WAVE, the name given to estate of disturbance propagated from one set of particles of a medium to the adjoining set, and so on; sometimes with, sometimes without, a small permanent displacement of these particles. But the essential characteristic is, that energy (see FORCE), not matter (q.v.), is on the whole transferred. The theory of wave-motion is of the utmost importance in physical science; since, besides the tide-wave, waves in. the sea, in ponds, or in canals, undulations in a stretched cord (such as a piano-forte wire), or in a solid (as sound-waves or earthquake-waves), we know that sounds in air are propagated as waves (see SOUND), and that even light (see UNDULATORY THEORY) is a form of wave-motion.
The general investigation of the form and rate of propagation of waves demands the application of the highest resources of mathematics; and the theory of even such com paratively simple cases as the wind-waves in deep water (the Atlantic roll, for instance), though easily enough treated to a first, and even to a second and third approximation, has not yet been thoroughly worked out, as fluid friction has not been taken account In this article, therefore, we will merely state some of the more important conclusions which mathematical analysis has established in the more difficult of these inquiries, com paring them with the observations of Scott Russell and others; while we give at full length the very simple investigations of the motion of a wave along a stretched cord, and of the propagation of a particular kind of sound-wave.
To find the rate at which an undulation runs along a stretched cord, as for instance, when a harp-string is sharply struck or plucked near one end, a very simple investiga tion suffices, Suppose a uniform cord to be stretched with a given tension in a smooth tube of any form whatever, we may easily show that there is a certain velocity with which the cord must be drawn through the tube in order to cease to press on it at any point, that is, to move independently of the tube altogether. For the pressure on the
tube is due to the tension of the cord; and is relieved by the so-called centrifugal force (see CENTRAL FORCES) when the cord is in motion.
If T be the tension of the cord, 9' the radius of curvature of the tube at any point, the pressure ou the tube per unit of length Is T r If m be the mass of unit length of the cord. v its velocity, the centrifugal force is These are equal in magnitude, and so destroy each other, if T =me.
Hence, if the cord be pulled through the tube with the velocity thus determined, there will be no pressure on the tube, and it may therefore be dispensed with. If we suppose the tube to have a form such as that in the figure, where the extreme portions are in one straight line, the cord will appear to be drawn with velocity v, along this, the curved part being occupied by each portion of the cord in succession; presenting something like the appearance of a row of sheep in In dian file, jumping over a hedge.
To a spectator moving in the direction of the arrow with velocity v, the straight parts of the cord will appear to be at rest, while an undulation of any definite form and size whatever runs along it with velocity ro, in the opposite direction. This is a very singu lar ease, and illustrates in a very clear manner the possibility of the propagation of a rolitary wave.
Thus we have proved that the velocity with which an undulation runs along such a cord is If .1 be the length of the cord in feet, to its whole weight, IV the appended weight by which it is stretched, g= 32.2 ft., the measure of the earth's gravity, this becomes • This formula is found to agree almost exactly with the results of experiment. We can easily see why it should be to some small extent incorrect, because we have supposed. the cord to be inextensible, and perfectly flexible, which it cannot be; and we have neg lected the effects of extraneous forces, such as gravity, the resistance of the air, etc.