String Standing Waves

For an ideal string of length L which is fixed at both ends, the solutions to the wave equation can take the form of standing waves:

This kind of solution can be verified by direct substitution into the wave equation:

Substituting:

These two expressions are equal for all values of x and t provided

Fourier Series

If a continuous function f(t) is periodic with period T, then it may be approximated arbitrarily well by a Fourier series

where the coefficients are given by: