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Derivation of Doppler Effect Velocity Equations
by Ron Kurtus
A common use for the Doppler Effect is to determine the velocity of the source of waves or the velocity of the observer.
The derivation of the Doppler Effect velocity equations starts with the general waveform and frequency equations. By setting the observer velocity to zero, the source velocity can then be found. Likewise, setting the source velocity to zero results in the observer velocity.
In the equations, it is assumed that the motion is constant and in the x-direction.
(See Conventions for Doppler Effect Equations for more information.)
Questions you may have include:
- What are the equations for a moving source and stationary observer?
- What are the equations for a moving observer and stationary source?
- What are the equations for reflecting off moving object?
This lesson will answer those questions. Useful tool: Units Conversion
Useful tools: Units Conversion.
Moving source and stationary observer
When the source is moving in the x-direction but the observer is stationary, you can find the velocity of the source by taking the general wavelength and frequency equations, setting vO = 0, and then solving for vS.
Source is moving toward stationary observer
Velocity with respect to wavelength
To determine the velocity with respect to wavelength, you can start with the general wavelength equation:
λO(c − vO) = λS(c − vS)
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 in the x-direction
- vO is the constant velocity of the observer in the x-direction
(See Derivation of Doppler Effect Wavelength Equations for more information.)
Set vO = 0 and solve for vS:
λOc = λS(c − vS)
λOc = λSc − λSvS
λvS = λSc − λOc
vS = c(λS − λO)/λS
Since the change is wavelength is Δλ = (λS − λO), the velocity of the source is:
vS = cΔλ/λS
Velocity with respect to frequency
To determine the velocity with respect to frequency, you can start with the general frequency equation:
fO = fS(c − vO)/(c − vS)
where
- fO is the observed frequency
- fS is the constant wave frequency from the source
(See Derivation of Doppler Effect Frequency Equations for more information.)
Set vO = 0 and solve for vS:
fO = fSc/(c − vS)
fO(c − vS) = fSc
fOc − fOvS = fSc
− fOvS = fSc − fOc
fOvS = − c(fS − fO)
Since the change is frequency is Δf = (fS − fO), the velocity of the source is:
vS = − cΔf/fO
Moving observer and stationary source
Suppose the source is stationary and the observer is moving in the x-direction from the source.
Observer moving away from oncoming waves
Velocity with respect to wavelength
Start with the general wavelength equation:
λO(c − vO) = λS(c − vS)
Set vS = 0, and solve for vO:
λO(c − vO) = λSc
λOc − λOvO = λSc
−λOvO = λSc − λOc
Multiply both sides of the equation by −1, factor out c, and divide by λO
vO = −c(λS − λO)/λO
Thus:
vO = −cΔλ/λO
Velocity with respect to frequency
Start with the general frequency equation:,
fO = fS(c − vO)/(c − vS)
Set vS = 0, and solve for vO:
fO = fS(c − vO)/c
fOc = fSc − fSvO
Add fSvO and subtract fOc from both sides of the equation:
fSvO = fSc − fOc
Factor out c and divide by fS:
vO = c(fS − fO)/fS
Thus:
vO = cΔf/fS
Reflection off moving object
One method to determine the velocity of an object is to reflect a wave off the object and measure the Doppler shift caused by the motion. In this case, both the velocity of the source and observer are zero: vS = 0 and vO = 0. The observer is usually nearby the source.
Waves moving toward moving object
Waves reflected off moving object
Waves "observed" by moving object
Let vR be the velocity of the object, moving in the x-direction. The wavelength and frequency "observed" by the object are:
λR = λSc/(c − vR)
fR = fS(c − vR)/c
where
- λR is the observed wavelength of the moving object
- λS is the original source wavelength
- vR is the constant object velocity in the x-direction
- fR is the observed frequency at the moving object
- fS is the original source frequency
Waves reflected to stationary observer
The object reflects the "observed" waves as if the object was a moving source.
Note: Although the motion is still in the positive direction, the wave is now moving in the negative direction. Thus, the sign of c must change.
Wavelength equation
The wavelength equation for a moving source and stationary observer is:
λO = λS(c − vS)/c
However, λR represents the reflected source wavelength and vR is the velocity of the reflecting object, acting as a source. Replace λS with λR and vS with vR in the equation. Also, change the sign of c since the wave is moving in the opposite direction.
Thus, the reflected wavelength equation is:
λO = λR(−c − vR)/(−c)
λO = λR(c + vR)/c
where λO is the wavelength measured by the stationary observer.
Using the equation λR = λSc/(c − vR), substitute for λR and then solve for vR:
λO = [λSc/(c − vR)]*[(c + vR)/c]
λO = λSc(c + vR)/(c − vR)c
λO = λS(c + vR)/(c − vR)
λO(c − vR) = λS(c + vR)
λOc − λOvR = λSc + λSvR
Subtract λOc and λSvR from both sides of the equaiton:
−λSvR− λOvR = λSc − λOc
−vR(λS+ λO) = c(λS − λO)
Divide both sides be −(λS+ λO), resulting in:
vR = −c(λS − λO)/(λS+ λO)
Since Δλ = (λS − λO), the velocity equation is:
vR = −cΔλ/(λS+ λO)
If the object is moving in the opposite direction, vR becomes negative, and the equation is:
vR = cΔλ/(λS+ λO)
Frequency equation
The frequency equation for a moving source and stationary observer is:
fO = fSc/(c − vS)
However fR represents the reflected source frequency and vR is the velocity of the reflecting object, acting as a source. Also, the sign of c changes.
The reflected frequency equation is:
fO = fR(−c)/(−c − vR)
fO = fRc/(c + vR)
Using the equation fR = fS(c − vR)/c, substitute for fR and then solve for vR:
fO = [fS(c − vR)/c]*[c/(c + vR)]
fO = fSc(c − vR)/(c + vR)c
fO = fS(c − vR)/(c + vR)
fO(c + vR) = fS(c − vR)
fOc + fOvR = fSc − fSvR
fSvR + fOvR = fSc − fOc
vR(fS+ fO) = c(fS− fO)
Thus:
vR = cΔf/(fS+ fO)
Summary
The derivation of the Doppler Effect velocity equations starts with the general waveform and frequency equations. By setting the observer velocity to zero, the source velocity can then be found. Likewise, setting the source velocity to zero results in the observer velocity.
Combining the two equations results in the equations for the velocity reflected off a moving object.
Moving source and stationary observer
vS = cΔλ/λS
vS = − cΔf/fO
Moving observer and stationary source
vO = −cΔλ/λO
vO = cΔf/fS
Reflection off moving object
vR = −cΔλ/(λS+ λO)
vR = cΔf/(fS+ fO)
Do your work methodically
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Derivation of Doppler Effect Velocity Equations