Doppler Effect for Light

Learning objectives

By the end of this section, you will be able to:

• Explain the origin of the shift in frequency and wavelength of the observed wavelength when observer and source moved toward or away from each other
• Derive an expression for the relativistic Doppler shift
• Apply the Doppler shift equations to real-world examples

The Relativistic Doppler Effect

Suppose an observer in S sees light from a source in $S\prime$ moving away at velocity v (CNX_UPhysics_38_07_DopplerFr). The wavelength of the light could be measured within $S\prime$—for example, by using a mirror to set up standing waves and measuring the distance between nodes. These distances are proper lengths with $S\prime$ as their rest frame, and change by a factor $\sqrt{1-v^2\text{/}{c}^2}$ when measured in the observer’s frame S, where the ruler measuring the wavelength in $S\prime$ is seen as moving.

If the source were stationary in S, the observer would see a length $c\text{Δ}t$ of the wave pattern in time $\text{Δ}t.$ But because of the motion of $S\prime$ relative to S, considered solely within S, the observer sees the wave pattern, and therefore the wavelength, stretched out by a factor of

as illustrated in (b) of CNX_UPhysics_38_07_DopplerFr. The overall increase from both effects gives

where ${\lambda }_{\text{src}}$ is the wavelength of the light seen by the source in $S\prime$ and ${\lambda }_{\text{obs}}$ is the wavelength that the observer detects within S.

Red Shifts and Blue Shifts

The observed wavelength ${\lambda }_{\text{obs}}$ of electromagnetic radiation is longer (called a “red shift”) than that emitted by the source when the source moves away from the observer. Similarly, the wavelength is shorter (called a “blue shift”) when the source moves toward the observer. The amount of change is determined by

where ${\lambda }_{\text{s}}$ is the wavelength in the frame of reference of the source, and v is the relative velocity of the two frames S and $S\prime .$ The velocity v is positive for motion away from an observer and negative for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written as

Notice that the signs are different from those of the wavelength equation.

Calculating a Doppler Shift

Strategy

Solution

1. Identify the knowns: $u=0.825c;{\lambda }_s=0.525\text{m.}$
2. Identify the unknown: ${\lambda }_{\text{obs}}.$
3. Express the answer as an equation:
   ${\lambda }_{\text{obs}}={\lambda }_s\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}.$
4. Do the calculation: $\begin{array}{}\\ \\ \\ \\ \\ {\lambda }_{\text{obs}}& ={\lambda }_s\sqrt[]{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}\\ & =(0.525\text{m)}\sqrt{\frac{1+\frac{0.825c}{c}}{1-\frac{0.825c}{c}}}\\ & =1.70\text{m.}\end{array}$

Significance

The relativistic Doppler effect has applications ranging from Doppler radar storm monitoring to providing information on the motion and distance of stars. We describe some of these applications in the exercises.

Summary

• An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation: ${\lambda }_{\text{obs}}={\lambda }_s\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}.$ where ${\lambda }_{\text{obs}}$ is the observed wavelength, ${\lambda }_s$ is the source wavelength, and v is the relative velocity of the source to the observer.

Conceptual Questions

Problems