# String vibration for inhomogeneous string

In case of the vibration of a homogeneous string under the boundary conditions that the string is fixed at both its ends, using fourier analysis we can show how the amplitudes of successive frequencies fall solving the wave equation.

But what modification will be required if the string is not homogeneous?

Say,half of it has double the density than the rest,or density increases linearly from one end to another end.

• The reason that Fourier analysis is such a powerful tool for analyzing systems like a homogenous vibrating string is that such systems have normal modes, which are nice simple vibrations with all parts moving at the same frequency and in a fixed phase relationship to each other. Don't think that these other systems you described have simple normal mode vibrations, though. – Samuel Weir Jul 18 '17 at 19:55
• The first device you mentionned seems to be similar to the following: a first mirror, a transparent medium of index $n_1$, a transparent medium of index $n_2$, a second mirror. – Spirine Jul 18 '17 at 20:13
• Then what will be the final solution? – user157588 Jul 21 '17 at 11:50