Weite out a generalized wave equation to describe the propagation of longitudinal
Added by Madison R.
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Longitudinal waves are waves in which the particle displacement is parallel to the direction of wave propagation. Examples include sound waves in air. Show more…
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A sound wave in a cylinder is described by Equations 17.2 through $17.4 .$ Show that $\Delta P=\pm \rho v \omega \sqrt{s_{\max }^{2}-s^{2}}$
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4. Longitudinal waves in a rod. Consider a uniform metal bar, of cross-sectional area A and mass per unit length Δ, with a stress distribution s(x,t) and a resulting longitudinal displacement u(x,t). Newton's second law applied to a small element of the rod between x and x + Δx yields A ∂s/∂x = σ ∂^2 u / ∂t^2. Assume a linear stress-strain relationship of the form s = E ∂u/∂x. a) Show that u(x,t) obeys a wave equation ∂^2u/∂x^2 = (1/c^2) ∂^2u/∂t^2. What is c in terms of the material properties of the rod? b) Next, assume that the rod is attached to a rigid wall at x = 0 and is free at x = L. Prior to time t = 0, we pull on the free end with a force F0 so that the rod is in static equilibrium, with a uniform stress σ0 = F0/A. At t = 0, we remove the force. The longitudinal displacement in the rod obeys ∂^2u/∂x^2 = (1/c^2) ∂^2u/∂t^2, 0 < x < L, t > 0. ∂u/∂x(0,t) = 0, t > 0. ∂u/∂x(L,t) = 0, t > 0. Find u(x,t) using separation of variables.
Madhur L.
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