Doppler Effect and Moving Sources and Observers

(Topic)

Doppler Effect and Moving Sources and Observers

The Doppler effect is the change in observed frequency produced by relative motion between a wave source and an observer.

For sound and other mechanical waves, one fact is essential:

The wave speed is measured relative to the material medium.

A moving observer changes the rate at which existing wavefronts are encountered.

A moving source changes the spacing of the wavefronts it emits.

Those are physically different mechanisms, although they combine into one compact formula.

For one-dimensional propagation in the positive

x

(1)

direction, let:

c be the wave speed relative to the medium,
v_s be the source velocity relative to the medium,
v_o be the observer velocity relative to the medium,
positive velocities point in the propagation direction from source toward observer.

Then the classical Doppler formula is

|--------------| | c −-vo | |fo = fsc − vs.| ---------------

(2)

With this convention:

a source moving toward the observer has v_s > 0 and raises the observed frequency,
an observer moving toward the source has v_o < 0 and raises the observed frequency.

The signs follow from geometry rather than from a memorized “approach means plus” rule.

Figure 1. A stationary source emits equally spaced wavefronts into a stationary medium. The source period fixes the spacing λ = c∕f_s.

1 Stationary source and stationary observer

Let a stationary source emit at frequency

fs.

(3)

The source period is

|--------| | 1- | |Ts = f .| -------s--

(4)

During one source period, a wavefront travels distance

cTs.

(5)

Therefore the wavelength is

|-----------c--| |λ = cTs = --.| ------------fs-|

(6)

A stationary observer is crossed by wavefronts at rate

|------------| fo = c-= fs.| ------λ-------

(7)

There is no Doppler shift.

2 Moving observer, stationary source

Now keep the source stationary and let the observer move.

The wavelength in the medium is unchanged:

|--------| |λ = c-.| ------fs-|

(8)

The observer changes only the rate at which wavefronts are crossed.

For a wave propagating in the positive

x

(9)

direction, the relative speed between the wavefronts and observer is

c − vo.

(10)

Therefore

f = c −-vo. o λ

(11)

Substituting

λ = c∕f s

(12)

gives

|------------------| | ( c − vo) | |fo = fs ------ .| ------------c-------

(13)

If the observer moves toward the source,

vo < 0,

(14)

c − v > c. o

(15)

The observed frequency rises.

Figure 2. A moving observer does not change the wavelength already established in the medium. Motion changes the wavefront encounter rate.

3 Example 1: observer moving toward a stationary source

Take

fs = 1000 Hz,

(16)

c = 343 m ∕s,

(17)

and let the observer move toward the source at

20.0 m∕s.

(18)

Since positive velocity points with the wave,

vo = − 20.0 m ∕s.

(19)

Then

f_o	= 1000	(20)
	≈ 1058.3 Hz.	(21)

4 Moving source, stationary observer

Now let the observer remain stationary while the source moves in the propagation direction.

The source emits one crest, then moves before emitting the next crest.

During one source period,

1 Ts = --, fs

(22)

the first crest moves forward distance

cTs,

(23)

while the source moves forward distance

vsTs.

(24)

Therefore the wavelength ahead of the source is

|------------------| λahead = (c − vs)Ts.| --------------------

(25)

Equivalently,

|----------------| | c −-vs | |λahead = f .| -------------s---

(26)

The stationary observer receives wavefronts at rate

--c--- fo = λahead.

(27)

Hence

|---------c----| |fo = fs------.| --------c −-vs-|

(28)

A source moving toward the observer has

vs > 0,

(29)

so the wavelength is compressed and the frequency rises.

5 Wavefronts behind a moving source

Behind the source, the source moves away from previously emitted rearward wavefronts.

The rear wavelength is

----------------- | c + vs | |λbehind = ------ | ------------fs---|

(30)

when

vs > 0

(31)

denotes the source speed toward the forward observer.

Thus a stationary observer behind the source measures

|-------------c----| |fbehind = fs------.| ------------c +-vs-|

(32)

The moving source compresses wavefronts ahead and stretches them behind.

Figure 3. A subsonic source moving through the medium compresses wavefront spacing ahead and expands it behind. The propagation speed relative to the medium remains c.

6 Example 2: source moving toward a stationary observer

Let

fs = 1000 Hz,

(33)

c = 343 m ∕s,

(34)

and

v = 30.0 m ∕s. s

(35)

Then

f_o	= 1000	(36)
	≈ 1095.8 Hz.	(37)

Behind the source,

f_behind	= 1000	(38)
	≈ 919.6 Hz.	(39)

7 Moving source and moving observer

The moving source first establishes the wavelength in the medium:

c − v λ = -----s. fs

(40)

The moving observer then encounters those wavefronts at relative speed

c − vo.

(41)

Therefore

f_o	=	(42)
	= .	(43)

Thus

|--------------| | c −-vo | |fo = fsc − vs.| ---------------

(44)

This equation should always be used together with a stated sign convention.

8 Sign convention

For the one-dimensional formula used here, the positive direction is the direction of wave propagation from source toward observer.

Therefore:

v_s > 0: source moves toward observer,
v_s < 0: source moves away from observer,
v_o > 0: observer moves away from source,
v_o < 0: observer moves toward source.

This is summarized geometrically in Figure 4.

Figure 4. One-dimensional sign convention for a wave propagating from source to observer. Positive velocity points with propagation. Source motion in the positive direction closes the separation, while observer motion in the positive direction increases it.

9 Example 3: both source and observer moving

Let the source move toward the observer at

30.0 m∕s,

(45)

and let the observer move toward the source at

20.0 m∕s.

(46)

Then

vs = +30.0 m∕s, vo = − 20.0m ∕s.

(47)

For

fs = 1000 Hz,

(48)

and

c = 343 m ∕s,

(49)

f_o	= 1000	(50)
	≈ 1159.7 Hz.	(51)

10 Why source and observer motion are not symmetric

For mechanical waves in a medium, source and observer velocities appear differently:

c − vo fo = fs------. c − vs

(52)

This asymmetry is physical.

The medium provides a preferred frame for propagation.

Observer motion changes encounter rate.

Source motion changes wavelength.

This is fundamentally different from the relativistic Doppler effect for light in vacuum, where no material propagation medium defines a preferred frame.

11 Low-speed approximation

Suppose

|vs| ≪ c, |vo| ≪ c.

(53)

Then

fo= 1-−-vo∕c. fs 1 − vs∕c

(54)

Using

--1-- ≈ 1 + 𝜖 1 − 𝜖

(55)

for small

𝜖,

(56)

	≈	(57)
	≈ 1 + .	(58)

Therefore

|--------------| |Δf--≈ vs-−-vo.| -fs-------c-----

(59)

The leading-order shift depends on the closing velocity along the propagation direction.

12 Closing speed

Define

|---------------| vclose =-vs −-vo.-

(60)

With the present sign convention,

vclose > 0

(61)

means the source and observer are approaching.

Then at low speed,

|--------------| | vclose | -Δf--≈-fs--c--.|

(62)

This approximation is useful for quick estimates, but the exact classical expression should be used when speeds are not very small compared with

c.

(63)

13 Three-dimensional motion and radial velocity

Only the component of velocity along the propagation direction contributes directly to the first-order Doppler shift.

Let

^n

(64)

point along the acoustic ray from the source emission point toward the observer.

Define the radial velocity components

|------------| |v = v ⋅n^, | --s,r----s-----

(65)

and

|------------| |vo,r = vo ⋅ ^n. --------------

(66)

The corresponding classical ray formula is

|---------------| | c-−-vo,r | fo = fsc − v . | ------------s,r---

(67)

For low speeds,

Δf-----v---−-v----| ----≈ -s,r----o,r. | -fs--------c-------

(68)

Transverse motion matters because it changes the line-of-sight direction with time, but the instantaneous first-order shift is governed by radial components.

Figure 6. In multidimensional motion, project source and observer velocities onto the propagation direction. The radial components determine the instantaneous classical Doppler shift along that ray.

14 Passing source

Consider a source moving at constant speed

V

(69)

along a straight road.

Let the observer be a perpendicular distance

b

(70)

from the road.

Choose

t = 0

(71)

when the source is at the point of closest approach.

The source position relative to the observer can be represented as

r(t) = (− V t)ex + bey.

(72)

The source-observer distance is

|------√------------| R-(t) =--V-2t2 +-b2.--

(73)

The source radial velocity toward the observer is

|------------------| | V 2(− t) | |vs,r = √---2-2---2-| ---------V--t-+-b--

(74)

with the sign determined by the chosen propagation direction.

The important physical result is:

before closest approach, the radial component is toward the observer and the received frequency is elevated,
at closest approach, the instantaneous radial component is zero,
after closest approach, the radial component is away from the observer and the received frequency is reduced.

In a realistic pass-by geometry, the frequency shift changes continuously rather than making the discontinuous jump of an idealized one-dimensional crossing.

Figure 5. A source passing an observer at nonzero lateral distance has a radial velocity that changes continuously from approaching to receding. At closest approach the instantaneous radial component vanishes.

15 Doppler shift and beats

M06-13 developed the beat frequency

fbeat = |f1 − f2|.

(75)

If one tone is an unshifted reference at

fs

(76)

and the other is Doppler shifted to

fo,

(77)

their superposition produces beats at

|----------------| |fbeat = |fo − fs|. ------------------

(78)

For small speeds,

|----------------| | |vclose| | |fbeat ≈ fs------.| ------------c-----

(79)

Thus beat measurements can be used to infer a small velocity.

16 Example 4: velocity from an acoustic beat shift

A reference tone is

500 Hz.

(80)

A Doppler-shifted return produces a beat frequency

5.0 Hz.

(81)

For a one-way low-speed shift in air,

c = 343 m ∕s,

(82)

the corresponding closing speed is approximately

v_close	≈ c	(83)
	= 343	(84)
	≈ 3.43 m∕s.	(85)

This example is one-way Doppler. Reflection introduces a second shift.

17 Reflection from a moving target

A wave reflected from a moving object undergoes two Doppler transformations.

First, the moving target acts as an observer.

Second, the target re-radiates or reflects the disturbance and therefore acts as a moving source for the return path.

Consider a stationary transmitter and receiver at the same location.

Let a target move toward them at speed

u > 0.

(86)

The target first receives

|--------------| | c-+-u | |f1 = f0 c .| ---------------

(87)

For the reflected return, the approaching target acts as a moving source.

Therefore the receiver measures

|---------c----| |fr = f1-----. | --------c-−-u--|

(88)

Combining,

|--------------| |f = f c-+-u. | --r----0c-−-u--|

(89)

This is the classical double Doppler shift for an approaching reflector in this one-dimensional geometry.

Figure 7. Reflection from a moving target produces two shifts. The target first receives the incident wave as a moving observer, then returns it as an effective moving source.

18 Low-speed double Doppler shift

For

u ≪ c,

(90)

fr-= 1-+-u-∕c. f0 1 − u ∕c

(91)

Using the small-speed expansion,

|--------------| |f − f 2u | |-r----0 ≈ ---.| ---f0-------c--

(92)

Thus

|----------------| | 2f0u- | Δfdouble ≈ c .| ------------------

(93)

The factor of two is a consequence of the outgoing and return Doppler transformations.

19 Example 5: reflected acoustic Doppler shift

Take

f0 = 40.0kHz,

(94)

u = 2.00 m ∕s,

(95)

and

c = 343 m ∕s.

(96)

The exact return frequency is

f_r	= 40000	(97)
	≈ 40469.2 Hz.	(98)

Thus

|----------------| -Δf--≈-469.2-Hz.-|

(99)

The low-speed approximation gives

Δf	≈	(100)
	≈ 466.5 Hz.	(101)

The approximation is already close because

u∕c

(102)

is small.

20 Effect of a moving medium

For sound, the velocity

c

(103)

is defined relative to the medium.

Therefore source and observer velocities in the classical formula should also be interpreted relative to the same medium.

If the air itself moves relative to the ground, ground speeds cannot simply be inserted without accounting for the medium velocity.

Let the medium velocity in the ground frame be

w.

(104)

Then the source velocity relative to the medium is

|----------------------| -vs,med-=-vs,ground −-w,-|

(105)

and similarly

|----------------------| |vo,med = vo,ground − w. | -----------------------

(106)

These medium-relative velocities belong in the acoustic Doppler formula.

21 Retarded emission time

When source motion is significant, the wave received at observation time

t

(107)

was emitted at an earlier time.

Call that time

te.

(108)

The propagation condition is

|----------------| | R-(te,t) | |t − te = c ,| ------------------

(109)

where

R

(110)

is the source-to-observer path length connecting the emission event to the reception event.

This is the classical retarded-time condition.

For simple constant-velocity one-dimensional motion, it reduces to the formulas already derived.

For arbitrary trajectories, solving the retarded geometry is the systematic approach.

22 The sonic limit

For a source moving toward a stationary observer,

--c--- fo = fsc − v . s

(111)

− vs → c

(112)

the ideal formula predicts

fo → ∞.

(113)

This divergence signals the breakdown of the simple separated-wavefront picture.

Wavefronts emitted in the forward direction pile up.

At and above the wave speed, the geometry changes qualitatively.

The relevant physics becomes:

wavefront accumulation,
Mach cones,
shock waves,
sonic booms,
nonlinear effects.

These topics are the natural continuation in M06-15.

Figure 8. As a source approaches the wave speed, forward wavefront spacing collapses. At supersonic speed the disturbance forms a Mach-cone geometry rather than an ordinary sequence of separated forward wavefronts.

23 Why the classical formula cannot be used blindly at supersonic speed

vs > c,

(114)

the expression

c − vs

(115)

becomes negative for the forward one-dimensional construction.

That does not mean the observer measures an ordinary negative frequency.

It means the assumptions behind the subsonic wavelength derivation no longer describe the geometry.

The source overtakes its own wavefronts.

The disturbance must be analyzed using the envelope of wavefronts and shock propagation.

24 Common mistakes

Treating the Doppler shift as a change in wave speed relative to the medium.
Using the same physical argument for source motion and observer motion.
Memorizing plus and minus signs without defining a coordinate direction.
Forgetting that a moving observer does not change the wavelength already present in the medium.
Forgetting that a moving source changes the emitted wavelength.
Assuming source and observer motion are symmetric for mechanical waves.
Using ground-frame velocities when the medium itself is moving.
Using total speed instead of the radial velocity component in multidimensional motion.
Assuming transverse velocity produces the same first-order shift as radial velocity.
Confusing source frequency with observed frequency.
Applying the low-speed approximation when velocities are not small compared with c.
Forgetting the second Doppler transformation in a moving-reflector problem.
Missing the factor of two in the low-speed reflected Doppler shift.
Assuming a passing source maintains a constant radial velocity.
Extending the ordinary subsonic formula through the sonic limit without changing the wavefront geometry.

25 Practice exercises

Derive the moving-observer formula from the rate at which wavefronts cross the observer.
Derive the wavelength ahead of a moving source.
Derive the wavelength behind a moving source.
Combine the moving-source and moving-observer effects to obtain the one-dimensional classical Doppler formula.
Reproduce Examples 1 through 3 numerically.
Derive the low-speed approximation for the one-dimensional Doppler shift.
Rewrite the low-speed shift in terms of closing speed.
Show why only radial velocity enters the first-order multidimensional shift.
For a source passing an observer at lateral distance b, derive the source-observer distance R(t).
Use the Doppler-beat connection to estimate velocity from a specified beat frequency.
Derive the exact double Doppler formula for an approaching reflector.
Derive the factor-of-two low-speed approximation for reflected Doppler.
Reproduce the 40 kHz reflected-shift example.
Explain how a uniform wind changes which source and observer velocities belong in the acoustic formula.
Explain physically why the subsonic Doppler formula breaks down as source speed approaches the wave speed.

26 Summary

For a stationary source and moving observer,

|--------------| |fo = fsc −-vo.| ----------c----|

(116)

For a moving source and stationary observer,

|--------------| | c | |fo = fs------.| --------c −-vs-

(117)

Combining both effects,

|--------------| |f = f c −-vo.| --o----sc −-vs-|

(118)

At low speeds,

|--------------| |Δf-- vs-−-vo | |fs ≈ c .| ----------------

(119)

In multidimensional motion, use radial components:

|------c-−-v----| fo = fs-----o,r. | -------c-−-vs,r---

(120)

For reflection from an approaching target,

|--------------| | c-+-u- | |fr = f0c − u. | ---------------

(121)

At low speed,

|----------------| Δfdouble ≈ 2f0u-.| -------------c----

(122)

The Doppler effect is therefore a direct consequence of wavefront timing, wavelength geometry, observer encounter rate, and the propagation frame defined by the medium.

References

[1] PhysicsLibrary, M06-13, Beats, Interference, and Coherence.

[2] PhysicsLibrary, M06-11, Acoustic Waves, Sound Intensity, and Acoustic Impedance.

[3] PhysicsLibrary, M06-10, Fourier Analysis, Wave Packets, and Dispersion.

[4] A. P. French, Vibrations and Waves, W. W. Norton, 1971.

[5] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics, 4th ed., Wiley, 2000.

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Doppler Effect and Moving Sources and Observers