Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Special relativity in the plane acoustic and electromagnetic wave

  1. Jan 8, 2009 #1
    Let us envision three frames of reference I, I’ and I” in standard configuration with I’ moving with velocity V relative to I and I” moving with velocity u relative to I and with velocity u’ relative to I’. Let O, O’ and O” be the spatial origins of the frames respectively; O, O’ and O” coinciding at the origin of time in the three frames (t=t’=0).
    Accept the following notations g(V)=1/sqr(1-VV/cc); g(u)=1/sqr(1-uu/cc) and g(u’)=1/sqr(1-u’u’/cc).
    Let I” be the rest frame of a rod located along the permanently overlapped axes x,x’x” where its length is L0. Observers from I detect simultaneously the space coordinates of the ends of the rod, subtract them obtaining that its length is L related to L0 by
    L=L0/g(u). (1)
    Following the same procedure observers from I’ obtain for the length of the moving rod
    L’=L0/g(u’). (2)
    Combining (1) and (2) we obtain that L and L’ are related by
    L=L’g(u’)/g(u). (3)
    Expressing the right side of (3) as a function of u’ via the addition law of parallel speeds
    u=(u’+V)/(1+Vu’/cc) (4)
    the result is
    L=L’/[g(V)(1+Vu’/cc)]. (5)
    I underline that (4) could be derived without invoking the Lorentz transformations.
    Introducing the physical quantities k=1/L and k’=1/L’ we obtain that they transform as
    k=g(V)k’(1+Vu’/cc). (6)
    If the rod is generated by a particle moving with speed u for a proper time interval t(0) i.e.
    L=ut(0) (7)
    in I and
    L’=u’t’(0) (8)
    then (6) becomes
    k=g(V)[k’+V/t’(0)cc]=g(V)[k’+Vf’/cc] (9)
    where f’=1/t’(0) has the physical meaning of frequency. If the rods are generated by light signals then L=ct(0) and L’=ct’(0) (6) becoming
    k=k’sqr[(1+V/c)/(1-V/c)] . (10)
    We obtain the transformation equations for the frequencies f and f’ starting with f=ku and f’=k’u’. Combining them and taking into account (4) the result is
    f=g(V)f’(1+V/u’)=g(V)(f’+Vk’) (11)
    which becomes in the case of rods generated by light signals
    f=f’sqr[(1+V/c)/(1-V/c)]. (12)
    Would you accept all that as a introduction in special relativity in the acoustic and in the electromagnetic plane waves?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted