WHAT IS WAVE?

Albert Einstein and Leopold Infeld
An expert from their book The Evolution of Physics,1961

A bit of gossip starting in Washington reaches New York very quickly, eventhough not a single individual who takes part in spreading it travels between these two cities. There are two different motions involved, that of the rumor, Washington to New York, and that of persons who spread the rumor. The wind, passing over the field grain, sets up a wave which spreads out across the field. Here again we must distinguish between the motion of wave n the motion of separate plants, which undergo only small oscillations. We have all seen the waves that spread in wider n wide circles when a stone is dropped in to a pool of water. The motion of the wave is very different fm that of the particles of water. The particles merely go up n down .The observed motion of the wave is that of a state of matter n not of matter itself. A cork floating on the wave shows this clearly, for it moves up n down in imitation of the actual motion of the water, instead of being carried along by the wave.........

In order to understand better mechanism of the wave, let us consider an idealized experiment .Suppose that a large space is filled quite uniformly with water, or air, or some other medium. Somewhere in the center there is a sphere. At the beginning of the experiment there is no motion at all. Suddenly the sphere begins to "breathe" rhythmically, expanding and contracting in volume, although retaining its spherical shape. What will happen in the medium? Let us begin our examination at the moment the sphere begins to expand .The particles of the medium in immediate vicinity of the sphere are pushed out, so that the density of the spherical shell of water, or air, as the case may be, is increased above its normal value. Similarly, when the sphere contracts, the density of that part of the medium immediately surrounding it will be decreased. These changes of density r propagated throughout the entire medium. The particles constituting the medium perform only small vibrations, but the whole motion is that of a progressive wave. The essentially new thing here is that for the first time we consider the motion of something which is not matter, but energy propagated throughout the matter.
Using the example of pulsating sphere, we may introduce two physical concepts, important for the characterization of waves. The first is the velocity with which the waves spread. This will depend on the medium, being different for water and air, for example. The second concept is of wavelength. In the case of waves on sea or river it is the distance from the trough of one wave to that of next, or the crest of one wave to that of next
. Thus the sea waves have greater wavelength than river waves. In the case of our waves set up by a pulsating sphere the wavelength is the distance, at some definite time, between two neighboring spherical shells showing maxima or minima of density. It is the evident that this distance will not depend on the medium alone .The rate of pulsation of the sphere will certainly have a great effect, making the wavelength shorter if the pulsation becomes more rapid, longer if the pulsation becomes slower.
This concept of a wave proved very successful in physics. It is definitely a mechanical concept. The phenomenon is reduced to the motion of particles which, according to the kinetic theory, are constituents of matter. Thus every theory which uses the concept of wave can, in general, be regarded as a mechanical theory. For example, the explanation of acoustical phenomena is based essentially on this concept. Vibrating bodies, such as vocal cords and violin strings, are sources of sound waves which are propagated through the air in the manner explained for the pulsating sphere. It is thus possible to reduce all the acoustical phenomena to mechanics by means of wave concept