x(car) = L + vt

x(wave 1) = cT + ct
x(wave 2) = 2cT + ct
x(wave 3) = 3cT + ct
Event 2 : wave 2 meets car x_{2} =  t_{2} =
Event 1 : wave 1 meets car x_{1} =  t_{1} =

T′ = t′_{1} − t′_{2}

\frac{1}{f} = T = t_{1} − t_{2}

T′ = T \begin{matrix}√ & \overline{1 + β} \\ √ & \overline{1 − β} \end{matrix}

f′ = \frac{1}{T′} = \frac{1}{T} \begin{matrix}√ & \overline{1 − β} \\ √ & \overline{1 + β} \end{matrix}

f_{obs} = f_{source} \begin{matrix}√ & \overline{1 − β} \\ √ & \overline{1 + β} \end{matrix}

f′ ≃ f( )

f′ ≃ f(1 − \frac{v}{c} )

Event 2 : wave 2 meets car x′_{2} =
t′_{2} =
Event 1 : wave 1 meets car x′_{1} =
t′_{1} =

\frac{1}{f′} = T′ = t′_{1} − t′_{2}

\begin{aligned}v_{s} & = fλ \\ c & = fλ \end{aligned}

f = \frac{1}{T}

T = \frac{1}{f}

(Δt)_{car} = \frac{1}{γ} (Δt)_{Fred}

T_{car} = \frac{1}{γ} T_{Fred}

f_{car} = γ f_{Fred}

f_{source} = γ f_{obs}

f_{obs} = \frac{1}{γ} f_{source}