WAVE disturbances. A moment s thought will convince the reader that there must be a limit to the frequency of the oscillations which can be transmitted along a system like this, though there was none such with the continuous wire. It is not difficult to find this limit. Let the transverse displacement, at time t, of the th pellet of the series, be called y n ; and let T be the tension of the wire, m the mass of a pellet, and a the distance from one pellet to the next. Then the equation of motion is obviously where D stands for g dx . [We remark in passing that, if a be very small, this equation tends to become d-y _T z d?y clt- a dx: 2 But in this case we have ultimately m^ctfj,; so that we recover the equation for transverse vibrations of a uniform wire.] Suppose (where x = na) to be a possible free motion of the system. When this value is substituted in the equation above it gives >p r~ (pt - qx) = a _ 2 cos (pt - 2T cos (pt - qx) . (1 - cos qa}, so that mp- - 4T = sin 2 qa/2. 1 1 appears from the form of this expression that the greatest value of p is 2/T/am ; or 2v/a, if v be the speed of a disturbance in a uniform wire under the same tension, and of the same actual mass per unit of length. The time of oscillation of a pellet is 2ir/p, and cannot therefore be less than tr/am/T, or ira/v. The result is that, if v be the speed of propagation of a disturbance in a uniform wire with the same tension and same mass, the period of the quickest simple harmonic transverse oscillation which can be freely transmitted in such a system is TT times the time of running from one pellet to the next with speed v. Instead of pellets on a tended wire, we might have a series of equal bar magnets, supported horizontally at proper distances from one another, in a line. The mag netic forces here take the place of the tension ; and by arranging the magnets with their like poles together, i.e., by inverting the alternate ones, we can produce the equivalent of pressure instead of tension along the series. If the magnets have each bifilar suspension, their masses will come in, as well as their moments of inertia, in the treatment of transverse disturbances. This question is closely connected with Stokes s explan ation of fluorescence (see LIGHT, vol. xiv. p. 602), for the effect of a disturbing force, of a shorter period than the limit given above, applied continuously to one of the pellets, would be to accumulate energy mainly in the immediate neighbourhood ; and this, if we suppose the disturbing force to cease, would be transmitted along the system in waves of periods equal at least to the limit. These would correspond to light of lower refrangibility than the inci dent, but having as characteristic a definite upper limit of refrangibility. Such investigations, with their results, prepare us to expect that the usual mode of investigating the propa gation of sound, to which we proceed, cannot be correct in the case of exceedingly high notes if the medium consist of discrete particles. (4) Waves of Compression and Dilatation in a Fluid ; $ound; Explosions. Consider the case of plane waves, where each layer of the medium moves perpendicularly to itself, and therefore may suffer dilatation or compres sion. The case is practically the same as that treated in (2) above, and can be represented by the same graphic method. For we may obviously consider only the matter contained in a rigid cylinder of unit sectional area, whose axis is parallel to the displacements. The only point of difference is in the law connecting pressure and conse quent compression, and that, of course, depends upon the properties of the medium considered. (a) If the medium be a liquid, such as water, for instance, the compression may be taken as proportional to the pressure. Thus the acceleration on unit length of the column, multiplied by its mass (which in this case is simply the density of the medium), is equal to the increase of pres sure per unit length, i.e., to the increase of condensation per unit length, multiplied by the resistance to compression, K. Thus the speed of the wave is v /E//a, which is exactly analogous to the forms of (1) and (2). The density, p, of water is 62 3 Bb per cubic foot, and for it II is about 20,000 atmospheres at C., so that the speed of sound at that temperature is about 4700 feet per second. That even intense differences of pressure take time to adjust themselves over very short distances in water was well shown by the damage sustained by the open copper cases of those of the " Challenger " thermometers which were crushed by pressure in the deep sea. When a strong glass shell (containing air only) is enclosed in a stout open iron tube whose length is two or three of its diameters, and is crushed by water pressure, the tube is flattened by excess of external pressure before the relief can reach the outside. (b) In the case of a gas, such as air, we must take the adiabatic relation between pressure and density. The pressure increases faster than, instead of at the same rate with, the density, as it would do if the gas followed Boyle s law. Thus the changes of pressure, instead of being equal to the changes of compression (multiplied by the modulus), exceed them in the proportion of the specific heat at constant pressure (K) to that at constant volume (N). Thus the speed of sound is ^K/N.jp/p, where p is the pressure and p the density in the undisturbed air. The ratio of the two latter quantities, as we know, is very approximately proportional to the absolute temperature. The questions of the gradual change of type or the dying away even of plane waves of sound, whether by reason of their form, by fluid friction, or by loss of energy due to radiation, are much too complex to be treated here. In all ordinary simple, sounds even of very high pitch the displacements are extremely small compared with the wave length, so that the approximate solution gives the speed with considerable accuracy. And a very refined experimental test that this speed is independent of the pitch consists in listening to a rapid movement played by a good band at a great distance. But there seems to be little doubt that, under certain conditions at least, very loud sounds travel a great deal faster than ordinary sounds. The above investigation gives the speed of sound relatively to the air. ^Relatively to the earth s surface, it has to be compounded with the motion of the air itself. But, as the speed of wind usually increases from the sur face upwards, at least for a considerable height, the front of a sound-wave, moving with the wind, leans forward, and the sound (being propagated perpendicularly to the front) moves downwards ; if against the wind, upwards. In the case of a disturbance in air due to a very sudden explosion, as of dynamite or as by the passage of a flash of lightning, it is probable that for some distance
from the source the motion is of a projectile character;