 Solution

Using the stipulated point-mass quadrupole, take $z(t)=L\cos \omega t$ and $Q_{ij}=m(x_i{x}_j-r^2{\delta }_{ij}/3)$. The only nonzero components areSince $z^2=L^2(1+\cos 2\omega t)/2$, its third derivative is $4L^2{\omega }^3\sin 2\omega t$. Thereforeand the total outward radiated power isQuadrupole radiation oscillates at twice the motion frequency. The approximation assumes slow motion $L\omega \ll c$ and a prescribed motion relative to the rest of the source. A lone freely moving point mass cannot execute this oscillation; a physical conserved source includes the accelerating apparatus or other matter, whose quadrupole would also need specification. The displayed value is the point-mass contribution requested by the supplied formula.