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 E = D the compressions and extensions of the air, as measured by the ratio of § 98, or by V dy in our present notation (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 n, 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 117. In general, for the propagation of any disturbance 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 or the pressure behind exceeds the pressure in front by As the mass of the layer per unit area is D dx, the accelera Similar reasoning will apply to the propagation of transverse waves along a string. It thus appears that the undulations which we have -been discussing in the previous chapters from a merely kinematical point of view, are dynamically possible as consequences of the laws of elasticity. CHAPTER XI. ENERGY OF VIBRATIONS. 119. AN ordinary pendulum affords a very good example of the transformation of energy. When we draw it aside from its lowest position we do work against gravity, the amount of this work being equal to the weight of the pendulum multiplied by the height of the new position of its centre of gravity above its lowest position. If we now release the pendulum it falls back, and in the fall gravity does as much work upon it as was previously done against gravity. This amount of work may be called the energy of vibration of the pendulum, and it is continually undergoing transformation. In the two extreme positions it is all in the shape of 'potential energy,' or as it may be better called 'statical energy,' because its amount is computed without any reference to the laws of motion. In the lowest position it is all in the form of 'kinetic energy' or energy of motion, the amount of which is measured by multiplying each element of the mass by half the square of its velocity and adding these products for the whole pendulum. In intermediate positions the energy is partly in the one form and partly in I |