  We obtain systematic approximations for the modes of vibration of a string of
variable density, which is held fixed at its ends. These approximations are
obtained iteratively applying three theorems which are proved in the paper and
which hold regardless of the inhomogeneity of the string. Working on specific
examples we obtain very accurate approximations which are compared both with
the results of WKB method and with the numerical results obtained with a
collocation approach. Finally, we show that the asymptotic behaviour of the
energies of the string obtained with perturbation theory, worked to second
order in the inhomogeinities, agrees with that obtained with the WKB method and
implies a different functional dependence on the density that in two and higher
dimensions.
