Cylindrical waves and cosmic strings of Petrov type D

A three-parameter family of (time-dependent) cylindrically symmetric solutions of the vacuum Einstein equations is obtained. They all are asymptotically flat, they admit a regular axis, and they are free from curvature singularities. For general values of the parameters the solution exhibits an angle deficit in going around the axis and it is interpreted as a cosmic string surrounded by gravitational waves. The angle deficit is eliminated for a two-parameter subfamily, leading to an everywhere-regular Petrov type-D spacetime describing the propagation of nonradiating cylindrical gravitational waves. A relationship between the linear mass density of the string and the C-energy of the spacetime is established