Attached to the shaft of the motor in the photograph is a cylinder that has a reed at the opening of the long end, and a counterweight plugging the short end. A swivel barb on the mount allows connection of an air line, and for the cylinder to rotate during operation. When you turn on the air supply, the reed sounds at a steady pitch. When you turn on the motor and the instrument begins to rotate, as the reed end moves towards the class, the students hear a higher pitch, and as it recedes from them, they hear a lower pitch. The pitch thus fluctuates, and the students hear a warble. As you increase the motor speed, the changes in pitch become greater, and the warble more greatly pronounced.

The apparatus in this demonstration resembles one built by Ernst Mach, which he built at the suggestion of Andreas von Ettingshausen, under whom he had recently finished his doctorate, with the hopes of settling an argument going on at the time regarding the existence of the Doppler effect. (A professor named Joseph Petzval denied the possibility of the effect, because it broke his “Law of Conservation of the Period of Oscillation.”) His device was significantly larger – the tube was six feet long (with the rotation axis in the middle) – and the tube rotated in a vertical plane, but it used a reed at one of the tube ends, set oscillating by a flow of air.^4,5

When a source, such as the reed instrument in this demonstration, emits a sound wave, and the source is stationary with respect to the medium in which the wave travels (in this case, air), the frequency an observer hears (the rate at which the wavefronts reach his ears), if he is also stationary with respect to the medium, corresponds to the rate at which the speaker oscillates. This frequency, ν, equals the speed of the wave in the medium, v (346 m/s in dry air at 298 K), divided by its wavelength, λ, or ν = v/λ. Otherwise, we can consider two types of situation: one in which the source is moving, and one in which the observer is moving. In either case, the observer hears a different frequency from that emitted at the source. Since for this demonstration, the students are sitting still and the instrument is moving, we consider first the situation with the stationary observer and the moving source. After we have examined both of these situations, we will then consider light waves, which do not require a medium, and whose motion-related frequency shift is important to astronomical observations. In the following analysis, we consider motion only along the line between the source and the observer.

If the source is moving towards the observer, because of its motion in the direction of the sound wave it is emitting, the wavefronts come closer together than they otherwise would (and, behind it, farther apart than otherwise). This shortening of the wavelength corresponds to the distance the source travels during a single vibration, which is v_s/ν, where the subscript “s” denotes “source.” The shortened wavelength is thus

λ′ = v/ν - v_s/ν.

Since the wavelength is shortened, the frequency is increased:

ν′ = v/λ′ = v/[(v - v_s)/ν] = ν[v/(v - v_s)].

This is what happens when the reed end of the instrument is swinging towards the class.

If the source is moving away from the observer, since it is traveling in the opposite direction to the sound wave, the wavefronts come farther apart than they otherwise would (and, again, in front of it they would be closer together than otherwise). The wavelength is thus greater than λ by v_s/ν, again, the distance the source travels during a single vibration, and we have

λ′ = v/ν + v_s/ν, and ν′ = v/λ′ = v/[(v + v_s)/ν] = ν[v/(v + v_s)].

This is what happens when the reed end is swinging away from the class.

For a stationary observer with respect to the medium, then, and a source that is moving through it, the frequency the observer hears is

ν′ = ν[v/(v ∓ v_s)],

where, as above, ν is the frequency of the sound emitted by the source, v is the speed of sound in the medium, and v_s is the speed of the source relative to the medium (and the observer). The minus sign in the denominator holds for the case where the source is moving towards the observer (higher observed frequency/shorter apparent wavelength), and the plus sign holds for the case in which the source is moving away from the observer (lower observed frequency/longer apparent wavelength). When the reed instrument is rotating, as its reed end repeatedly travels towards and away from the class, the students hear, respectively, a higher and a lower pitch, which they hear as a warble.

In the case where the source is stationary and the observer is moving, the traveling wavefronts are one wavelength apart, but because the observer is moving, they appear closer together or farther apart, depending on whether he is moving towards or away from the source. Because ν = v/λ, over a given time, t, a stationary observer hears vt/λ waves. If he is moving towards the source, he hears v_ot/λ additional waves in that time. Thus, the (increased) frequency he hears is

ν′ = (vt/λ + v_ot/λ)/t = (v + v_o)/λ = (v + v_o)/(v/ν), or

ν′ = ν[(v + v_o)/v] = ν[1 + (v_o/v)].

If the observer is moving away from the source, he hears v_ot/λ fewer waves in a particular time, and the (decreased) frequency he hears is

ν′ = (vt/λ - v_ot/λ)/t = (v - v_o)/λ = (v - v_o)/(v/ν), or

ν′ = ν[(v - v_o)/v] = ν[1 - (v_o/v)].

For an observer moving with respect to the medium, and a stationary source, then, the frequency the observer hears is

ν′ = ν[(v ± v_o)/v)],

where ν is the frequency of the sound emitted by the source, v is the speed of sound in the medium, and v_o is the speed of the observer relative to the medium (and the source). The plus sign in the numerator holds for the case where the observer is moving towards the source (higher frequency/shorter apparent wavelength), and the minus sign holds for the case in which the observer is moving away from the source (lower frequency/longer apparent wavelength).

We can combine the two general equations above to find the observed frequency when both the source and the observer are moving through the medium. The result is ν′ = ν[(v ± v_o)/(v ∓ v_s)], where the plus sign in the numerator and the minus sign in the denominator hold when the source and observer are moving towards each other, and the minus in the numerator and the plus in the denominator hold when they are moving away from each other. If, of course, either the source or the observer is stationary, this equation reduces to one of the two general equations above.

We also see that, because the sound wave travels through a medium (on which its speed depends), these two general equations give slightly different results for the magnitude of the frequency shift caused by the relative motion of the source and the observer to each other, depending on which one is stationary with respect to the medium. It is possible to show, however, that for source and observer moving at speeds much less than the speed of sound in the medium, the difference between them becomes small.

This phenomenon of a perceived increase in pitch when a source is moving towards an observer, or vice-versa, and a decrease in pitch when one is moving away from the other, is called the Doppler effect, after Christian Johann Doppler, who first noted that observed color of luminous bodies, and hence, stars, must be affected by their motion with respect to an observer. In this case, the wave is light, which does not require a medium. In addition, the speed of light is constant (c = 2.998 × 10⁸ m/s), regardless of the motion (at constant velocity) of either the source or the observer. This is a postulate of Einstein‘s special theory of relativity, and it has been confirmed by experiment. Thus, for any case in which the source is receding from the observer at a particular speed, that is, whether or not either is stationary with respect to some reference frame, or which one is receding from the other, the frequency shift must be the same. Similarly, if they are moving towards each other at a particular speed, whichever is moving relative to some reference frame, or if they are both moving, the frequency shift must be the same. Technically, the Doppler shift predicted by the theory of relativity is ν′ = ν[(1 - (v/c)]/√[1 - (v/c)²], where v is the relative speed of the source with respect to the observer (or vice versa), when they are moving away from each other (and c is the speed of light). (Replacing v with -v gives the case in which they are moving towards each other.) The speed of light is so great, that for most cases, we could use any of the three equations above to calculate the Doppler shift for a light-emitting source moving with respect to an observer. Only in cases where the sources are moving at a significant portion of the speed of light, would we notice a difference in the results between the relativistic and the classical equations.

Astronomers use the Doppler effect to determine how quickly astronomical objects are moving towards or away from us, along our line of sight to them. They refer to these line-of-sight velocities as radial velocities. Depending on the type of object, and its physics, the spectrum of light from an astronomical body can manifest dark lines, from the absorption of certain wavelengths of light as it passes through layers that contain atoms of elements that can absorb at those wavelengths to become excited, or it can contain bright lines, from the emission of excited atoms of elements that emit at those wavelengths when they relax to lower-energy states. (See demonstration 84.21 or 88.15 – Spectra.) The patterns of absorption or emission lines we observe in the light from astronomical bodies, correspond to those we measure in the laboratory for various elements, except that the lines are shifted. How much they are shifted, and in which direction, depends on how fast the body is moving away from, or towards, us. The equation for this shift is λ′/λ = (1 + v/c), where λ′ is the apparent wavelength, λ is the true wavelength (what we measure in the laboratory), v is the speed at which the object is receding, and c is the speed of light. For a receding object, the apparent wavelength is longer than the true wavelength, and the light coming from it is said to be redshifted. If an object is coming towards us, (v/c) becomes negative, and the apparent wavelength is shorter than the true wavelength; the light is blueshifted.

More mundane examples of the Doppler effect are the change in pitch of a train engine’s horn, heard by someone waiting at a crossing as the train first approaches and then recedes (or the similar change in pitch of the clanging bells on the gate, heard by a passenger as the train approaches the gate and goes past it), or the change in pitch of a blaring car horn or wailing siren as a car or emergency vehicle approaches and goes by. To check for speeding drivers, police officers use the Doppler effect to measure how fast cars are going. They use an instrument that emits a a radio wave in the microwave region of the electromagnetic spectrum (frequency ~10-30 GHz, wavelength ~1-3 cm). When the wave is reflected back to the instrument from a passing car, its frequency is shifted according to the speed of the car. Based on the frequency difference between the outgoing wave and the incoming reflected wave, the instrument calculates the speed of the car. Similar devices are used in roadside speed indicators, and to measure the speed of balls thrown by baseball pitchers.

References:

1) Halliday, David and Resnick, Robert. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 445-449.
2) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1977), pp. 929-932.
3) Chaisson, Eric and McMillan, Steve. Astronomy Today (Upper Saddle River, New Jersey: Prentice-Hall, 1999) pp. 70-72, 78-93.
4) Blackmore, John T. Ernst Mach; His Work, Life, and Influence (Berkeley: University of California Press, 1972) pp. 17-19.
5) Nolte, David D. Physics Today 73 (3) (2020), 30-35.