The string of variable density: Further results

We analyze the problem of calculating the solutions and the spectrum of a string with arbitrary density and fixed ends. We build a perturbative scheme which uses a basis of WKB-type functions and obtain explicit expressions for the eigenvalues and eigenfunctions of the string. Using this approach we show that it is possible to derive the asymptotic (high energy) behavior of the string, obtaining explicit expressions for the first three coefficients (the first two can also be obtained with the WKB method). Finally, using an iterative approach we also obtain analytical expressions for the low energy behavior of the eigenvalues and eigenfunctions of a string with rapidly oscillating density, recovering (in a simpler way) results in the literature.

► We devise a perturbative approach to finding the modes of a string with arbitrary density. ► We obtain explicitly the first three coefficients of the asymptotic high energy expansion. ► We apply our findings to a series of examples, obtaining both analytical and numerical results.