Derivation of longitudinal Doppler shift equation between two moving   bodies in a reference frame at rest using the particle property of photons
  The equation of the Doppler shift of two bodies in inertial motion in a reference frame at rest (i.e., stationary reference frame) is derived. In this derivation, the wave-particle duality of photons in the theory of special relativity is considered. The dilation of time of a moving clock, which is derived from the Lorentz transformation that depends on the velocity of the moving clock in the reference frame at rest, and wavefront counting by geometrical drawing are used to describe the longitudinal Doppler shift. We show that the equation of the Doppler shift depends on the velocities of the two bodies in the reference frame at rest. 
Masanori Sato