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Unformatted text preview: Chapter 14 Vibration and Diffusion in OneDimensional Media In this chapter, we study the solutions, both analytical and numerical, to the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffusion of thermal energy in a body; here, we analyze the case of a one-dimensional bar. The wave equation describes vibrations and waves in continuous media, including sound waves, water waves, elastic waves, electromagnetic waves, and so on. Again, we restrict our attention here to the case of waves in a one-dimensional medium, e.g., a string, or a bar, or a column of air. The two- and three-dimensional versions of these fundamental equations will be analyzed in the later Chapters 17 and 18. As we saw in Section 12.1, the basic solution strategy is inspired by our eigenvalue- based methods for solving linear systems of ordinary differential equations. Substituting the appropriate exponential or trigonometric ansatz will effectively reduce the partial dif- ferential equation to a one-dimensional boundary value problem. The linear superposition principle implies that general solution can then be expressed as a infinite series in the resulting eigenfunction solutions. In both cases considered here, the eigenfunctions of the one-dimensional boundary value problem are trigonometric, and so the solution to the partial differential equation takes the form of a time-dependent Fourier series. Although we cannot, in general, analytically sum the Fourier series to produce a simpler formula for the solution, there are a number of useful observations that can be gleaned from it. In the case of the heat equation, the solutions decay exponentially fast to thermal equi- librium, at a rate governed by the smallest positive eigenvalue of the associated boundary value problem. The higher order Fourier modes damp out very rapidly, and so the heat equation can be used to automatically smooth and denoise signals and images. It also implies that the heat equation cannot be run backwards in time determining the initial temperature profile from a later measurement is an ill-posed problem. The response to a concentrated unit impulse leads to the fundamental solution, which can then be used to construct integral representations of the solution to the inhomogeneous heat equation. We will also explain how to exploit the symmetry properties of the differential equation in order to construct new solutions from known solutions. In the case of the wave equation, each Fourier mode vibrates with its natural fre- quency. In a stable situation, the full solution is a linear combination of these fundamen- tal vibrational modes, while an instability induces an extra linearly growing mode. For one-dimensional media, the natural frequencies are integral multiples of a single lowest fre- quency, and hence the solution is periodic, which, in particular, explains the tonal qualities 2/25/07 752 c circlecopyrt 2006 Peter J. Olver of string and wind instruments. The one-dimensional wave equation admits an alternativeof string and wind instruments....
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- Fall '09