Derivation of the Doppler effect formulas

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

Derivation of Eq. (1) in the following section is for sound waves and when the motion of source and observer is along the line joining the two.

Case 1: Both source and observer are at rest

Doppler effect, sound waves, source and observer at rest

Figure 1:Wave fronts (grey dotted lines) of a sound wave emitted at source S (red dot) reaching an observer O (blue dot). v_s and v_o are speeds of the source and the observer, respectively. They are zero since the source and the observer are stationary. λ_s is the wavelength of the sound wave emitted by the source.

The speed of a sound wave $v$ can be expressed as

$\texrm{(2)}\hspace{40px}\displaystyle{v = \lambda f}$

where $\lambda$ is the wave wavelength, that is the distance between the two consecutive wave fronts (the yellow double arrow in Fig. 1) and $f$ is the wave frequency. Rearranging Eq (2), $\lambda$ can be expressed as

$\texrm{(3)}\hspace{40px}\displaystyle{\lambda = \frac{v}{f}}$

Let $\lambda_s$ , $f_s$ , $\lambda_o$ , and $f_o$ be the wavelength and frequency of the sound wave emitted by the source and the wavelength and frequency heard by an observer from a distance from the source, respectively. If both source and observer are at rest, then

$\begin{alignat*}{3} &\qquad \qquad \lambda_o &&= \lambda_s\\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} \\ \\ \end{align*}$

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

The observer in this case will hear a sound of the same pitch (same frequency) as that of the source. $\lambda_s$ is the source wavelength, depicted by the yellow double arrow in all Figures in this Section.

Case 2: Source approaching observer

Figure 2:Source approaching an at-rest observer. Grey dotted lines: sound wave fronts emitted from the source at start; dark red dotted lines: sound wave fronts emitted from the moving source; yellow double arrow: the distance the sound emitted from source traveling per wave period, i.e., source wavelength λ_s; red double arrow: distance source traveling per wave period, ds_s; blue double arrow: observed wavelength, λ_o; blue arrow: source’s velocity showing direction of source.

As shown in Fig. 2, when the source is approaching an observer at a constant speed $v_s$ , the sound wave fronts (dark red dotted lines) will travel a distance $ds_s$ (red double arrow) ahead of the wave fronts already emitted (grey dotted lines). Note that, sound wave motion after leaving the source is governed by the medium (e.g., air) properties, and is not affected by the source’s motion. Since in one wave period, sound waves travel one $\lambda_s$ , distance $ds_s$ can be related to source wavelength as $ds_s = \frac{v_s}{v}}\lambda_s$ . Therefore

$\displaystyle{ds_s = \frac{v_s}{v}\lambda_s = \frac{v_s}{v}\frac{v}{f_s}}$

$\texrm{(5)}\hspace{40px}\displaystyle{ds_s = \frac{v_s}{f_s}}$

On the observer end, the distance between the consecutive wave fronts, i.e., the new wavelength $\lambda_o$ (depicted by the blue double arrow) now becomes $(\lambda_s - ds_s)$ . As $\lambda_o = \frac{v}{f_o}$ , we have

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

$\texrm{(6)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v}{v-v_s}\right)}$

As $v > (v - v_s)$ , $f_o$ is greater than $f_s$ , which means observer hears sounds of a higher pitch than it actually is.

Case 3: Source moving away from observer

Doppler effect, sound waves, source receding from observer

Figure 3:Source moving away from observer at constant speed v_s. Grey dotted lines: sound wave fronts emitted from source at start; orange dotted lines: sound wave fronts emitted from receding source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; red double arrow: distance source traveling per wave period, ds_s; blue double arrow: observed wavelength, λ_o; blue arrow: source’s velocity showing direction of source.

As seen in Fig. 3, when source moves away from observer at a constant speed $v_s$ , the new sound wave fronts (orange dotted lines) also move further away from observer a distance $ds_s$ . As a result, the new observed wavelength $\lambda_o$ becomes $\lambda_s + d_s}$ , combined that with Eq. (5), we have

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

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

In Eq. 7, $v < (v + v_s)$ , therefore, $f_o < f_s$ , the observer hears sounds at a lower pitch than it is at the source.

Case 4: Observer approaching an at-rest source

Doppler effect, sound waves, observer approaching source

Figure 4:Observer approaching an at-rest source at constant speed v_o. Grey dotted lines: sound wave fronts emitted from source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; bright green double arrow: distance observer traveling per observed wave period, ds_o; blue double arrow: observed wavelength, λ_o; green arrow: observer’s velocity showing direction of observer

In this case, source is stationary, observer moves towards it at a constant speed $v_o$ . For one wave period, observer travels a distance $ds_o$ (bright green double arrow). As wavelength $\lambda_o$ is the distance the observed wave travels for one wave period, $ds_o$ can be related to $\lambda_o$ as $ds_o = \frac{v_o}{v}\lambda_o$ , therefore

$\displaystyle{ds_o = \frac{v_o}{v}\lambda_o = \frac{v_o}{v}\frac{v}{f_o}}$

$\texrm{(8)}\hspace{40px}\displaystyle{ds_o = \frac{v_o}{f_o}}$

Now, on observer end, $\lambda_o$ becomes $(\lambda_s - ds_o)$ , we have

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

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

Since $(v + v_o) > v$ , $f_o > f_s$ , observer hears sounds of a higher pitch than it is at the source.

Case 5: Observer moving away from a stationary source

Doppler effect, sound waves, observer moving away from source

Figure 5:Observer moving away from a stationary source at constant speed v_o. Grey dotted lines: sound wave fronts emitted from a stationary source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; bright green double arrow: distance observer traveling per observed wave period, ds_o; blue double arrow: observed wavelength, λ_o; green arrow: observer’s velocity showing direction of observer.

In this case, source is stationary and observer is approaching source at a constant speed $v_o$ . As shown in Fig. 5, observer’s new wavelength $\lambda_o$ now becomes ( $\lambda_s + ds_o$ ), combined that with Eq. (8) we have

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

$\texrm{(10)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v - v_o}{v}\right)}$

Since $(v - v_o) < v$ , $f_o < f_s$ , observer hears a lower pitch sound.

Case 6: Both source and observer moving towards each other

Doppler effect, sound waves, both source and observer approach each other

Figure 6:Both source and observer are moving towards each other at constant speeds v_s and v_o, respectively. Grey dotted lines: sound wave fronts emitted from source at start; dark red dotted lines: sound wave fronts emitted from approaching source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e, source wavelength λ_s; red double arrow: distance source traveling per source wave period, ds_s; bright green double arrow: distance observer traveling per observed wave period, ds_o; blue double arrow: observed wavelength, λ_o; green arrow: observer’s velocity showing direction of observer; blue arrow: source’s velocity showing direction of source.

In this case, source is approaching observer at constant speed $v_s$ and observer is moving towards the source at a constant speed $v_o$ . As seen in Fig. 6, observer’s new wavelength $\lambda_o$ is equal to ( $\lambda_s - ds_s- ds_o$ ), and we have

$\begin{alignat*}{3} &\qquad \qquad \lambda_o &&= \lambda_s - ds_s - ds_o\\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s} - \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} + \frac{v_o}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v + v_o}{f_o} &&= \frac{v - v_s}{f_s} \end{align*}$

$\texrm{(11)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v + v_o}{v - v_s}\right)}$

Since $(v + v_o) > v > (v - v_s)$ , $f_o > f_s$ , observer hears sounds of a higher pitch.

Case 7: Both source and observer moving away from each other

Doppler effect, sound waves, both source and observer moving away from each other

Figure 7:Both source and observer are moving away from each other at constant speeds v_s and v_o, respectively. Grey dotted lines: sound wave fronts emitted from source at start; orange dotted lines: sound wave fronts emitted from receding source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; red double arrow: distance source traveling per source wave period, ds_s; blue double arrow: observed wavelength, λ_o; bright green double arrow: distance observer traveling per observed wave period, ds_o; green arrow: observer’s velocity showing direction of observer; blue arrow: source’s velocity showing direction of source.

In this case, source is receding from observer at a constant speed $v_s$ and observer is also moving away from source at a constant speed $v_o$ . As seen in Fig. 7, observer’s new wavelength $\lambda_o$ is now equal to ( $\lambda_s + ds_s + ds_o$ ), so we have

$\begin{alignat*}{3} &\qquad \qquad \lambda_o &&= \lambda_s + ds_s + ds_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s} + \frac{v_o}{f_o}\\ \\ &\rightarrow \qquad \frac{v}{f_o} - \frac{v_o}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v - v_o}{f_o} &&= \frac{v + v_s}{f_s} \end{align*}$

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

Since $(v - v_o) < v < (v + v_s)$ , $f_o < f_s$ , observer hears a lower pitch sound.

Case 8: Source approaching, observer moving away

Doppler effect, sound waves, source approaching, observer moving away

Figure 8:Source approaching observer at constant speed v_s, observer moving away at constant speed v_o. Grey dotted lines: sound wave fronts emitted from source at start; dark red dotted lines: sound wave fronts emitted from approaching source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; red double arrow: distance source traveling per wave period, ds_s; bright green double arrow: distance observer traveling per observed wave period, ds_o; blue double arrow: observed wavelength, λ_o; green arrow: observer’s velocity showing direction of observer; blue arrow: source’s velocity showing direction of source.

In this case, source is approaching observer at a constant speed $v_s$ but observer is moving away from source at a constant speed $v_o$ . As seen in Fig. 8, observer’s new wavelength $\lambda_o$ is now equal to ( $\lambda_s - ds_s + ds_o$ ), so we have

$\begin{alignat*}{3} &\qquad \qquad \lambda_o &&= \lambda_s - ds_s + ds_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s} + \frac{v_o}{f_o}\\ \\ &\rightarrow \qquad \frac{v}{f_o} - \frac{v_o}{f_o} &&= \frac{v}{f_s} - \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v - v_o}{f_o} &&= \frac{v - v_s}{f_s} \end{align*}$

$\texrm{(13)}\hspace{40px}\displaystyle{f_o = f_s\left(\frac{v - v_o}{v - v_s}\right)}$

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

case 9: source receding, observer approaching

Doppler effect, sound waves, source receding, observer approaching

Figure 9:Source receding from observer at constant speed v_s, observer approaching source at constant speed v_o. Grey dotted lines: sound wave fronts emitted from source at start; orange dotted lines: sound wave fronts emitted from receding source; yellow double arrow: distance sound waves emitted from source traveling per wave period, i.e., source wavelength λ_s; red double arrow: distance source traveling per wave period, ds_s; bright green double arrow: distance the observer traveling per wave period, ds_o; blue double arrow: observed wavelength, λ_o; green arrow: observer’s velocity showing direction of observer; blue arrow: source’s velocity showing direction of source.

In this case, source is receding from observer at constant speed $v_s$ while observer is moving towards source at a constant speed $v_o$ . As shown in Fig. 9, observer’s new wavelength $\lambda_o$ is now equal to ( $\lambda_s + ds_s - ds_o$ ), and we have

$\begin{alignat*}{3} &\qquad \qquad \lambda_o &&= \lambda_s + ds_s - ds_o \\ \\ &\rightarrow \qquad \frac{v}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s} - \frac{v_o}{f_o} \\ \\ &\rightarrow \qquad \frac{v}{f_o} + \frac{v_o}{f_o} &&= \frac{v}{f_s} + \frac{v_s}{f_s}\\ \\ &\rightarrow \qquad \frac{v + v_o}{f_o} &&= \frac{v + v_s}{f_s} \end{align*}$

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

In this case, if $v_o > v_s$ , then the net effect is receding, therefore, $f_o > f_s$ , observer hears a higher pitch sound. The opposite occurs if $v_o < v_s$

Summary

Combining Eqs. (6), (7), and (9) – (14) we have

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

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

You can find tips on quickly extracting the Doppler effect equation for a specific relative motion between the sound source and an observer from this generalized equation here.