  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.
