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# Doppler Effect Equations for Light

by Ron Kurtus (updated 6 January 2022)

The Doppler Effect equations for light show the change in the observed wavelength or color compared with that emitted from a moving source.

Note: Typically, the observed frequency is measured in the Doppler Effect. However in same cases, the change in wavelength is measured.

The source of light or electromagnetic radiation must travel at a high speed for the Doppler effect to cause an observable shift in the wavelength. Since the speed of light is much greater than the speed of the source, an approximate equation can be used to determine the shift of the radiation.

The shift in wavelength is used in astronomy to tell when a distant galaxy or star is moving toward the Earth (blue-shift) or away (red-shift). Equations are available for determining the new frequency and wavelength, as well as the velocity of the source.

Questions you may have include:

• What are the equations for calculating frequency?
• How do you calculate the wavelength shift?
• What are the equations for velocity?

This lesson will answer those questions. Useful tool: Units Conversion

## Wavelength equations

The general Doppler Effect wavelength equation when the source of waves and the observer are both moving along the x-axis is:

λO = λS(c − vS)/(c − vO)

The equation for the change in wavelength is:

Δλ = λS(vS − vO)/(c − vO)

where

• λO is the observed wavelength
• λS is the constant wavelength from the source
• c is the constant velocity of the wavefront in the x-direction
• vS is the constant velocity of the source along the x-axis
• vO is the constant velocity of the observer along the x-axis
• vO is the projection of the source velocity in the x-direction
• Δλ is the change in wavelength (λS − λO)

Note: Although c often denotes the speed of light, it is also used for the speed or velocity of other waveforms.

## Wavelength equations

A shift in frequency of electromagnetic radiation is not readily measured. Instead, devices such as a spectroscope is used to measure a change in wavelength of the light. Knowing the velocity of the moving source of light (vs), you can use the equations c = fλ and f = c/λ to convert the frequency equations to solve for wavelength.

### Blue-shift wavelength equation

The blue-shift equation for wavelength is:

λb = λc/(c + vb)

where

• λb is the observed blue-shift wavelength
• λ is the emitted wavelength (Greek symbol lambda)

### Red-shift wavelength equation

The red-shift equation for wavelength is:

λr = λc/(c − vr)

### Example

When heated, the various chemical elements give off light in a specific series of wavelengths or spectral lines. Astronomers study a number of spectral lines, but for our purposes, we will use the Hydrogen spectral line of λ = 434 nm, which is in the violet region of the visible spectrum.

If a distant galaxy is moving away from us at approximately 50,000 km/s and we approximate the speed of light as c = 300,000 km/s, then the resulting wavelength will be:

λr = λc/(c − vr)

Substitute values into equation.

λr = (434 nm)*(300000 km/s)/[(300000 − 50000) km/s]

λr = (434 nm)*(300000 km/s)/(250000 km/s)

λr = (434 nm)*(1.2)

λr = 520.8 nm

The line has shifted from violet toward green. Other spectral lines would also have shifted toward the red end of the spectrum.

## Velocity equations

Typically, an astronomer would study the spectrum of a distant star or galaxy and measure the new observed spectral lines of the various elements on the object. Then the direction and velocity of the star or galaxy would be determined.

The equation for velocity is derived from the above wavelength equations.

### Blue-shift velocity equation

The blue-shift equation for velocity is:

vb = c(λ/λb − 1)

### Red-shift velocity equation

The red-shift equation for velocity is:

vr = c(1 − λ/λr)

### Example

If the astronomer saw that the Hydrogen 434 spectral line of a distant star had shifted to 482 nm, what would be the velocity of the star?

Since the wavelength increased, you would use the red-shift equation.

vr = c(1 − λ/λr)

Substitute the values.

vr= (300000 km/s)*(1 − 434/482)

vr = 300000*(1 − 0.9) km/s

vr = 30,000 km/s

## Summary

When the source of light or electromagnetic radiation is traveling, the Doppler effect will cause a shift in the observed frequency and wavelength and an approximate equation can be used to determine that shift. The shift in wavelength is used in astronomy to tell when a distant galaxy or star is moving toward the Earth (blue-shift) or away (red-shift). Equations are available for determining the new frequency and wavelength, as well as the velocity of the source.

Think of some clever new ways of doing things

## Resources and references

Ron Kurtus' Credentials

### Websites

Doppler Shift for Sound and Light - From book "Reflections on Relativity"

Electromagnetic Waves Resources

Physics Resources

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www.school-for-champions.com/science/
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