal-boat is only comprehensible if it is clearly seen that a water-surface wave has a certain velocity determined by its wave-length. If the wave-speed is small, the waves are short. As the speed increases the waves get longer. Or the matter may be put in another way. We may say that just as a pendulum has a certain rate of vibration depending on its length, so a water wave has a certain frequency, and therefore, a certain speed of propagation dependent upon the wave-length, or shortest distance from one wave-crest to the next. ”

“ Now, the velocity of wave propagation can be seen, without the aid of any mathematical analysis, to depend on the elasticity of the medium and its density; for we can see that if a medium is highly elastic the disturbance would be propagated at a great speed. ”

“ The speed with which the surface waves move outwards in all directions, varies not only with the force of the wave, but also with the kind of material through which they pass. This velocity may be in the neighborhood of twenty miles per second, while in others the velocity is as great as 140 miles per second. ”

“ If a wave meets obliquely a boundary separating two regions, in one of which it moves slower than in the other, then, for the same reason that the direction of the row of soldiers in the above illustration is bent by reason of the retardation of velocity experienced by each man in turn as he steps over the dividing line, so the wave-line or wave-front is bent by passing from a place where it moves quickly to a place where it moves more slowly. The ratio of the velocities or speeds of the wave in the two regions is called the index of refraction. ”

“ We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W. J. M. Rankine (Phil. Trans., 1870, p. 277) .
Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any “external” force we liked to each particle, in addition to the “internal” force called into play by the compressions or extensions. For instance, if we have a wave with displacement curve of form ABC (fig. 8) , and we require it to travel
Fig. 8. ”

“ If we cast aside the mythology introduced into the subject by the light radiation specialists, and treat the question as it should be treated, as a matter of ordinary dynamics, it is evident that it is the displacement of matter (if any) that gives rise to the momentum of a wave, and it is the momentum of the wave that gives rise to the pressure at reflection, and by equating the two, as has been done in the foregoing demonstration, we have the simplest known method of obtaining the expression for the velocity of sound. ”

“ In those places at which the pressure variation is greatest, the velocity changes are least, and vice versa. Outside the tube, as a result of these air motions in it, we have a hemispherical air wave produced, which travels out from the mouthpiece as a centre; and if we could examine the distribution of air pressure and velocity through all external space, we should find a distribution which is periodic in space as well as time, constituting the familiar phenomenon of an air wave. ”

“ The wave of velocity is only the wave which happens in uniform motion.
As to the wave of acceleration, this is a perturbation altogether analogous to light waves, which starts from the electron at the instant when it undergoes an acceleration, and which is then [Pg 510] propagated by successive spherical waves with the velocity of light. Whence follows: in a straight and uniform motion, the energy is wholly conserved; but, when there is an acceleration, there is loss of energy, which is dissipated under the form of luminous waves and goes out to infinity across the ether. ”

“ The most simple plan for determining the velocity of waves—and it may be hourly practised on board of every vessel—is the plan which is followed by Captain Ginn, of the American ship John Knox, one of our co-operators. When he heaves the log with the 8eas following, instead of hauling in the line immediately, he leaves the chip to tow, watching till he observes it on the crest of a wave; he then turns the glass, or notes his watch, and marks the time it takes the wave to reach the ship. The usual velocity of the waves in the Atlantic is 22-3 miles an hour, off Cape Horn 26-8. ”

“ Distribution of Velocity in a Wave.—If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve. Let the wave AQBTC (fig. 5) travel to A′QB′TC in a very short time. In that short time the displacement at H decreases from HP to HP′ or by PP′. The motion of the particle is therefore backwards towards A. At J the displacement remains the same, or the particle is not moving. At K it increases by RR′ forwards, or the motion is forwards towards B. At L the displacement backward decreases, or the motion is forward. ”

“ If the steepness gets very great our investigation ceases to apply, and neither experiment nor theory has yet shown what happens. Probably there is a breakdown of the wave somewhat like the breaking of a water-wave when the, crest gains on the next trough. In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small. ”

“ The longer waves travel faster than the shorter, so that each wave is continually being drawn out in length, and its velocity of propagation therefore continually increases as it advances. If we fix our attention on a particular point of the surface, the level there will rise and fall with increasing rapidity and increasing amplitude. ”

“ But those whose wave-length is near to that of the wave of minimum velocity will diverge less than any of the others, so that the most marked feature at a distance from the body will be the two long lines of ripples of minimum velocity. ”

“ In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to a′a′b′b′ without deformation. ”

“ In this case the velocity of each particle of the water is proportional to its distance from the axis, and the whole mass, when relative equilibrium is set up, turns, as if it were rigid, about the axis of the vessel. Each element of the fluid in this "forced vortex," as it is called, is in rotation, and, like the moon, makes one turn in one revolution about the centre of its path. This is, therefore, a true, though very simple, case of vortex-motion. ”