If a vibrating string satisfying (4.4.1)–(4.4.3) is initially unperturbed, f(x) = 0, wit...

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)