Abstract Using general relativity we study gravitational waves from isolated, axially-symmetric sources. We start with a metric due to Bondi, and use the double-series approximation method. In the linear approximation we obtain a general solution for the 2s axially-symmetric multipole field. Passing to the non-linear approximations, we demonstrate that the source loses mass on account of the quadrupole-quadrupole interaction, and that it recoils because of the quadrupole-octupole interaction. The mass and momentum changes of the source agree with the results obtained by means of the pseudo tensor of energy and momentum. We explain why we believe that these waves have tails, and discuss this in relation to a paper by Bondi, Van der Burg & Metzner.