Reflection of light from a moving mirror: derivation of the relativistic   Doppler formula without Lorentz transformations
  A special case of the relativistic Doppler effect, which occurs when light reflects from a moving mirror, is discussed. The classic formula for the Doppler shift is derived in a fully non-relativistic fashion using basic facts from geometry. The derivation does not involve Lorentz transformations, length contractions and time dilations, and therefore is conceptually simpler than the standard derivations in physics textbooks. This discussion can be useful for teaching introductory physics and also for understanding the foundations of special relativity. 
Malik Rakhmanov