Making use of the relationship between the corresponding field configurations, we derive the metric around a straight global cosmic string with traveling waves in terms of the static metric (without traveling waves) in the weak-field limit. We discuss under which conditions the effect of the traveling waves may overcome the repulsive gravitational potential of the static straight global string. We also extend the calculation beyond the weak-field limit combining our result with the recent observation made by Garfinkle and Vachaspati that the exact solution must be of the generalized Kerr-Schild type.