Stretching Wiggly Strings

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 $ϵ$ 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.