The field equations of the general theory of relativity are solved in the linear approximation for all cases of spherical waves with quadrupole symmetry. Energy is radiated outward by all these waves as determined by the canonical expression for the energy flux. A qualitative check of the validity of this method of calculation is made by the application of the same approximation to cylindrical gravitational radiation, for which an exact solution is known. In this case the exact and the linearized calculations lead to corresponding results.