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Derivation of Doppler Effect Velocity Equations

by Ron Kurtus (revised 27 March 2016)

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:

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

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

(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

(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 = fScfOc

fOvS = − c(fSfO)

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

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 moving toward moving object

Waves reflected off 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

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

−vRS+ λO) = c(λS λO)

Divide both sides be −(λS+ λO), resulting in:

vR = −c(λS λO)/S+ λO)

Since Δλ = (λS − λO), the velocity equation is:

vR = −cΔλ/S+ λO)

If the object is moving in the opposite direction, vR becomes negative, and the equation is:

vR = cΔλ/S+ 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(fSfO)

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/S+ λO)

vR = cΔf/(fS+ fO)


Do your work methodically


Resources and references

Ron Kurtus' Credentials

Websites

Wave Motion Resources

Books

Top-rated books on the Doppler Effect


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Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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