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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 Equationsfor 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 **v _{O}** = 0, and then solving for

**v**.

_{S}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:

λ_{O}(c − v_{O}) = λ_{S}(c − v_{S})

where

**λ**is the observed wavelength_{O}**λ**is the constant wavelength from the source_{S}**c**is the constant velocity of the wavefront in the**x**-direction**v**is the constant velocity of the source in the_{S}**x**-direction**v**is the constant velocity of the observer in the_{O}**x**-direction

(See

Derivation of Doppler Effect Wavelength Equationsfor more information.)

Set **v _{O}** = 0 and solve for

**v**:

_{S}

λ_{O}c = λ_{S}(c − v_{S})

λ_{O}c = λ_{S}c − λ_{S}v_{S}

λv_{S}= λ_{S}c − λ_{O}c

v_{S}= c(λ_{S}− λ_{O})/λ_{S}

Since the change is wavelength is **Δλ = (λ _{S} − λ_{O})**, the velocity of the source is:

v_{S}= cΔλ/λ_{S}

### Velocity with respect to frequency

To determine the velocity with respect to frequency, you can start with the general frequency equation:

f_{O}= f_{S}(c − v_{O})/(c − v_{S})

where

**f**is the observed frequency_{O}**f**is the constant wave frequency from the source_{S}

(See

Derivation of Doppler Effect Frequency Equationsfor more information.)

Set **v _{O}** = 0 and solve for

**v**:

_{S}

f_{O}= f_{S}c/(c − v_{S})

f_{O}(c − v_{S}) = f_{S}c

f_{O}c − f_{O}v_{S}= f_{S}c

− f_{O}v_{S}= f_{S}c−f_{O}c

f_{O}v_{S}= − c(f_{S}−f_{O})

Since the change is frequency is **Δf = (f _{S} − f_{O})**, the velocity of the source is:

v_{S}= − cΔf/f_{O}

## 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:

λ_{O}(c − v_{O}) = λ_{S}(c − v_{S})

Set **v _{S}** = 0, and solve for

**v**:

_{O}

λ_{O}(c − v_{O}) = λ_{S}c

λ_{O}c − λ_{O}v_{O}= λ_{S}c

−λ_{O}v_{O}= λ_{S}c − λ_{O}c

Multiply both sides of the equation by **−**1, factor out **c**, and divide by **λ _{O}**

v_{O}= −c(λ_{S}− λ_{O})/λ_{O}

Thus:

v_{O}= −cΔλ/λ_{O}

### Velocity with respect to frequency

Start with the general frequency equation:,

f_{O}= f_{S}(c − v_{O})/(c − v_{S})

Set **v _{S}** = 0, and solve for

**v**:

_{O}

f_{O}= f_{S}(c − v_{O})/c

f_{O}c = f_{S}c − f_{S}v_{O}

Add ** f _{S}v_{O}** and subtract

**f**from both sides of the equation:

_{O}c

f_{S}v_{O}= f_{S}c − f_{O}c

Factor out **c** and divide by **f _{S}**:

v_{O}= c(f_{S}− f_{O})/f_{S}

Thus:

v_{O}= cΔf/f_{S}

## 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: **v _{S}** = 0 and

**v**= 0. The observer is usually nearby the source.

_{O}Waves moving toward moving object

Waves reflected off moving object

### Waves "observed" by moving object

Let **v _{R}** be the velocity of the object, moving in the

**x**-direction. The wavelength and frequency "observed" by the object are:

λ_{R}= λ_{S}c/(c − v_{R})

f_{R}= f_{S}(c − v_{R})/c

where

**λ**is the observed wavelength of the moving object_{R}**λ**is the original source wavelength_{S}**v**is the constant object velocity in the_{R}**x**-direction**f**is the observed frequency at the moving object_{R}**f**is the original source frequency_{S}

### 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 ofcmust change.

#### Wavelength equation

The wavelength equation for a moving source and stationary observer is:

λ_{O}= λ_{S}(c − v_{S})/c

However, **λ _{R}** represents the reflected source wavelength and

**v**is the velocity of the reflecting object, acting as a source. Replace

_{R}**λ**with

_{S}**λ**and

_{R}**v**with

_{S}**v**in the equation. Also, change the sign of

_{R}**c**since the wave is moving in the opposite direction.

Thus, the reflected wavelength equation is:

λ_{O}= λ_{R}(−c − v_{R})/(−c)

λ_{O}= λ_{R}(c + v_{R})/c

where **λ _{O}** is the wavelength measured by the stationary observer.

Using the equation **λ _{R} = λ_{S}c/(c − v_{R})**, substitute for

**λ**and then solve for

_{R}**v**:

_{R}

λ_{O}= [λ_{S}c/(c − v_{R})]*[(c + v_{R})/c]

λ_{O}= λ_{S}c(c + v_{R})/(c − v_{R})c

λ_{O}= λ_{S}(c + v_{R})/(c − v_{R})

λ_{O}(c − v_{R}) = λ_{S}(c + v_{R})

λ_{O}c − λ_{O}v_{R}= λ_{S}c + λ_{S}v_{R}

Subtract **λ _{O}c** and

**λ**from both sides of the equaiton:

_{S}v_{R}

−λ_{S}v_{R}− λ_{O}v_{R}= λ_{S}c_{ }−λ_{O}c

−v_{R}(λ_{S}+ λ_{O}) = c(λ_{S}_{ }−λ_{O})

Divide both sides be **−(λ _{S}+ λ_{O})**, resulting in:

v_{R}= −c(λ_{S}_{ }−λ/_{O})(λ_{S}+ λ_{O})

Since **Δλ = (λ _{S} − λ_{O})**, the velocity equation is:

v_{R}= −cΔλ/(λ_{S}+ λ_{O})

If the object is moving in the opposite direction, **v _{R}** becomes negative, and the equation is:

v_{R}= cΔλ/(λ_{S}+ λ_{O})

#### Frequency equation

The frequency equation for a moving source and stationary observer is:

f_{O}= f_{S}c/(c − v_{S})

However **f _{R}** represents the reflected source frequency and

**v**is the velocity of the reflecting object, acting as a source. Also, the sign of

_{R}**c**changes.

The reflected frequency equation is:

f_{O}= f_{R}(−c)/(−c − v_{R})

f_{O}= f_{R}c/(c + v_{R})

Using the equation **f _{R} = f_{S}(c − v_{R})/c**, substitute for

**f**and then solve for

_{R}**v**:

_{R}

f_{O}= [f_{S}(c − v_{R})/c]*[c/(c + v_{R})]

f_{O}= f_{S}c(c − v_{R})/(c + v_{R})c

f_{O}= f_{S}(c − v_{R})/(c + v_{R})

f_{O}(c + v_{R}) = f_{S}(c − v_{R})

f_{O}c + f_{O}v_{R}= f_{S}c − f_{S}v_{R}

f_{S}v_{R}+ f_{O}v_{R}= f_{S}c −f_{O}c

v_{R}(f_{S}+ f_{O})_{}= c(f_{S}−f_{O})

Thus:

v_{R}_{}= cΔf/(f_{S}+ f_{O})

## 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

v_{S}= cΔλ/λ_{S}

v_{S}= − cΔf/f_{O}

### Moving observer and stationary source

v_{O}= −cΔλ/λ_{O}

v=_{O}cΔf/f_{S}

### Reflection off moving object

v_{R}= −cΔλ/(λ_{S}+ λ_{O})

v_{R}_{}= cΔf/(f_{S}+ f_{O})

Do your work methodically

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## Derivation of Doppler Effect Velocity Equations