 Solution

Use retarded time $t_r$ and reserve $r$ for the large observation distance. The source velocity is nonrelativistic, $|\mathbf{u}|/c\ll 1$. There are two errors in the original PDF that must be corrected before the radiation zone calculation: the vector-potential prefactor contains an extraneous $1/r$, and the displayed magnetic radiation field has the wrong sign for the stated outward unit vector $\mathbf{n}$. The extra distance factor is dimensionally wrong; the magnetic sign would make the Poynting vector point inward. The following derivation uses dimensionally consistent retarded potentials and outgoing radiation.