20 INTRODUCTION Chap. 1 force acting at the lower end is uA, where A is the cross-sectional area of the wire. The force acting at the upper end is A(a + da). For a continuous function a(.) the differential da is equal to (da/dx)dx. The acceleration of the element, d2u/dt2, must be caused by the difference in axial forces. Now, the mass of the element is pAdx, with p denoting the density of the material. Hence, equating the mass times acceleration with the axial load Ada, and canceling the nonvanishing factor Adz, we obtain the equation of motion (3) a2u aa p-=- at2 ax A substitution of Eq. (2) yields the wave equation (4) in which the constant c is the wave speed: (5) The wave speed c is a characteristic constant of the material. The general solution of Eq. is (6) u = f(~ - Ct) + F(x + ct) , where
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