Derivation of the Doppler effect formulas

The Doppler effect

Derivation of the Doppler effect formulas
1. Both source and observer are at rest
2. Source approaching a stationary observer
3. Source moving away from a stationary observer
4. Observer approaching a stationary source
5. Observer moving away from a stationary source
6. Both source and observer moving towards each other
7. Both source and observer moving away from each other
8. Source approaching, observer receding
9. Source receding, observer approaching
10. Summary

The Doppler effect

The Doppler effect is the change in frequency when there is relative motion between a source of waves and an observer.

In the case of sound waves, observed frequency $f_o$ (i.e., the frequency that is heard by an observer from a distance), is given by

$\texrm{(1)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v \pm v_o}{v \pm v_s}\right)}$

where $f_s$ is the frequency emitted by the source, $v$ is the speed of sound ( $v = 343 m/s$ in air at 20° C), $v_o$ is the speed of the observer, and $v_s$ is the speed of the source. Equation (1) was derived on conditions that the source frequency is constant, and both source’s speed and observer’s speed must also be constant. In addition, the motion of source and observer has to be along the line joining them

Derivation of the Doppler effect formulas

Equation Eq. (1) is a combined form of nine separate scenarios: (1) Both source and observer are at rest; (2) Source approaching a stationary observer; (3) Source moving away from a stationary observer; (4) Observer approaching a stationary source; (5) Observer moving away from a stationary source; (6) Source and observer moving towards each other; (7) Source and observer moving away from each other; (8) Source approaching, observer receding; and (9) Source receding, observer approaching. Following sections describe how you can derive each of these equations.

Case 1: Both source and observer are at rest

Sound waves are emitted from the source, with wavelength $\lambda_s$ , frequency $f_s$ , and travel at speed $v$ in the air (the medium) towards the observer. We can use the general formula

$\hspace{80px}\displaystyle{Distance = Speed \times Time}$

to describe the relationship between these quantities.
Thus, for one period $T_s$ , a sound wavefront travels a distance $\lambda_s$ , which can be expressed as

$\texrm{(2)}\hspace{40px}\displaystyle{\lambda_{s} = v T_{s}}$

where $T_{s} = \frac{1}{f_{s}}$

Sound source and observer both stationary; wavefronts evenly spaced — Diagram showing the absence of the Doppler Effect when there is no relative motion. Consecutive wavefronts reach the observer with the same wavelength as the source.

The perceived frequency $f_o$ , as heard by the observer can be obtained from the wave period $T_o$ — the time between two consecutive wavefronts that reach the observer, since $f_{o} = \frac{1}{T_{o}}$ .

After a wavefront reaches the observer, the next will reach the observer after a time $T_o$ , having travelled a distance $\lambda_{o}$

$\texrm{(3)}\hspace{40px}\displaystyle{\lambda_{o} = v T_{o}}$

When both the source and the observer are stationary, there is no relative motion between them, $\lambda_o = \lambda_s$ and $T_o = T_s$ ; as a result, no Doppler effect is observed.

Thus,

$\texrm{(4)}\hspace{40px}\displaystyle{f_o = f_s}$

The observer hears sounds of the same pitch (same frequency) as they are at the source.

Case 2: Source approaching a stationary observer

Assume the first wavefront is emitted at $t=0$ s. Each consecutive wavefront is emitted every $T_s$ after that, regardless of whether the source is stationary or moving. After emitting a wavefront, if the source is moving toward the stationary observer with speed $v_s$ , then before emitting the next wavefront, it travels a distance $d_s$ , where

$\texrm{(5)}\hspace{40px}\displaystyle{d_s = v_sT_s}$

As shown in the figure below, to reach the observer, the following wavefront (the next wavefront) needs to cover a distance $\lambda_o$ , which is shorter than $\lambda_s$

$\texrm{(6)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s - d_s}$

Source moving toward a stationary observer; the distance between consecutive wavefronts in the observer’s direction is compressed — Illustration of the Doppler Effect. The approaching source produces wavefronts closer together in front, leading to a higher perceived frequency for the stationary observer.

Substituting $\lambda_s$ , $\lambda_o$ , $d_s$ in Eq. (6) with their equivalents from Eqs. (2), (3) and (5), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s - v_s T_s\\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s} \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v - v_s}{f_s} \end{align*}$

therefore,

$\texrm{(7)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v}{v-v_s}\right)}$

As $v > (v - v_s)$ , $f_o$ is greater than $f_s$ , the observer hears sounds at a higher pitch than they actually are at the source.

Case 3: Source moving away from a stationary observer

Similar to Case 2, but this time, after emitting a wavefront, the source moves a distance $d_s$ away from the observer before releasing the next wavefront. Consequently, the distance $\lambda_o$ that the subsequent wavefront must cover to reach the observer is

$\texrm{(8)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s + d_s}}$

as illustrated in the figure below,

Sound source moving away from a stationary observer; wavefronts stretched behind the source — Illustration of the Doppler Effect showing a sound source receding from a stationary observer. Wavefronts behind the source are farther apart, resulting in a lower perceived frequency.

Substituting $\lambda_s$ , $\lambda_o$ , $d_s$ in Eq. (8) with their equivalents from Eqs. (2), (3) and (5), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s + v_s T_s\\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s} \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v + v_s}{f_s} \end{align*}$

therefore,

$\texrm{(9)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v}{v + v_s}\right)}$

As $v < (v + v_s)$ , $f_o < f_s$ , the observer hears sounds at a lower pitch than they are at the source.

Case 4: Observer approaching a stationary source

In this case, the source is stationary and the observer moves toward it with speed $v_o$ . After encountering one wavefront, the observer meets the next after a time $T_o$ , having travelled a distance $d_o$ , where

$\texrm{(10)}\hspace{40px}\displaystyle{d_o = v_oT_o}$

During that same time, the next wavefront must travel a distance $\lambda_o$ to reach the observer. As shown in the figure below,

$\texrm{(11)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s - d_o}$

Stationary sound source; observer moving toward it — The observer moves toward a stationary source, encountering consecutive wavefronts in a shorter time period, and thus perceiving a higher frequency.

Substituting $\lambda_s$ , $\lambda_o$ , $d_o$ in Eq. (11) with their equivalents from Eqs. (2), (3) and (10), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s - v_o T_o\\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} + \frac{v_o}{f_o} &&= \frac{v}{f_s} \\ \\ &\rightarrow \qquad \frac{v + v_o}{f_o} &&= \frac{v}{f_s} \end{align*}$

therefore,

$\texrm{(12)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v + v_o}{v}\right)}$

Since $(v + v_o) > v$ , $f_o > f_s$ , the observer hears sounds at a higher pitch than they actually are at the source.

Case 5: Observer moving away from a stationary source

In this case, the source is stationary and the observer moves away from it with speed $v_o$ . As illustrated in the figure below, once a wavefront passes the observer, the next wavefront must travel a distance $\lambda_s + d_o$ over a time interval $T_o$ to reach the observer. Therefore

$\texrm{(13)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s + d_o}$

Stationary sound source; observer moving away from it — Diagram showing a stationary sound source while the observer moves away, encountering consecutive wavefronts over a longer time period and perceiving a lower pitch.

Substituting $\lambda_s$ , $\lambda_o$ , $d_o$ in Eq. (11) with their equivalents from Eqs. (2), (3) and (10), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s + v_o T_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} - \frac{v_o}{f_o} &&= \frac{v}{f_s} \\ \\ &\rightarrow \qquad \frac{v - v_o}{f_o} &&= \frac{v}{f_s} \end{align*}$

therefore,

$\texrm{(14)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v - v_o}{v}\right)}$

Since $(v - v_o) < v$ , $f_o < f_s$ , the observer hears sounds at a lower pitch than they are at the source.

Case 6: source and observer moving towards each other

This case combines Cases 2 and 4, in which the source approaches the observer with speed $v_s$ , while the observer simultaneously moves toward the source with speed $v_o$ . As shown in the figure below, the effective spacing $\lambda_o$ between consecutive wavefronts can be expressed as

$\texrm{(15)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s - d_s - d_o}$

Sound source and observer moving toward each other; wavefronts compressed between them — Illustration of a sound source and an observer moving toward each other. Their relative motion further compresses the wavefronts between them, resulting in a greater increase in perceived frequency.

Substituting $\lambda_s$ , $\lambda_o$ , $d_s$ , $d_o$ in Eq. (15) with their equivalents from Eqs. (2), (3), (5) and (10), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s - v_s T_s - v_o T_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s} - \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} + \frac{v_o}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v + v_o}{f_o} &&= \frac{v - v_s}{f_s} \end{align*}$

therefore,

$\texrm{(16)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v + v_o}{v - v_s}\right)}$

Since $(v + v_o) > v > (v - v_s)$ , $f_o > f_s$ , the observer hears sounds at a much higher pitch than they actually are at the source.

Case 7: source and observer moving away from each other

Since the source recedes from the observer with speed $v_s$ while the observer simultaneously moves away from the source with speed $v_o$ , the distance between consecutive wavefronts $\lambda_o$ is stretched as illustrated below.

Sound source and observer moving apart; wavefronts stretched between them — Diagram showing both the sound source and observer moving away from each other, resulting in wavefronts stretched between them and a lower observed frequency.

Thus,

$\texrm{(17)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s + d_s + d_o}$

Substituting $\lambda_s$ , $\lambda_o$ , $d_s$ , $d_o$ in Eq. (17) with their equivalents from Eqs. (2), (3), (5) and (10), we obtain

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s + v_s T_s + v_o T_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s} + \frac{v_o}{f_o}\\ \\ &\rightarrow \qquad \frac{v}{f_o} - \frac{v_o}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v - v_o}{f_o} &&= \frac{v + v_s}{f_s} \end{align*}$

therefore,

$\texrm{(18)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v - v_o}{v + v_s}\right)}$

Since $(v - v_o) < v < (v + v_s)$ , $f_o < f_s$ , the observer hears sounds at a much lower pitch than they are at the source.

Case 8: Source approaching, observer receding

Sound source moving toward the observer while the observer moves away — The source approaches while the observer moves away. Their opposing motions partially offset each other, resulting in only a small change in the perceived frequency.

As shown above, when the source approaches the observer with speed $v_s$ while the observer moves away with speed $v_o$ , the distance between consecutive wavefronts $\lambda_o$ can be expressed as

$\texrm{(19)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s - d_s + d_o}$

hence

$\begin{alignat*}{3} &\qquad \qquad v T_o &&= v T_s - v_s T_s + v_o T_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s} + \frac{v_o}{f_o}\\ \\ &\rightarrow \qquad \frac{v}{f_o} - \frac{v_o}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v - v_o}{f_o} &&= \frac{v - v_s}{f_s} \end{align*}$

therefore,

$\texrm{(20)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v - v_o}{v - v_s}\right)}$

In this case, if $v_o < v_s$ , then the net effect is that the source is approaching, therefore, $f_o > f_s$ , and the observer hears a higher-pitched sound. The opposite occurs if $v_o > v_s$ .

Case 9: Source receding, observer approaching

Sound source moving away from the observer while the observer moves toward it — The source moves away while the observer moves toward it. The net frequency shift depends on their relative speeds.

As shown above, when the source moves away from the observer with speed $v_s$ while the observer approaches the source with speed $v_o$ , the distance between consecutive wavefronts $\lambda_o$ can be expressed as

$\texrm{(21)}\hspace{40px}\displaystyle{\lambda_o = \lambda_s + d_s - d_o}$

hence

$\begin{alignat*}{3} &\qquad \qquad v T_o && v T_s + v_s T_s - v_o T_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s} - \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} + \frac{v_o}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v + v_o}{f_o} &&= \frac{v + v_s}{f_s} \end{align*}$

therefore,

$\texrm{(22)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v + v_o}{v + v_s}\right)}$

In this case, if $v_s > v_o$ , then the net effect is that the source is receding, therefore, $f_o < f_s$ , the observer hears a lower-pitched sound. The opposite occurs if $v_s < v_o$ .

Summary

By combining all nine equations, 4, 7, 9, 12, 14, 16, 18, 20, and 22, we obtain

$\displaystyle{f_o = f_s\left(\frac{v \pm v_o}{v \pm v_s}\right)}$

This is the same general equation (Eq. 1) mentioned earlier in the introduction to the Doppler effect, which combines all relative motions between a sound source and an observer along the straight line joining them.

Identifying the appropriate Doppler-effect equation for a specific scenario using the general Doppler effect formula can be confusing. A simple way to make the task easier is to note that the ratio on the right hand side of the general formula represents the relative speed of the sound wave and the observer, (| $v$ |–| $v_o$ |), over the relative speed of the sound wave and the source, (| $v$ |–| $v_s$ |).

If we choose the propagation direction of the sound wave toward the observer as the positive direction, then the value of $v$ is always positive, so | $v$ | $=v$ .

If the source moves in the same direction as $v$ (the positive direction), then | $v_s$ | $=v_s$ , therefore

| $v$ |–| $v_s$ | $=v-v_s$ (source approaching)

If the source moves in the opposite direction, then | $v_s$ | $=-v_s$ , and therefore

| $v$ |–| $v_s$ | $=$ | $v$ |–( $-v_s$ ) $=v+v_s$ (source receding)

Similarly, for the observer’s motion: if the observer moves in the same direction as $v$ , then | $v_o$ | $=v_o$ , and therefore

| $v$ |–| $v_o$ | $=v-v_o$ (observer receding)

If the observer moves in the opposite direction, then | $v_o$ | $=-v_o$ , and therefore

| $v$ |–| $v_o$ | $=$ | $v$ |–( $-v_o$ ) $=v+v_o$ (observer approaching)

A more detailed explanation on how to extract the correct Doppler-effect equation for a given relative motion between the source and the observer from the general formula can be found here.

Derivation of the Doppler effect formulas

Contents

The Doppler effect

Derivation of the Doppler effect formulas

Case 1: Both source and observer are at rest

Case 2: Source approaching a stationary observer

Case 3: Source moving away from a stationary observer

Case 4: Observer approaching a stationary source

Case 5: Observer moving away from a stationary source

Case 6: source and observer moving towards each other

Case 7: source and observer moving away from each other

Case 8: Source approaching, observer receding

Case 9: Source receding, observer approaching

Summary