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.

Earthquakes radiate waves with periods of tenths of seconds to several minutes. Rocks behave like elastic solids at these frequencies. Elastic solids allow a variety of wave types and this makes the ground motion after an earthquake or explosion (called an event) quite complex. There are two basic types of elastic wave: one involving compression and rarefaction of the elastic material in the direction of propagation of the wave, and one involving no compression but shear of the elastic material perpendicular to its direction of propagation. These are called P and S waves respectively, for primary and secondary since the P wave travels fastest and arrives first.

The propagation problem is the problem of explaining by the structural properties of the swinging system what the character of the swings would be in case the system was started in some initial situation

Earthquakes generate elastic waves when one block of material slides against another; the break between the two blocks being called a fault. Explosions generate elastic waves by an impulsive change in volume in the material. Small explosive charges are used in controlled-source seismic experiments in which the waves penetrate only a few kilometres into the earth.

The theory of waves has been much simplified, and somewhat extended, and their motions have been illustrated by experiments... A similar method of reasoning has been applied to the circulation of the blood, to the propagation of sound, either in fluids or in solids, and to the vibrations of musical chords; the general principle of a velocity, corresponding to half the height of a certain modulus, being shown to be applicable to all these cases: and a connexion has been established between the sound to be obtained from a given solid, and its strength in resisting a flexure of any kind; or, in the case of ice and water, between the sound in a solid and the compressibility in a fluid state. The doctrine of sound and of sounding bodies in general has also received some new illustrations, and the theory of music and of musical intervals has been particularly discussed.

What properties are essential to a medium capable of transmitting wave motion? Roughly, we may say two: elasticity and inertia.

A pure elastic fluid is one the constituent particles of which are all alike, or in no way distinguishable. Steam, or aqueous vapour, hydrogenous gas, oxygenous gas... and several others are of this kind. ...Whatever ...may be the shape or figure of the solid atom abstractedly, when surrounded by such an atmosphere it must be globular; but as all the globules in any small given volume are subject to the same pressure, they must be equal in bulk, and will therefore be arranged in horizontal strata, like a pile of shot.

The attempts to try to represent the electric field as the motion of some kind of gear wheels, or in terms of lines, or of stresses in some kind of material have used up more effort of physicists than it would have taken simply to get the right answers about electrodynamics. It is interesting that the correct equations for the behavior of light were worked out by MacCullagh in 1839. But people said to him: 'Yes, but there is no real material whose mechanical properties could possibly satisfy those equations, and since light is an oscillation that must vibrate in something, we cannot believe this abstract equation business'.

The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

When a constant electric current flows along a cylindrical wire, its strength is the same at every part of the section of the wire. But if the current is variable, self-induction produces a deviation from this... induction opposes variations of the current in the centre of the wire more strongly than at the circumference, and consequently the current by preference flows along the outer portion of the wire. When the current changes its direction... this deviation increases rapidly with the rate of alternation; and when the current alternates many million times per second, almost the whole of the interior of the wire must, according to theory, appear free from current, and the flow must confine itself to the very skin of the wire. Now in such extreme cases... preference must be given to another conception of the matter which was first presented by Messrs. 0. Heaviside and J. H. Poynting, as the correct interpretation of Maxwell's equations as applied to this case. According to this view, the electric force which determines the current is not propagated in the wire itself, but under all circumstances penetrates from without into the wire, and spreads into the metal with comparative slowness and laws similar to those which govern changes of temperature in a conducting body. ...Inasmuch as I made use of electric waves in wires of exceedingly short period in my experiments on the propagation of electric force, it was natural to test by means of these the correctness of the conclusions deduced. As a matter of fact the theory was found to be confirmed by the experiments...

The mathematical theory of the Navier-Stokes equations has centered upon basic questions of the existence, uniqueness, and regularity of solutions of the initial value problem for fluid motions in all of space or in a subdomain of finite or infinite extent. Such solutions, when they can be constructed or shown to exist, represent flows of a viscous incompressible fluid. In two space dimensions the theorem of existence, uniqueness and regularity was essentially completed thirty years ago by the work of Leray ..., Lions ... and Ladyzhenskaya ... who showed that a smooth solution of the initial value problem exists for arbitrary square-integrable initial data. For viscous, incompressible fluid motions in three space dimensions, ... the theorem of existence uniqueness and regularity has been proved only for sufficiently small initial data or in special cases such as cylindrical symmetry that essentially reduce the problem to two space dimensions in some sense.

Si vous demandez à tout mathématicien si dans son esprit il fait une distinction les théories de l'élasticité et celles de l'électrodynamique, il vous dira qu'il n'en fait pas, car les types de équations différentielles qu'il rencontre, et les méthodes qu'il doit employer pour résoudre les problèmes qui se présentent, sont tout à les mêmes dans le deux cas. (If you ask any mathematician if in his mind he makes a distinction between the theories of elasticity and those of electrodynamics, he will tell you that he does not, because the types of differential equations he encounters, and the methods which he must employ to solve the problems which arise, are all the same in the two cases.)

A solid stores energy in the vibrations of the atoms about their equilibrium positions. In the simplest approximation, which Einstein considered, each atom vibrates with the same frequency, ν, in each of three dimensions, so a solid containing N atoms can be thought of as 3N one-dimensional simple harmonic oscillators.

... Let a perturbation be produced anywhere, like sound; it is not immediately perceived at every other point. There are then points in space which the action has not reached in any given time. Therefore the wave, in that sense a surface, separates the medium into two portions (regions): the part which is at rest, and the other which is in motion due to the initial vibration. These two portions of space are contiguous. It was only in 1887 that Hugoniot, a French mathematician, who died prematurely, showed what the surface of the wave can be; and even his work was not well known until Duhem pointed out its importance in his work on mathematical physics.