Doppler Shift | Primer on Geometric Algebra | David Hestenes

Doppler Shift

Previous section: Lorentz Contraction.

Next sectionLorentz Transformations.


A distant source with velocity V = γ (1 + v/c) emits light signals with frequency
′ = ω′ / 2π = 1 / t′  that are received with frequency  f = ω / 2π = 1 / t, as shown in the figure.

 

X₀ X₂ X₁ receiver history c∆t' β receding source history c∆t' c∆t X₀ X₁ ' '



Exercise:
From the figure derive the equation

                         λ(X2 X1)  =  D − V ,

               where λ is a scale factor and

                         D   ′ /  f

               is the Doppler factor. Derive and discuss the result

\[\Large{D\,\,\, = \,\,\,\frac{{f'}}{f}\,\,\, = \,\,\,\gamma \,\left( {1\,\, \pm \,\,v/c} \right)\,\,\, = \,\,\,\sqrt {\frac{{c \pm v}}{{c \mp v}}} \,\,\, = \,\,\,\frac{1}{{\gamma \,\left( {1\,\, \mp \,\,v/c} \right)}}}\]


               (
Click here for a solution to this exercise.)




Previous sectionLorentz Contraction.

Next section: Lorentz Transformations.


© David Hestenes 2005, 2014