Wave motion

“ If we could see the particles of the air in the concert-room, and fasten our attention upon any one of them, we should see it executing a most complicated motion under the combined action of these air-wave-producing instruments. We should be fascinated by the amazing dance of molecules to and fro and from side to side, as the medley of waves of compression or rarefaction embraced them and drove them hither and thither in their resistless grasp. ”

“ If the movement is rapid enough, the appearance of a wave moving along will be seen. [4] In all these exhibitions of wave-motion the movement of the particles is due to a common cause, but the moving particles do not control each other’s motion. There is no connection or tie between them. Suppose, however, that we suspend a series of heavy balls like pendulums, and interconnect them by elastic threads (see Fig. 7) , then we have an arrangement along which we can propagate a true wave. ”

“ In more learned language, we may define a wave-motion by saying that a wave-motion exists in any medium when the separate portions of it along any line execute in order any kind of cyclical or repeated motion, the particles along this line performing the movement one after the other, and with a certain assigned delay between each adjacent particle as regards their stage in the movement. ”

“ If we imagine all the velocities in any progressive system of plane waves to be reversed at a given instant without affecting the displacements, and the system of wave-motion thus obtained to be superposed upon the original system, we obtain a system of stationary waves having the same wave-length and period of oscillation as the original progressive system. If we then reduce the magnitude of the displacements in the uniform ratio of two to one, they will be identical, at an instant of maximum displacement, with those of the original system at the same instant. ”

“ If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. ”

“ This is superficial science, for it explains nothing. Waves are simply a form of motion, and a form of motion creates nothing. Vibrations of what? Wave motions of what? Our learned friend does not inform us. Vibrations and wave motions are like heat sensations, they are not realities; the force that creates them is the reality, and they are but the mode or law of operation. ”

“ Anything thus doubly periodic is a wave; and all waves, whether in air as sound waves, or in ether as light waves, or on the surface of water as ocean waves, are comprehended in the definition.
What properties are essential to a medium capable of transmitting wave motion? Roughly we may say two—elasticity and inertia. Elasticity in some form, or some equivalent of it, in order to be able to store up energy and effect recoil; inertia, in order to enable the disturbed substance to overshoot the mark and oscillate beyond its place of equilibrium to and fro. ”

“ Whatever view we take of the motions which constitute light, we can hardly suppose them (disturbed as they are by the presence of the ponderable molecules) to be in strictness represented by the equations of wave-motion. Yet in a certain sense a wave-motion may and does exist. If, namely, instead of the actual displacement at any point, we consider the average displacement in a space large enough to contain an immense number of molecules, and yet small as measured by a wave-length, such average displacements may be represented by the equations of wave-motion ”

“ A most important factor in determining the propagation of any wave-motion, through a gas or solid, is the relationship of the elasticity of the gas or solid to its density. Suffice to say, that the velocity of any wave-motion is determined by the relation of the elasticity to the density. For example, sound, which is a wave-motion of the air, can not only be transmitted through gaseous bodies as air, but also through liquids and solids. ”

“ Hence the particular group of balls which are the seat of the visible wave-motion is continually being shifted along. The rate at which the centre of this active group of vibrating balls is displaced may be called the velocity of the wave-train. The velocity of the wave is, however, something greater, since the waves are all the time moving through the group. This wave-velocity is numerically estimated by taking the product of the wave-length and frequency of the motion.
At this stage it is necessary to explain that waves are not merely a mode of motion; they are a means of conveying energy. ”

“ As a wave progresses over the surface, wave-energy is continually being imparted to portions of the water in front, and it is transferred away from others in the rear. In the very act of setting a fresh particle of water in oscillation, the portions already vibrating must diminish their own motion. They may hand on the whole of their energy or only a part of it to their neighbours. This distinction is a very important one, and it determines whether a single act of disturbance shall create a solitary wave or wave-train in a medium. ”

“ As in nature all is ebb and tide, all is wave motion, so it seems that; ”

“ Light gives precisely the same kinds of information, and hence it is only natural to infer that light also is a wave motion. We know, in fact, that it is so; but before giving the evidence to prove it, it will be well to make a little preliminary study of the chief characteristics of wave motion.
FIG. 1 One of the difficulties encountered in studying wave motion is the rapidity of the propagation of the waves. A fairly moderate speed is attained by the waves propagated along a spiral spring. ”

“ Let us consider a wave-motion model such as that represented in Fig. 7, in which a number of suspended heavy balls are connected to one another by elastic threads. Let one ball in the centre be drawn on one side and then released. It will swing to and fro, and will start a wave outwards in both directions. If the row of balls is sufficiently long, it will be seen that the ball by which the wave was started soon comes to rest, and that the wave-motion is confined to a certain group of balls on either side. ”

“ We see, therefore, that each ball must oscillate, or swing to and fro, and that its movement is gradually communicated to its neighbours. A wave-motion is thus started, and a true wave is propagated along the line of balls, in consequence of the presence of elasticity and inertia. The necessary conditions for the production of a true wave in a medium of any kind are therefore: (1) that the medium must elastically resist some sort of deformation ”

“ Since the incoming wave affects water which is at the same time under the influence of the undertow, it gives to that current a pulsating character, for the wave-motion sometimes supports and sometimes opposes the undertow, and thus endows it with a higher transporting power than belongs to its mean velocity. ”

“ If, therefore, we can determine the direction of motion of the wave particle in the first semiphase, and its maximum velocity, we can obtain, from any selected point, a line (that of emergence of the shock) somewhere in which, if prolonged beneath the earth, the focus must have been; and if we can obtain like results for two or more selected points, we decide the position and the depth of the focus, which must be in the intersection of the several lines of direction of the wave particle motion at each point, when prolonged downwards. ”

“ In all departments of natural science we find ourselves confronted by the phenomena of wave-motion. In the study of earthquakes and tides, telegraphs and telephones, as well as terrestrial temperature, no less than in the examination of water waves and ripples, sound, music, or light and heat, we are bound to consider waves of some particular kind. ”

“ If a wave is made and travels away over water, it carries with it energy, and hence it can only be created if we have a store of energy to draw upon. If we suppose that skin friction is absent, and that the ship floats upon a perfect fluid, it would nevertheless be true that, if the moving object creates waves, it will thereby reduce its own movement and require the application of force to it to keep it going. We may say therefore that if any floating object creates waves on a liquid over which it moves, these waves rob the floating body of some of its energy of motion. ”

“ Accordingly, the experiment of producing interference between two sets of light rays so that they destroy each other is a strong argument in favour of the view that light must consist in some kind of wave-motion existing in a medium susceptible of supporting it, filling all space, and existing in all transparent bodies. ”

“ Does he question the inference that some of the motion embodied in each wave is from instant to instant dissipated, partly in this way and partly in the heat evolved by fluid friction? Can he show any reason for doubting that, when the sound-waves have become too feeble to affect our senses, their motion still continues to undergo this transformation and diminution until it is all lost? If not, why does he implicitly deny that the molar motion constituting sound eventually disappears in producing the molecular motion constituting heat? ”

“ We are led, therefore, to infer that, even when agitated by waves, each particle of water never moves far from its position when at rest, and that the real movement of the water is something very different from its apparent motion. If we place on the surface of water a number of corks or pieces of paper, and then watch them as a wave passes over them, we shall notice that the corks or bits of paper rise and fall successively, that is, one after the other, and not all together. ”

“ Vibrations or a Clamped Plate.
does not move forward. The floating straw is not borne along, but merely rises and falls in its place, and so the particles of water only oscillate up and down in circles, and, communicating their motion to the adjacent particles, there is an outward transference of force by wave-action, and the water-particles move up and down while the wave moves forward. Air-waves exemplify the same principle, but in a different way. A vibrating body throws the contiguous air into movement, and produces the wave. ”

“ If we can send out a spherical wave front, and have it lengthen rapidly as it proceeds, we will have a wave front that is, at all points, different. Any entering wave would, sooner or later, meet a wave that was half a phase out, no matter what the motion was, nor what the frequency, as long as it lies within the comparatively narrow molecular wave band. ”

“ We understand, then, that there is motion of particles as well as a transit-wave; that the "travelling zone or shell of vibration" is a zone or shell of "elastic compression."
Fig. 9 illustrates the "shells" as they move away from B, the focus of disturbance. The transit-wave, with its interstitial vibrations, reaches the surface in the manner shown in the diagram Fig. 10. ”

“ Similarly a sound wave in air is reflected upon striking the surface of a body. If the wave strikes perpendicularly it returns along the line from which it comes, if, however, it strikes at some other angle it does not return along the same line, but as in other cases of reflected motion, the direction of the reflected wave is described by the Law of Reflected Motion as follows: The angle of reflection is always equal to the angle of incidence. This law is illustrated in Fig. ”

“ Finally, it may be noticed that the conditions of wave-propagation without change of type may be investigated in another manner. If we impress on the whole medium a velocity equal and opposite to that of the wave we obtain a “steady” or “stationary” state in which the circumstances at any particular point of space are constant. Thus in the case of the vibrations of an inextensible string we may, in the first instance, imagine the string to run through a fixed smooth tube having the form of the wave. ”

“ 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. ”