We derive a novel time-domain asymptotic solution for a transient whispering-gallery (WG) mode radiation from the aperture plane of a cylindrical concave conducting boundary. The WG mode is excited by a Gaussian-type modulated pulse source. We show that the WG mode radiation field propagates with the phase velocity coincident with the speed of light and the newly derived group velocity. The validity of the time-domain asymptotic solution is confirmed by comparing with the reference solution calculated numerically. We show that the asymptotic solution proposed here is easy to understand the physical phenomena of the transient WG mode radiation field.