A new proof of the non-constancy of speed of light in vacuum is obtained in the process of solving the damped wave equation with a mirror boundary in uniform rectilinear motion with respect to a simple, lossy medium and with specific conditions for the other boundary, using Laplace transform. Lorentz transformation is also needed in the proof. However because of the need to assign different speeds of light in vacuum to the two moving media, the resulting solution has further to be stretched in time, in addition to the hyperbolic rotation of the Lorentz transformation. The existing methods in literature to solve the involved damped wave equation are employed and a simple solution is proposed when the boundary conditions are of a special class. In the solution Laplace transform technique is used to convert the partial differential equation into an ordinary one. The uniform rectilinear motion of the mirror boundary and the particular type of conditions treated for the other boundary, permit a solution by this technique. The presentation of the work is organized as a series of three papers with the same title but with an extension in the title as Part (I), Part (II) and Part (III). In Part (I) the moving mirror boundary condition is imposed and the new proof for the non-constancy of speed of light in vacuum is introduced. In Part (II) which is the present paper, different speeds of light in vacuum for K and K' are incorporated in the differential equation. In Part (III), an example is worked out that illustrates the ideas developed in the first two parts. In the present paper we discuss a simple general solution of the differential equation involved making use of a simplification that applies when the boundary conditions for the non-moving boundary are of a special class. In particular we assume E-x vertical bar z' = 0 and partial derivative Ex/partial derivative z'vertical bar z' = 0 have no essential singularities or branch points and furthermore the former tends to zero while the latter remains bounded as s' tends to infinity. Ex(z1, t') is the unknown function of the differential equation, s' is the complex frequency, z' is the space coordinate and t' is the time variable, the last three of which are measured in K'.