Derivation of Doppler Effect Velocity Equations

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.

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

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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.

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_O is the observed wavelength
• λ_S is the constant wavelength from the source
• c is the constant velocity of the wavefront in the x-direction
• v_S is the constant velocity of the source in the x-direction
• v_O is the constant velocity of the observer in the x-direction

(See Derivation of Doppler Effect Wavelength Equations for more information.)

Set v_O = 0 and solve for v_S:

λ_Oc = λ_S(c − v_S)

λ_Oc = λ_Sc − λ_Sv_S

λv_S = λ_Sc − λ_Oc

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:

vS = cΔλ/λS

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_O is the observed frequency
• f_S is the constant wave frequency from the source

(See Derivation of Doppler Effect Frequency Equations for more information.)

Set v_O = 0 and solve for v_S:

f_O = f_Sc/(c − v_S)

f_O(c − v_S) = f_Sc

f_Oc − f_Ov_S = f_Sc

− f_Ov_S = f_Sc − f_Oc

f_Ov_S = − c(f_S − f_O)

Since the change is frequency is Δf = (f_S − f_O), 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.

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_Sc

λ_Oc − λ_Ov_O = λ_Sc

−λ_Ov_O = λ_Sc − λ_Oc

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:

vO = −cΔλ/λO

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_Oc = f_Sc − f_Sv_O

Add f_Sv_O and subtract f_Oc from both sides of the equation:

f_Sv_O = f_Sc − f_Oc

Factor out c and divide by f_S:

v_O = c(f_S − f_O)/f_S

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: v_S = 0 and v_O = 0. The observer is usually nearby the source.

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_Sc/(c − v_R)

f_R = f_S(c − v_R)/c

where

• λ_R is the observed wavelength of the moving object
• λ_S is the original source wavelength
• v_R is the constant object velocity in the x-direction
• f_R is the observed frequency at the moving object
• f_S is the original source frequency

Waves reflected to stationary observer

The object reflects the “observed” waves as if the object was a moving source.

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_R is the velocity of the reflecting object, acting as a source. Replace λ_S with λ_R and v_S with v_R in the equation. Also, change the sign of 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_Sc/(c − v_R), substitute for λ_R and then solve for v_R:

λ_O = [λ_Sc/(c − v_R)]*[(c + v_R)/c]

λ_O = λ_Sc(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_Oc − λ_Ov_R = λ_Sc + λ_Sv_R

Subtract λ_Oc and λ_Sv_R from both sides of the equaiton:

−λ_Sv_R− λ_Ov_R = λ_Sc − λ_Oc

−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:

vR = −cΔλ/(λS+ λO)

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_Sc/(c − v_S)

However f_R represents the reflected source frequency and v_R is the velocity of the reflecting object, acting as a source. Also, the sign of c changes.

The reflected frequency equation is:

f_O = f_R(−c)/(−c − v_R)

f_O = f_Rc/(c + v_R)

Using the equation f_R = f_S(c − v_R)/c, substitute for f_R and then solve for v_R:

f_O = [f_S(c − v_R)/c]*[c/(c + v_R)]

f_O = f_Sc(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_Oc + f_Ov_R = f_Sc − f_Sv_R

f_Sv_R + f_Ov_R = f_Sc − f_Oc

v_R(f_S+ f_O) = c(f_S− f_O)

Thus:

vR = cΔf/(fS+ fO)

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)

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Do your work methodically

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