This is the resultant force upon the element dx, and the acceleration will be found by dividing by the mass. m dx. The acceleration will therefore be Substituting for F T m its value (+), this expression becomes y, which was to be proved. 116. Very similar reasoning applies to the stationary undulation of a cylindrical column of air. Let the motion of the particles of air be specified by the equation y denoting the longitudinal displacement of the particle whose undisturbed position was x, so that x+y is its actual dy d'y position at time t. Then and will still denote redt2 spectively the velocity and acceleration of the particle, and we have to show that the value of day dt 29 as deduced is precisely the acceleration due to the elastic force of the air, if the coefficient of elasticity E (see § 98) and the density D fulfil the relation the compressions and extensions of the air, as measured dy by the ratio of § 98, or by in our present notation V dx (see § 51), being everywhere so small that their squares are negligible. Let p be the undisturbed pressure. Then the actual pressure at time t, at the particle whose undisturbed coordinate was x, is At the particle whose undisturbed co-ordinate was x+dx, the pressure is given by this expression, together with the additional term A layer of air of original thickness dx, of unit area and of original density D, is therefore subjected on its two faces to two opposite forces whose resultant is a backward force and dividing by the mass of the layer, which is D dx, we find for the acceleration the value E which, when we replace by its value (4), becomes D 117. In general, for the propagation of along a cylindrical column of air, we have as the expression for the pressure of the particle of air whose undisturbed ordinate was x. The expression for the pressure at the particle whose undisturbed ordinate was x+dx is Hence the pressure in front of a layer of original thickness dx exceeds the pressure behind it by |