Consider a vertically hanging cable of length L and weight
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Consider a vertically hanging cable of length L and weight w per unit length, with fixed upper end at x D L and free lower end at x D 0, as shown in Fig. 10.4.8. When the cable vibrates transversely, its displacement function y.x; t / satisfies the equation w g @2y @t2 D @ @x wx @y @x because the tension is T .x/ D wx. Substitute the function y.x; t / D X.x/sin !t, then apply the theorem of Section 8.6 to solve the ordinary differential equation that results. Deduce from the solution that the natural frequencies of vibration of the hanging cable are given by !n D n 2 r g L (rad=s); where f ng 1 1 are the roots of J0.x/ D 0. Historically, this problem was the first in which Bessel functions appeared.
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