  How does the amplitude of a wiggle on a string change when the string is
stretched? We answer this question for both longitudinal and transverse wiggles
and for arbitrary equation of state, {\it i.e.}, for arbitrary relation between
the tension $\tau$ and the energy per unit length $\epsilon$ of the string.
This completes our derivation of the renormalization of string parameters which
results from averaging out small scale wiggles on a string. The program is
presented here in its entirety.
