If a vibrating string satisfying (4.4.1)–(4.4.3) is initially unperturbed, f(x) = 0, with the initial velocity given, show that
where G(x) is the odd-periodic extension of g(x). [Hints:
See the comment after Exercise 4.4.7.
Reference Exercise 4.4.7:
If a vibrating string satisfying (4.4.1)–(4.4.3) is initially at rest, g(x) = 0, show that
where F(x) is the odd-periodic extension of f(x). [Hints:
Comment: This result shows that the practical difficulty of summing an infinite number of terms of a Fourier series may be avoided for the one-dimensional wave equation.
Reference (4.4.1)–(4.4.3)
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