If the values of r and s can be found, those of u and p follow at once from (29). Now (30) and (31) shew that r is constant for a geometrical point moving with velocity whilst s is constant for a point moving with velocity Hence any given value of r travels forward with the velocity dp /dp dp +u, and any given value of s backwards with the velocity These results enable us, if not to calculate, still to understand the character of the motion in any given case. Thus let us suppose that the initial disturbance is confined to the space between the values a and b of x; so that we have initially for 1 where ", "2, 8, 8, are constant. The region within which r is variable advances, that within which s is variable recedes, so that after a time these regions separate, and leave between them a space in which r=r1, s=s,, and which is therefore free from disturbance. The original disturbance is thus split up into two waves travelling in opposite directions. In the advancing wave s=s,, and therefore u=f(p) - 2s,, so that both density and particle-velocity advance at the rate /dp dp +u. This velocity of propagation is greater, the greater the value of p. The law of progress of the wave may be illustrated by drawing a curve with x as abscissa and p as ordinate, and making every point of this curve move forward with the above velocity. It appears that those parts move fastest for which the ordinates are greatest, so that finally points with larger overtake points with smaller ordinates, and the curve becomes at some point perpendicular dp du to x. The functions are then infinite, and the above dx' dx formulæ are no longer applicable. We have in fact a 'bore,' or wave of discontinuity. Compare Art. 152. Earnshaw's Method. 172. The same results follow from Earnshaw's investigation; which is however somewhat less general in that it embraces waves travelling in one direction only. If for simplicity we suppose p and ρ to be connected by Boyle's law or, writing y=x+, so that y denotes the absolute position at time t of the particle x, To obtain the complete solution of (32) we must* eliminate a so that if u be the velocity of the particle x, we have In the parts of the fluid not yet reached by the wave we have ppo, u = 0. Hence we have C=0, and therefore To obtain results independent of the form of the wave let us take two particles, which we distinguish by suffixes, so related that the value of p which obtains for the first particle at the time t, is found at the second particle at the time t. * Boole, Differential Equations, c. 14. is the same for both these particles, so that which shews that any value p of the density is propagated from particle to particle with the velocity c. The rate of propagation Po in space is given by 2, viz. it is ±c loga, or For a wave travelling in the positive direction we must take the lower signs throughout. If it be one of condensation (i.e. p>P.), u is, by (35), positive. We see as before that the denser parts of the wave gain continually on the rarer, and at length overtake them, when a bore is formed, and the subsequent motion is beyond the scope of this analysis. Eliminating x between (36) and (37), and writing for clog a its value – u, we find for a wave travelling in the positive direction, y= (c+u)t + F(x), which may, in virtue of (35), be written. u = ƒ{y - (c+u)t}. This formula is due to Poisson*. Its interpretation, leading of course to the same results as before, was discussed and illustrated by Stokes and Airy, in the Philosophical Magazine, in 1848-9. The theoretical result that violent sounds are propagated faster * Journ. de l'Ecole Polyt. t. 12, cahier 14, p. 319. Quoted by Earnshaw and Rankine. than gentle ones is confirmed by the remarkable experiments of Regnault on the motion of sound in pipes. 173. The above investigations shew that a plane wave of air experiences in general a gradual change of type as it proceeds. It is worth while to inquire under what circumstances a wave could be propagated without change. Let A, B be two points of an ideal tube of unit section drawn in the direction of propagation, and let the values of the pressure, the density, and the particlevelocity at A and B be denoted by P1, P1, u1, and P2, P2, u2, respectively. If as in Art. 152 we impress on everything a velocity c equal and opposite to that of the waves we reduce the problem to one of steady motion. Since the same amount of matter now crosses in unit time each section of the tube, we have 29 P1(cu1) = P(cu2) = const.=m, say....................... (40). The quantity m, denoting the mass swept past in unit time by a plane moving with the wave, in the original motion, is called by Rankine† the 'mass-velocity' or 'somatic velocity' of the wave. Again, the total force acting on the mass included between A and B is p2- P1, and the gain per unit time of momentum of this where, still following Rankine, we denote by s [= p ̃1] the 'bulkiness,' i.e. the volume per unit mass, of the substance. Hence the variations in pressure and bulk experienced by any small portion of the medium as the wave passes over it must be such that p+m's const..... = .(43). This condition is not fulfilled by any known substance, whether at constant temperature, or when free from gain or loss of heat by * Mém. de l'Acad. t. 37. Quoted in Wüllner's Experimental physik, t. 1, p. 685. An abstract of the experiments is printed at the end of the second edition of Tyndall's Sound. + Phil. Trans. 1870. |