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# Doppler Effect Frequency Equations

by Ron Kurtus

The Doppler Effect causes the observed frequency of a waveform to change according to the velocity of the source and/or observer. The * Doppler Effect frequency equations* can derived by starting with the general wavelength equation.

After the general frequency equation is determined, you can find the frequency equations for a moving source and stationary observer and moving observer with a stationary source.

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 is the general frequency equation?
- What are the equations for a moving source and stationary observer?
- What are the equations for a moving observer and stationary source?

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

## General frequency equation

In order to establish the general Doppler Effect frequency equation—where both the source and observer are moving—you start with the previously derived general wavelength equation and put it in terms of frequency. Start with:

λ_{O}= λ_{S}(c − v_{S})/(c − v_{O})

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.)

Substitute **λ _{O} = c/f_{O}** and

**λ**in the equation to get:

_{S}= c/f_{S}

c/f]_{O}= (c/f_{S})[(c − v_{S})/(c − v_{O})

c/f_{O}= c(c − v_{S})/f_{S}(c − v_{O})

where

**f**is the constant observed frequency_{O}**f**is the constant source frequency_{S}

Divide both sides by **c** and reciprocate the equation. This results in the general frequency equation:

f_{O}= f_{S}(c − v_{O})/(c − v_{S})

The equation is often written in the convenient format:

f_{O}/(c − v_{O}) = f_{S}/(c − v_{S})

### Change in frequency

The change in frequency or Doppler frequency shift is:

Δf = f_{S}− f_{O}

Substitute in **f _{O} = f_{S}(c − v_{O})/(c − v_{S})**:

Δf = f_{S}− f_{S}(c − v_{O})/(c − v_{S})

Combine terms:

Δf = [f_{S}(c − v_{S}) − f_{S}(c − v_{O})]/(c − v_{S})

Δf = (f_{S}c − f_{S}v_{S}− f_{S}c + f_{S}v_{O})/(c − v_{S})

Thus:

Δf = f_{S}(v_{O}−v_{S})/(c − v_{S})

## Moving source and stationary observer

When the source is moving in the **x**-direction but the observer is stationary, you can take the general frequency equation, set **v _{O}** = 0, and solve for

**f**.

_{O}Source is moving toward stationary observer

The general frequency equation is:

f_{O}= f_{S}(c − v_{O})/(c − v_{S})

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

**f**:

_{O}

f_{O}= f_{S}c/(c − v_{S})

The equation is often seen in the form:

f_{O}= f_{S}/(1 − v_{S}/c)

### Change in frequency

The change in frequency or Doppler frequency shift is:

Δf = f_{S}− f_{O}

Notethat when the source is moving toward the observer,fand_{S}> f_{O}Δfis negative.

Substitute for **f _{O}**:

Δf = f_{S}−f_{S}c/(c − v_{S})

Factor out **f _{S}**:

Δf = f_{S}[1 −c/(c − v_{S})]

Simplify:

Δf = f_{S}[(c − v_{S})/(c − v_{S}) −c/(c − v_{S})]

Δf = f_{S}(c − v_{S}−c)/(c − v_{S})

Δf = f_{S}(− v_{S})/(c − v_{S})

Thus:

Δf = −f_{S}v_{S}/(c − v_{S})

## Moving observer and stationary source

When the observer is moving in the **x**-direction but the source is stationary, you can take the general frequency equation, set **v _{S}** = 0, and solve for

**f**.

_{O}Observer moving away from oncoming waves

Set **v _{S}** = 0 in the general frequency equation:

f_{O}= f_{S}(c − v_{O})/(c − v_{S})

Thus:

f_{O}= f_{S}(c − v_{O})/c

or

f_{O}= f_{S}(1 − v_{O}/c)

Note: Stating the direction convention is very important. Often the equation will be written with the observer movingtowardthe source, resulting in the equation:f. Make sure you know the convention used._{O}= f_{S}(1 + v_{O}/c)

### Change in frequency

The change in frequency is:

Δf = f_{S}− f_{O}

Substitute for **f _{O}**:

Δf = f_{S}− f_{S}(c − v_{O})/c

Simplify:

Δf = cf_{S}/c − f_{S}(c − v_{O})/c

Δf = (cf_{S}− cf_{S}+ v_{O}f_{S})/c

Thus:

Δf = v_{O}f_{S}/c

Note: According to our direction convention,vbecomes_{O}−vwhen the observer is moving toward the source._{O}

## Summary

The Doppler Effect frequency equations can be readily determined from the derived general wavelength equation. The resulting general Doppler Effect frequency equation is:

f_{O}/(c − v_{O}) = f_{S}/(c − v_{S})

From the general equation, the equation for the case when the observer is stationary can be found be setting **v _{O}** = 0.

f_{O}= f_{S}c/(c − v_{S})

Δf = −f_{S}v_{S}/(c − v_{S})

Likewise, the equation when the source is stationary can be found be setting **v _{S}** = 0.

f_{O}= f_{S}(c − v_{O})/c

Δf = v_{O}f_{S}/c

Sometimes you need to improvise

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## Derivation of Doppler Effect Frequency Equations