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# Derivation of Doppler Effect Wavelength Equations

by Ron Kurtus

The * derivation of the Doppler Effect equations* is the most straightforward by starting with the derivation of the wavelength equations. Frequency and velocity equations will then follow.

Note: Before the derivations, you should first establish the conventions for direction and velocities. (SeeConventions for Doppler Effect Equationsfor more information.)

The method used is to first derive the equations for a moving source and stationary observer by considering the observed distance the wave travels with the motion of the source.

For a moving observer and stationary source, you consider the frequency for the difference in velocities of the wavefront and the moving observer and then convert to wavelength. By combining the equations for both situations, you can derive the general Doppler Effect equation.

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 when both are moving?

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

## Moving source and stationary observer

Consider the Doppler Effect when the the observer is stationary and the source of the wavefront is moving tpward it in the **x-**direction.

Source is moving toward stationary observer

Note: According to our conventions, the source velocity is constant and less than the wave velocity, thex-direction is positive, and only motion along thex-axis is considered.

### Finding observed wavelength

The wave velocity is:

c = λ_{S}/T

where:

**c**is the wave velocity**λ**is the wavelength of the source or the distance between crests_{S}**T**is the time it takes a wave to move one wavelength**λ**_{S}

Solving for **T**:

T = λ_{S}/c

If the source is moving at a velocity **v _{S}** toward a stationary observer, then the distance

**that the source moves in time**

**T**is:

d = v_{S}T

where

**d**is the distance the source moves in time**T****v**is the velocity of the source toward a stationary observer_{S}

When the source is moving in the **x**-direction, it is "catching up" to the previously emitted wave when it emits the next wavefront. This means the wavelength *reaching the observer*, **λ _{O}**, is shortened.

Note: If the source was moving in the opposite direction,λwould be lengthened._{O}

The observed wavelength **λ _{O}** is then:

λ_{O}= λ_{S}− d

Observed wavelength as a function of source velocity

Substitute **T = λ _{S}/c** into

**d = v**:

_{S}T

d = v_{S}λ_{S}/c

Substitute this value for **d** into **λ _{O} = λ_{S} − d**:

λ_{O}= λ_{S}− v_{S}λ_{S}/c

Factoring out **λ _{S}** gives you:

λ_{O}= λ_{S}(1 − v_{S}/c)

The equation is also often written as:

λ_{O}= λ_{S}(c − v_{S})/c

If the source is moving away from the observer, the sign of **v _{S}** changes.

### Change in wavelength

Define the change in wavelength as:

Δλ = λ_{S}− λ_{O}

Since **λ _{O} = λ_{S} − d**:

Δλ = λ_{S}− (λ_{S}− d)

Also since ** d = v _{S}λ_{S}/c**:

Δλ = λ_{S}− (λ_{S}− v_{S}λ_{S}/c)

Δλ = λ_{S}v_{S}/c

## Moving observer and stationary source

Suppose the source is stationary and the observer is moving in the **x**-direction away from the source.

Observer moving away from oncoming waves

### Finding observed wavelength

In this situation, the observed wave frequency is a combination of the wave velocity and observer velocity, divided by the actual wavelength:

f_{O}= (c − v_{O})/λ_{S}

where

**f**is the observed frequency_{O}**v**is the observer velocity_{O}

But also **f _{O} = c/λ**

**:**

_{O}

c/λ_{O}= (c − v_{O})/λ_{S}

Reciprocating both sides of the equation:

λ_{O}/c = λ_{S}/(c − v_{O})

λ_{O}= λ_{S}c/(c − v_{O})

Multiply by **c**:

λ_{O}= λ_{S}/[(c − v_{O})/c]

Thus:

λ_{O}= λ_{S}c/(c − v_{O})

or

λ_{O}= λ_{S}/(1 − v_{O}/c)

### Change in wavelength

The change in wavelength is defined as:

Δλ = λ_{S}− λ_{O}

Substitute **λ _{O} = λ_{S}c/(c − v_{O})**:

Δλ = λ_{S}− λ_{S}c/(c − v_{O})

Multiply **λ _{S}** times

**(c − v**:

_{O})/(c − v_{O})

Δλ =[ λ_{S}(c − v_{O}) − λ_{S}c]/(c − v_{O})

Reduce and simplify:

Δλ =[ λ_{S}c − λ_{S}v_{O}− λ_{S}c]/(c − v_{O})

Thus:

Δλ = −λ_{S}v_{O}/(c − v_{O})

or

Δλ = λ_{S}/(1_{}− c/v_{O})

## General wavelength equation

When both the source and observer are moving in the **x**-direction, you can combine the individual equations to get a general Doppler Effect wavelength equation.

Let **λ _{O1}** be the wavelength equation for a moving source and stationary observer:

λ_{O1}= λ_{S}(c − v_{S})/c

For the case when both the source and observer moving, substitute **λ _{O1}** for

**λ**

**in the**

_{S}**λ**:

_{O}= λ_{S}c/(c − v_{O})

λ_{O}= λ_{O1}c/(c − v_{O})_{}

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

Simplify:

λ_{O}= λ_{S}c(c − v_{S})/c(c − v_{O})

Thus:

λ_{O}= λ_{S}(c − v_{S})/(c − v_{O})

or

λ_{O}(c − v_{O}) = λ_{S}(c − v_{S})

### Change in wavelength

The general change in wavelength is:

Δλ = λ_{S}− λ_{O}

Substitute for **λ _{O}**:

Δλ = λ_{S}− λ_{S}(c − v_{S})/(c − v_{O})

Δλ = [λ_{S}(c − v_{O}) − λ_{S}(c − v_{S})]/(c − v_{O})

Δλ = (λ_{S}c − λ_{S}v_{O}− λ_{S}c + λ_{S}v_{S})/(c − v_{O})

Thus:

Δλ = λ_{S}(v_{S}− v_{O})/(c − v_{O})

## Summary

The derivation of the Doppler Effect equations is the most straightforward by starting with wavelength. The Doppler Effect equations for the change in wavelength or in frequency as a function of the velocity of the wave source and/or observer can be determined though simple and logical derivations.

You can start with a moving source and stationary observer by considering the observed distance the wave travels with the motion of the source. For a moving observer and stationary source, you consider the frequency for the difference in velocities of the wavefront and the moving observer and then convert to wavelength. By combining the equations for both situations, you can derive the general Doppler Effect equation.

### General wavelength equation

λ_{O}= λ_{S}(c − v_{S})/(c − v_{O})

Δλ = λ_{S}(v_{S}− v_{O})/(c − v_{O})

### Moving source and stationary observer

λ_{O}= λ_{S}(c − v_{S})/c

Δλ = λ_{S}v_{S}/c

### Moving observer and stationary source

λ_{O}= λ_{S}c/(c − v_{O})

Δλ = −λ_{S}v_{O}/(c − v_{O})

Move with the flow

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## Derivation of Doppler Effect Wavelength Equations