# Length contraction and wave length transformation

• bernhard.rothenstein
In summary: And also note that the rod is not the medium, so it does not have the same rest frame as the medium. In summary, the conversation discusses the relationship between the wavelengths of an acoustic wave propagating with speeds U and U' relative to different inertial reference frames, and a rod with a proper length that moves at different speeds in the same reference frames. The derived equation (4) shows the relationship between the measured lengths of the rod in the two frames. The conversation also mentions the possibility of considering the rod as a representation of the acoustic wave and finding the derivation in literature.
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

bernhard.rothenstein said:
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.

## 1. What is length contraction?

Length contraction is a phenomenon that occurs when an object is moving at high speeds relative to an observer. The object appears to be shorter in the direction of its motion compared to its length when at rest.

## 2. How does length contraction occur?

Length contraction occurs due to the effects of special relativity, which states that the laws of physics are the same for all observers in uniform motion. As an object's speed increases, its perception of time and space changes, resulting in the observed contraction of length.

## 3. What is the formula for calculating length contraction?

The formula for calculating length contraction is L = L₀/γ, where L is the contracted length, L₀ is the rest length, and γ is the Lorentz factor, which is equal to 1/√(1-v²/c²), with v being the relative velocity between the object and the observer, and c being the speed of light.

## 4. How does length contraction relate to wavelength transformation?

Length contraction and wavelength transformation are closely related as they both result from the effects of special relativity. As an object's length appears to contract, the wavelength of any waves emitted by the object also appears to shorten. This can be observed in the phenomenon of relativistic Doppler shift.

## 5. What are some real-world examples of length contraction and wavelength transformation?

Some real-world examples of length contraction and wavelength transformation include the contraction of a spacecraft's length as it travels close to the speed of light, the observed change in the wavelengths of light emitted from stars due to their high speeds relative to Earth, and the contraction of particle accelerators to accommodate for the increase in mass of particles as they approach the speed of light.

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