Taking into account that the wave phase `omegat - kx` is an invariant, i.e. it retains its value on transition from one interial frame to another, determine how the frequency `omega` and the wave number `k` entering the expression for the wave phase are transformed. Examine the unidimensional case.

Explanation

From the invatiance of phase under Lorentz transformations we get
`omega-kx = omega' t' - k' x'`
Here `omega = ck`. The primed corrdinates refer to the frame `S'` which is moving to the right with velocity `v`:-
Then `x' = gamma (x-vt)`
`t' = gamma(t-(vx)/(c^(2)))`
where `gamma = (sqrt(1-v^(2)//c^(2)))^(-1)`
Substituting and equating the coefficients of `t& x`
`omega = gamma omega' + gammak' v = omega' (1+v//c)/(sqrt(1-v^(2)//c^(2)))`
`k=gamma(omega'v)/(c^(2)) +gamma k' = k' (1+v//c)/(sqrt(1-v^(2)//c^(2)))`