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which answers to a given mode of drop-vibration. When a drop vibrates in a given mode, its position at any assigned moment during its vibration is of course known. If we also know the amount by which drops further on in the level-line are later in their starts [§ 9] than drops less advanced in that line, we can assign the positions of any number of given drops at any given instant of time.
Suppose that each drop makes one complete vibration per second about its position of original rest in the level-line. The law of its vibration is roughly indicated in Fig. 11.
AB is the path described by any drop; O its position when in the level line; 1, 2, 3, 4...16 its positions after 16 equal intervals of time each one-sixteenth of a second: 16 coincides with 0, i.e. the drop has returned to its starting-point.
Next, select a series of drops originally at rest in equidistant positions along the level-line, and so situated that each commences a vibration, identical with that above laid down in Fig. 11, one-sixteenth of a second after the drop next it has started on an equal oscillation. Fig. 12 shows the rest-positions of the series of drops
in the level-line, and their contemporaneous positions
αν, αν, αν, ας...16,
during a subsequent vibration.
The moment selected for the figure is that in which the first of the series, a,, is on the point of commencing its vibration in a vertical direction. Since the second drop started one-sixteenth of a second after the first, its position in the figure will be below the level-line at a, making the line
a1 a equal to the line 015 in Fig. 11. The next drop, which is two-sixteenths behind a, in its path, will be at a making a, a, equal to 014 in the same figure. In this way the positions of all the points aí ag ag', &c., in Fig. 12 are determined from Fig. 11. They give us, at once, a general idea of the form of the resulting wave. By laying down more points along the line AB in Fig. 11, we can get as many more points on the wave as we please, and should thus ultimately arrive at a continuous curved line. This is the wave-form resulting from the given vibration-mode with which we started, and, since only one wave-form can be obtained from it, we infer that each distinct mode of drop-vibration gives rise to a special form of wave.
It has now been sufficiently shown that corresponding to the three elements of a wave, amplitude, length, and form, there are three elements of its proper drop-vibration, extent, rate, and mode.
13. We have seen that a sea-wave consists of a state of elevation and depression of the surface above and below the level-plane. The same thing holds of the small ripples set up by throwing a stone into a pond, and the non-progressive nature of the motion of individual drops on the surface can be as easily made out on a small, as on a large sheet of water. Moreover the characteristic phenomenon on which
we have been engaged, viz. a uniformly progressive motion arising out of a number of oscillatory movements, is by no means confined to liquid bodies. Thus, when a carpet is being shaken, bulging forms, exactly like water-waves, are seen running along it. A flexible string, one end of which is tied to a fixed point, and the other held in the hand, exhibits the same phenomenon when the loose end is suddenly twitched. It has accordingly been found convenient to extend the term 'wave' beyond its original meaning, and to designate as wave-motion any movement which comes under the definition just laid down. We proceed to an instance of such motion which is important from its similarity to that to which the transmission of Sound is due.
14. Any one who has looked down from a slight elevation on a field of standing corn on a gusty day, must have frequently observed a kind of thrill running along its surface. As each ear of corn is capable of only a slight swaying movement, we have here necessarily an instance of wave-motion, the earvibrations corresponding to the drop-vibrations in water-waves. There is, however, this important difference between the cases, that the ears' movements are mainly horizontal, i. e. in the line of the wave's advance, whereas the drop-vibrations are entirely perpendicular to that line. The advancing wave
is therefore no longer exclusively a state of elevation or depression of surface, but of more tightly, or less tightly, packed ears. The annexed figure gives a
rough idea how this takes place. The wind is supposed to be moving from left to right and to have just reached the ear A. Its neighbours to the right are still undisturbed. The stalk of C has just swung back to its erect position. The ears about B are closer to, and those about C further apart from, each other, than is the case with those on which the wind has not yet acted.
After this illustration, it will be easy to conceive a kind of wave-motion in which there is no longer (as in the case of the ears of corn) any movement transverse to the direction in which the wave is advancing.
15. Let a series of points, originally at rest in equidistant positions along a straight line, as in (0), Fig. 14, be executing equal periodic vibrations in that line, in such a manner that each point is a