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

 

[Dale]

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?

Yes. You could consider a rod whose length is one wavelength in the frame where the acoustic medium is at rest and whose velocity is equal to the speed of sound in that frame. Such a rod will have a worldline such that the front of the rod follows the crest of one wave and the rear of the rod follows the crest of another. Thus its length in any given frame corresponds to one wavelength in that frame.

Note, that the rest frame of the rod is not the same as the rest frame of the medium.