length contraction and wave length transformation

bernhard.rothenstein

consider a rod of proper length Lo located along the overlapped OX(O'X') axes of the I and I' inertial reference frames in the standard arrangement I' moving relative to I' with speed V. The rod moves with speed U relative to I and with speed U' relative to I'. The measured length of the rod is
L=g(U)Lo (1)
L'=g(U')Lo (2)
in I and in I' respectively. Eliminating Lo between (1) and (2) we obtain
L=L'g(U)/g(U'). (3)
Expressing the right side of (3) as a function of U' only, via the addition law of relativistic velocities we obtain
L=L'g(V)/1+u'V/cc (4)
an equation that relates two non-proper lengths.
Is it correct to consider that the derivation presented above represents a derivtion for the relationship betwee the wavelengths of the same acoustic (mechanical) wave that propagates with speeds U and U' relative to I and I' respectively?
Did you find that derivation somewhere in the literature of the subject?
Thanks.
sine ira et studio