WAVES AND Wave is the name commonly given to a disturbance of the surface of a body in the form of a ridge and trough, propagated by forces tending to restore the surface to its figure of equilibrium, the particles not advancing with the wave. A more complete and scientific definition of a wave is that particular form of aggregation assumed by the molecules between the nearest two consecutive surfaces in which similar phases simultaneously exist throughout, The theory of wave-motion is of the utmost im portance in allphrical science, and the general investigation of t e form and rate of propaga tion of waves demands the application of the highest resources of mathematics. The theory of even such comparatively 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 approxima nen, has not been thoroughly worked out, as fluid friction has not been fully taken into ac count. 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 investigation suf fices. 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 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 inde pendently of the tube altogether. For the press ure on the tune is doe to the tension of the ce rd; and is relieved by the centrifugal force when the cord is in motion. If T he the tension of the cord, r the radius of cut-Nature of the tube at any point, the pressure on the tube per unit of length is T. If rn be the mass of unit length of the card, v its velocity, the centrifugal force is m — Tlusc arc equal in magnitude, and so destroy each other, if T Hence, if the cord be pulled through the tube with the velocity thus determined, these will be no more pressure on the tube, and it may therefore he dispensed teeth. If we suppose the tube to have a form such a, that in Fig. 1, where the extreme portions are in one straight line, the cord will appear to ht drawn with veh,city along this, the curved part being occupied by each por tion of the cord in succession ; presenting some thing like the appearance of a row of sheep in Indian file. jumping over a hedge. To a specta
tor mosing in the direction of the arrow with velocity t., 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 v, in the opposite direction. This is a very singular case, and illustrates in a very clear manner the possibility of the propa gation of A solitary wave. Thus we prove that the velocity with which an undulation runs along such a cord is the square root of T divided by on.
If I be the length of the cord in feet, w its whole weight, H' the appendedweight by which it is stretched, g = feet, the measure of the earth's gravity, this becomes (--/). This s formula is found to agree almost exactly with the results of experiment. We can easily see why it should be to some small extent incor rect, because we have supposed the cord to be inextensible and perfectly flexible, which it can not be; and we have neglected the effects of ex traneous forces, such as gravity, the resistance of the air, etc.
In gometrical investigations it has beer, proved that the velocity with which a sound wave travels is proportional to (-01 where p is the pressure, and d the density of the air. The easiest mode of doing this is to express, in terms of these and other quantities, the tion given by the laws of motion, Mass X celeration = Difference of Pressures, and to assume that Hooke's law holds, even ing the sudden compression of air. The going formula shows that the velocity of is not affected by the pressure of air (the height of the barometer) since, iu still air, p is proportional to d. The velocity does depend on the temperature, being, in fact, proportional to the square root of the ture measured from absolute zero. \\c also s-c from the formula that the velocity is inversely as the square root of the density of the gas the pressure being the same. Thus a sou wave travels about four times faster in gen than in air. Also, within the limits of proximation we have used, the velocity does not depend upon the intensity, pitch, or quality of the sound.