All Problems Optics of Moving Sources
Problem 5.231
Taking into account that the wave phase \(\omega t-k x\) is an invariant, i.e. it retains its value on transition from one inertial 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.
5.231. Substituting the expressions for \(t^{\prime}\) and \(x^{\prime}\) (from the Lorentz transformation) into the equation \(\omega t-k x=\omega^{\prime} t^{\prime}-k^{\prime} x^{\prime},\) we obtain \[ \omega=\omega^{\prime}(1+\beta) / \sqrt{1-\beta^{2}}, k=k^{\prime}(1+\beta) / \sqrt{1-\beta^{2}} \] where \(\beta=V / c .\) Here it is taken into account that \(\omega^{\prime}=c k^{\prime}\)