Previous work which negates Special Relativity Theory and which was carried out only for a monochromatic plane wave is extended to an impulsive plane wave. The media considered consist in a linear dispersive medium with absorption (the Lorentz medium) that fills the space x > -infinity (medium (I)) and a simple but lossy medium (II) initially filling the half space x > 0. Medium (I) has frequency dependent conductance and dielectric permittivity in distinction from the original work for the monochromatic wave. Inertial frames K and K' are attached to medium (I) and (II), and the above description of these two media are true when they are observed from K and K' respectively. K and K' have coincident origins in space and time initially, and at t = t' = 0, K' starts a uniform rectilinear motion with respect to K. An impulsive in time plane wave is assumed to impinge on the plane x = x(0), at the time t(0), when -x(0) and t(0) are both measured from K. Complex phase invariance principle is applied to the resulting elementary plane waves whose superposition yields the impulse response of the system. Transformation relations for the wave numbers and frequencies for incident and reflected waves for the interface of medium (I) and (II), are obtained using a 'modified Lorentz transformation' which incorporates different speeds of light in vacuum c and c' for K and K', as was done for the monochromatic plane wave case. Next using the dispersion relations for the incident and reflected waves an algebraic relation is derived between constitutive parameters of the Lorentz medium, the frequency, c and the relative speed of K' with respect to K. This result implies that the speed of light in vacuum c is dependent on the reference frame, a violation of the basic assumption of Special Relativity Theory.