Physics · Ch 11 — Waves

Doppler Effect

11.11

Doppler Effect

Standing on a railway platform and listening to the whistle of an approaching, then departing, train reveals a striking effect: the pitch (frequency) heard while the train approaches is noticeably higher than the pitch heard once it has passed and is moving away -- even though the train's whistle itself never actually changes its own frequency at any point. This phenomenon, called the Doppler effect after the Austrian mathematician and physicist Johann Christian Doppler (1803-1853) who first studied it, occurs whenever there is relative motion between a sound source and a listener: the frequency actually observed by the listener differs from the frequency genuinely produced by the source. The Doppler effect is a general wave phenomenon, occurring for light and other electromagnetic waves just as much as for sound, though sound's version has one distinctive asymmetry: because sound's speed is fixed relative to the medium (air) rather than relative to either party, the frequency shift caused by a moving SOURCE and a stationary listener is not numerically identical to the shift caused by an equal-speed moving LISTENER and a stationary source -- an asymmetry that does not arise for light, whose propagation is independent of any medium. This section derives the observed frequency for three cases in turn: a stationary source with a moving listener, a moving source with a stationary listener, and finally both source and listener moving simultaneously, consolidating every combination into a single reference table (Table 11.4). Case (i), stationary source and moving listener: with the source S at rest emitting sound of frequency $f$ and wavelength $\lambda=v/f$ , a listener L moving directly toward S at speed $v_L$ experiences the sound waves arriving at an effectively increased relative speed $v'=v+v_L$ (since the wavelength itself is unaffected by the listener's own motion), giving an observed frequency $f'=f\!\left(\dfrac{v+v_L}{v}\right)$ -- greater than $f$ ; if the listener instead moves away from the stationary source at the same speed, the sign of $v_L$ simply flips, giving $f'=f\!\left(\dfrac{v-v_L}{v}\right)$ -- less than $f$ . Case (ii), moving source and stationary listener: a source S moving directly toward a stationary listener L at speed $v_S$ crowds its successively emitted compressions closer together ahead of it, compressing the wavelength to $\lambda'=\lambda-v_ST$ , giving an observed frequency $f'=f\!\left(\dfrac{v}{v-v_S}\right)$ -- again greater than $f$ ; a source instead moving away gives $f'=f\!\left(\dfrac{v}{v+v_S}\right)$ -- less than $f$ . Case (iii), both source and listener moving: combining both effects gives the fully general formula $f'=f\!\left(\dfrac{v\pm v_L}{v\mp v_S}\right)$ , with each of $v_L,v_S$ taken positive when that party moves toward the other and negative when moving away, correctly reproducing every specific combination of source and listener motion listed in Table 11.4 -- including the case where the medium (such as air in a wind) is itself also moving, which can be folded dire …

Figure 11.45Listener moves toward the stationary source

What this figure shows. A stationary point source S is drawn at rest, emitting sound as a series of concentric circular wavefronts labelled "Compressions of sound waves", spaced evenly apart by the ordinary wavelength lambda since the source itself is not moving. A listener L is drawn moving toward the source with velocity $v_L$ , shown crossing successive wavefronts faster than a stationary listener would, since the listener's own motion adds directly to the wave's approach speed relative to them. The figure is the geometric basis for the moving-listener Doppler formula $f'=f(v+v_L)/v$ : because the wavelength emitted by the stationary source is completely unaffected by the listener's motion, but the listener rushes to meet each successive wavefront sooner than they otherwise would, the RATE at which wavefronts cross the listener's ear …

Figure 11.46Source moves toward the stationary listener

What this figure shows. Panel (a) shows a stationary source S and a stationary listener L, with two successive compressions already emitted and drawn as concentric circles spaced by the ordinary wavelength lambda. Panel (b) shows the source S now moving toward the listener at speed $v_S$ : the compression emitted when the source was at an earlier position A is drawn as a larger circle, while the compression emitted slightly later when the source has advanced to position B is drawn as a smaller circle much closer behind it, with the reduced spacing between the two compressions labelled as a shorter wavelength lambda-prime, and the distance the source itself moved in that same time interval marked as $v_ST$ . The figure is the geometric basis for the moving-source Doppler formula $f'=fv/(v-v_S)$ : because the source is chasing after its own previously-emitted wavefronts, each successive compression is emitted closer to the one before it, physically compressing the wavelength ahead of the source (and correspondingly stretching it behind) …

Table 11.4Observed frequency for different situations

S.No.	Situation	Observed frequency
1	L moves toward the stationary S	$f' = f\left(\dfrac{v+v_L}{v}\right)$
2	L moves away from the stationary S	$f' = f\left(\dfrac{v-v_L}{v}\right)$
3	S moves toward the stationary L	$f' = f\left(\dfrac{v}{v-v_S}\right)$
4	S moves away from the stationary L	$f' = f\left(\dfrac{v}{v+v_S}\right)$
5	S and L move toward each other	$f' = f\left(\dfrac{v+v_L}{v-v_S}\right)$
6	S and L recede from each other	$f' = f\left(\dfrac{v-v_L}{v+v_S}\right)$
7	S chases the L	$f' = f\left(\dfrac{v-v_L}{v-v_S}\right)$
8	L chases the S	$f' = f\left(\dfrac{v+v_L}{v+v_S}\right)$

Misc Example 11.30Direct and cliff-reflected Doppler frequencies from a receding source

Worked out. A source emits sound at 1500 Hz while moving away from an observer, directly toward a cliff, at 6 m/s; taking the speed of sound as 330 m/s, the task is to find (a) the frequency heard directly from the source, and (b) the frequency heard by the observer of the sound after it reflects off the cliff. For part (a), the source recedes from the stationary observer, so using the source-moving-away formula $f'=f\,v/(v+v_S)=1500\times330/(330+6)=1500\times330/336\approx1473\ \text{Hz}$ -- a lowered pitch, as expected for a receding source. For part (b), relative to the (stationary) cliff the source is instead APPROACHING, so the sound reflecting off the cliff and returning to the observer uses the source-moving-toward formula $f'=f\,v/(v-v_S)=1500\times330/(330-6)=1500\times330/324\approx1528\ \text{Hz}$ -- a raised pitch, since as far as the cliff (and the reflected wave) is concerned, the source is closing in on it. The observer therefore simultaneously hears two different frequenc …

Misc Example 11.31Beat frequency heard between an arriving and a departing train

Worked out. An observer stands on a platform between two trains, one arriving at the station and one leaving, each moving at the same speed of 8 m/s and each sounding a whistle of the same source frequency 240 Hz; taking the speed of sound as 330 m/s, the task is to find the number of beats the stationary observer hears. For the arriving train (source approaching the stationary observer), the observed frequency is $f_{in}=f\,v/(v-v_S)=240\times330/(330-8)=240\times330/322\approx246\ \text{Hz}$ . For the departing train (source receding from the stationary observer), the observed frequency is $f_{out}=f\,v/(v+v_S)=240\times330/(330+8)=240\times330/338\approx234\ \text{Hz}$ . Since the observer hears both whistles simultaneously, superposed, the beat frequency is simply their difference, $|f_{in}-f_{out}|=|246-234|=12$ beats per second -- a striking demonstration of how the SAME source frequency, split into a raised and a lowered copy purely by opposite relative motions, produces an …

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