  We consider a vortex line in the B phase of superfluid $^3$He under uniformly
precessing magnetization. The magnetization exerts torque on the vortex,
causing its order parameter to oscillate. These oscillations generate spin
waves, which is analogous to an oscillating charge generating electromagnetic
radiation. The spin waves carry energy, causing dissipation in the system.
Solving the equations of spin dynamics, we calculate the energy dissipation
caused by spin wave radiation for arbitrary tipping angles of the magnetization
and directions of the magnetic field, and for both vortex types of $^3$He-B.
For the double-core vortex we also consider the anisotropy of the radiation and
the dependence of the dissipation on twisting of the half cores. The radiated
energy is compared with experiments in the mid-temperature range $T \sim 0.5
T_c$. The dependence of the calculated dissipation on several parameters is in
good agreement with the experiments. Combined with numerically calculated
vortex structure, the radiation theory produces the order of magnitude of the
experimental dissipation. The agreement with the experiments indicates that
spin wave radiation is the dominant dissipation mechanism for vortices in
superfluid $^3$He-B in the mid-temperature range.
