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# General Doppler Effect Equations

by Ron Kurtus (revised 10 April 2015)

Doppler Effect equations show the changes in observed wavelength or frequency of a waveform, when the source and/or observer are moving with respect to the wave medium.

The general equations take into account the possible velocities of the source and observer at angles to the line between the objects or x-axis.

Note: It is important that you are aware of direction conventions and motion assumptions in examining these Doppler Effect equations.

Questions you may have include:

• What are the velocities with respect to x-axis?
• What are the Doppler Effect wavelength equations?
• What are the Doppler Effect frequency equations?

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

## Velocities with respect to x-axis

The velocity of the wave source, as well as the velocity of the observer may not be along the line between the object or x-axis.

Suppose the velocity of the source vS is at some angle a with respect to the x-axis. Then its velocity in the x-direction is vScos(a).

Likewise, if the velocity of the observer vO is at some angle b with respect to the x-axis, its velocity in the x-direction is vOcos(b). Source and observer moving at angles with respect to x-axis

When angle a = 0°, cos(a) = 1. Likewise, when b = 0°, cos(b) = 1

## Wavelength equations

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

λO = λS[c − vScos(a)]/[c − vOcos(b)]

The equation for the change in wavelength is:

Δλ = λS[vScos(a) − vOcos(b)]/[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
• a is the angle between vS and the x-axis
• vScos(a) is the projection of the source velocity in the x-direction
• vO is the constant velocity of the observer
• b is the angle between vO and the x-axis
• vOcos(b) 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.

In cases where the velocity of the wavefront, source, or observer is in the opposite direction, the sign in front of the velocity changes.

Note: Some textbooks use an equation where the source and observer are moving toward each other. Although they are not following a scientific convention, you need to be aware of what direction convention they are using.

## Frequency equations

Since frequency equals velocity divided by wavelength (f = c/λ), the general Doppler Effect frequency equations are:

fO = fS[c −vOcos(b)]/[c − vScos(a)]

Δf = fS[vOcos(b) − vScos(a)]/[c − vScos(a)]

where

• fO is the observed wave frequency
• fS is the wave frequency of the source
• Δf is the change infrequency (fS − fO)

## Summary

When the velocities of the source and observer are at angles to the x-axis, the Doppler Effect equations for wavelength are:

λO = λS[c − vScos(a)]/[c − vOcos(b)]

Δλ = λS[vScos(a) − vOcos(b)]/[c − vO]

The equations for frequency are:

fO = fS[c −vOcos(b)]/[c − vScos(a)]

Δf = fS[vOcos(b) − vScos(a)]/[c − vScos(a)]

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