y = a sin (x − vt).
And he might possibly refuse to give any other answer.
And in refusing to give any other answer than this, or its equivalent in ordinary words, he is entirely justified; that is what is meant by the term wave, and nothing less general would be all-inclusive.
Translated into ordinary English the phrase signifies, with accuracy and comprehensive completeness, the full details of "a disturbance periodic both in space and time." 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—can be 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. Any medium possessing these two properties can transmit waves, and unless a medium possesses these properties in some form or other, or some equivalent for them, it may be said with moderate security to be incompetent to transmit waves. But if we make this latter statement one must be prepared to extend to the terms elasticity and inertia their very largest and broadest signification, so as to include any possible kind of restoring force, and any possible kind of persistence of motion, respectively.
These matters may be illustrated in many ways, but perhaps a simple loaded lath, or spring, in a vice will serve well enough. Pull it to one side, and its elasticity tends to make it recoil; let it go, and its inertia causes it to overshoot its normal position. That is what inertia is,—power of overshooting a mark, or, more accurately, power of moving for a time even against driving force,—power to rush uphill. Both causes together make it swing to and fro till its energy is exhausted. This is a disturbance simply periodic in time. A regular series of such springs, set at equal intervals and started vibrating at regular intervals of time one after the other, would be periodic in space too; and so they would, in disconnected fashion, typify a wave. A series of pendulums will do just as well, and if set swinging in orderly fashion will furnish at once an example and an appearance of wave motion, which the most casual observer must recognise as such. The row of springs obviously possesses elasticity and inertia; and any wave-transmitting medium must similarly possess some form of elasticity and some form of inertia.
But now proceed to ask what is this Ether which in the case of light is thus vibrating? What corresponds to the elastic displacement and recoil of the spring or pendulum? What corresponds to the inertia whereby it overshoots its mark? Do we know these properties in the ether in any other way?
The answer, given first by Clerk Maxwell, and now reiterated and insisted on by experiments performed in every important laboratory in the world, is:—
The elastic displacement corresponds to electrostatic charge,—roughly speaking, to electricity.
The inertia corresponds to magnetism.
This is the basis of the modern electromagnetic theory of light.