In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0<|v|<1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases.