Dynamics of a string coupled to gravitational waves II - Perturbations   propagate along an infinite Nambu-Goto string
  The perturbative modes propagating along an infinite string are investigated within the framework of the gauge invariant perturbation formalism on a spacetime containing a self-gravitating straight string with a finite thickness. These modes are not included in our previous analysis. We reconstruct the perturbation formalism to discuss these modes and solve the linearized Einstein equation within the first order with respect to the string oscillation amplitude. In the thin string case, we show that the oscillations of an infinite string must involve the propagation of cosmic string traveling wave. 
Kouji Nakamura and Hideki Ishihara