General Doppler Effect Equations
About 3 min read
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.
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).
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.
(See Derivation of Doppler Effect Wavelength Equations for more information.)
In cases where the velocity of the wavefront, source, or observer is in the opposite direction, the sign in front of the velocity changes.
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)
(See Derivation of Doppler Effect Frequency Equations for more information.)
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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