Partial Differential Equation

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4.2 The wave equation (4.5) is an appropriate model for the longitudinal vibrations of a spring. In this application u(t, x) represents displacement parallel to the spring. Suppose that spring has length 2 and is free at the ends. This corresponds to the Neumann boundary conditions Bu an 8:\\ffor g E C 201%) and h E C 1(R), the wave equation (4.5) admits a unique solution 1 1 x+ct u(t, x) = [g(x + ct) + g(x 00] + if h(T) d7. (4.8)