All Problems

Optics of Moving Sources

Problem 5.231

Taking into account that the wave phase ωt−kx\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 ω\omega and the wave number kk entering the expression for the wave phase are transformed. Examine the unidimensional case.

Reveal Answer
5.231. Substituting the expressions for t′t^{\prime} and x′x^{\prime} (from the Lorentz transformation) into the equation ωt−kx=ω′t′−k′x′,\omega t-k x=\omega^{\prime} t^{\prime}-k^{\prime} x^{\prime}, we obtain ω=ω′(1+β)/1−β2,k=k′(1+β)/1−β2 \omega=\omega^{\prime}(1+\beta) / \sqrt{1-\beta^{2}}, k=k^{\prime}(1+\beta) / \sqrt{1-\beta^{2}} where β=V/c.\beta=V / c . Here it is taken into account that ω′=ck′\omega^{\prime}=c k^{\prime}
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