Sound Waves

Lessons

Senior 6 · Physics

Sound Waves

5 lessons

Lesson 4 of 5

Beats and the Doppler Effect

25 min

The note that drops as a moto passes

Stand at the side of a road and listen to a moto-taxi go by. As it approaches, the engine note is steady and high; at the moment it passes, the pitch drops suddenly and stays low as it leaves. The engine itself did not change. The change is in the waves reaching your ear, and it is called the Doppler effect. This lesson also explains a second effect of two waves meeting: the slow throbbing called beats, which musicians use to tune instruments.

Experiment 4.1: hearing beats

Apparatus: two tuning forks of the same frequency on resonance boxes, a small lump of plasticine, a rubber hammer (Figure 4.1).

Figure 4.1: Loading one fork with plasticine lowers its frequency slightly.

Strike both unloaded forks together. You hear one steady note.
Press a small lump of plasticine onto one prong of fork B. Strike both again: the sound now swells and fades regularly, "wah-wah-wah". These are beats.
Add more plasticine: the beats become faster.

Warning

Safety. Strike the forks only with the rubber hammer, never against the bench. Keep the plasticine lump small and pressed on firmly so it cannot fly off. If you use two phone tone-generator apps instead, keep the volume moderate: loud tones held for a long time can damage hearing.

Explaining beats

Two waves of nearly equal frequencies $f_{1}$ and $f_{2}$ and equal amplitude $A$ arrive at the same point. Using $\sin P + \sin Q = 2\cos\frac{P - Q}{2}\sin\frac{P + Q}{2}$ :

y = A\sin 2\pi f_1 t + A\sin 2\pi f_2 t = \underbrace{2A\cos\left(2\pi\,\frac{f_1 - f_2}{2}\,t\right)}_{\text{slowly varying amplitude}}\;\sin\left(2\pi\,\frac{f_1 + f_2}{2}\,t\right)

You hear a note of the average frequency $\frac{f_1 + f_2}{2}$ whose amplitude rises and falls. The loudness is greatest whenever the cosine factor is $+ 1$ OR $- 1$ , which happens twice in each of its cycles. So the number of loud moments per second is

f_{\text{beat}} = |f_1 - f_2|

Figure 4.2: Where the two waves are in phase they reinforce (loud); where they are in antiphase they cancel (quiet).

Definition

Beats are the regular rise and fall in loudness heard when two sound waves of slightly different frequencies superpose. The beat frequency is the number of loudness maxima per second: $f_{\text{beat}} = |f_1 - f_2|$ . The ear can count beats up to roughly 10 per second; faster beats are heard as a rough sound.

Worked example 1: tuning a string

An inanga player tunes a string against a 440 Hz fork and hears 3 beats per second. When she tightens the string slightly, the beats slow to 1 per second. What was the string's frequency?

With 3 beats per second the string is $440 \pm 3$ : either 437 Hz or 443 Hz. Tightening raises the frequency ( $f \propto \sqrt{T}$ ). If it were 443 Hz, raising it would INCREASE the beat frequency. The beats slowed, so it was 437 Hz, now about 439 Hz. She keeps tightening until the beats disappear.

Worked example 2: a loaded fork

In Experiment 4.1, fork A is 256 Hz and the loaded fork B gives 4 beats per second. Find the frequency of B, and the frequency of the note you hear.

Plasticine adds mass, so B's frequency falls: $f_B = 256 - 4 = 252\,\mathrm{Hz}$ . The note heard has the average frequency $\frac{256 + 252}{2} = 254\,\mathrm{Hz}$ , and it swells and fades 4 times per second.

The Doppler effect

The Doppler effect is the change in the observed frequency of a wave when the source or the observer moves relative to the medium. Let the source emit frequency $f$ and let sound travel at $v$ in still air.

1. Source moving towards a stationary observer

Figure 4.3: A moving source crowds its wavefronts ahead of it.

In one period $T = \frac{1}{f}$ a wavefront moves a distance $v T$ forwards, but the source also moves $v_{s} T$ forwards before emitting the next one. So the wavelength ahead of the source is shortened:

\lambda' = vT - v_s T = \frac{v - v_s}{f}

The observer is at rest, so the waves pass at speed $v$ and

f' = \frac{v}{\lambda'} = f\,\frac{v}{v - v_s} \qquad \text{(source approaching)}

Behind the source the wavelength is $\frac{v + v_s}{f}$ , so a receding source gives $f' = f\,\frac{v}{v + v_s}$ .

2. Observer moving towards a stationary source

Now the wavelength is unchanged, $\lambda = \frac{v}{f}$ , but an observer moving towards the source at $v_{o}$ meets the wavefronts at a relative speed $v + v_{o}$ :

f' = \frac{v + v_o}{\lambda} = f\,\frac{v + v_o}{v} \qquad \text{(observer approaching)}

3. Both moving (along the same line)

f' = f\,\frac{v \pm v_o}{v \mp v_s}

Choose the upper signs when that motion brings source and observer closer, and the lower signs when it takes them apart. The speeds $v_{o}$ and $v_{s}$ are measured relative to the air.

Definition

The Doppler effect is the apparent change in frequency of a wave caused by relative motion of the source and the observer. Approach raises the observed frequency; separation lowers it. For sound, $f' = f\,\frac{v \pm v_o}{v \mp v_s}$ .

Worked example 3: an ambulance siren

An ambulance siren emits 700 Hz and the ambulance travels at $25\,\mathrm{m\,s^{-1}}$ (90 km/h). Take $v = 340\,\mathrm{m\,s^{-1}}$ . Find the frequency heard by a person standing at the roadside (a) as it approaches, (b) as it moves away. Find also the wavelength in front of the ambulance.

\text{(a)}\; f' = 700 \times \frac{340}{340 - 25} = 756\,\mathrm{Hz} \qquad \text{(b)}\; f' = 700 \times \frac{340}{340 + 25} = 652\,\mathrm{Hz}

\lambda' = \frac{v - v_s}{f} = \frac{340 - 25}{700} = 0.450\,\mathrm{m} \quad \left(\text{compared with } \frac{340}{700} = 0.486\,\mathrm{m} \text{ at rest}\right)

The pitch drops by about 100 Hz as the ambulance passes.

Worked example 4: finding a speed from the drop in pitch

A car horn sounds at a steady frequency. A roadside microphone records 1060 Hz as the car approaches and 940 Hz as it recedes ( $v = 340\,\mathrm{m\,s^{-1}}$ ). Find the speed of the car.

Divide the two Doppler equations to eliminate the unknown $f$ :

\frac{f_a}{f_r} = \frac{f\,\frac{v}{v - v_s}}{f\,\frac{v}{v + v_s}} = \frac{v + v_s}{v - v_s} \quad\Rightarrow\quad v_s = v\,\frac{f_a - f_r}{f_a + f_r}

v_s = 340 \times \frac{1060 - 940}{1060 + 940} = 340 \times \frac{120}{2000} = 20.4\,\mathrm{m\,s^{-1}} \approx 73\,\mathrm{km\,h^{-1}}

Worked example 5: source and observer both moving

A bus moving at $20\,\mathrm{m\,s^{-1}}$ sounds a 500 Hz horn. A cyclist ahead of the bus rides towards it at $10\,\mathrm{m\,s^{-1}}$ . What frequency does the cyclist hear? ( $v = 340\,\mathrm{m\,s^{-1}}$ )

Both motions bring them closer, so both upper signs apply:

f' = 500 \times \frac{340 + 10}{340 - 20} = 500 \times \frac{350}{320} = 547\,\mathrm{Hz}

Important

The effect is not symmetric. A source approaching at 34 m s⁻¹ gives $f\frac{340}{306} = 1.11f$ , but an observer approaching at 34 m s⁻¹ gives $f\frac{374}{340} = 1.10f$ . For sound, it matters which one moves relative to the air. When a source approaches at the speed of sound, $v - v_s \to 0$ : the wavefronts pile up into a shock wave, heard as a sonic boom from a supersonic aircraft.

Warning

Common mistakes. (1) Saying the source's frequency changes: it does not; only the observed frequency does. (2) Thinking the pitch rises as an approaching source gets closer: for a source moving straight towards you at a steady speed the pitch stays constant (only the loudness grows), and it drops as the source passes. (3) Swapping signs: check the answer is HIGHER than $f$ for approach. (4) Giving the beat frequency as half the difference.

Tip

Put the observer's speed on top and the source's speed underneath: "O over S", $f' = f\,\frac{v \pm v_o}{v \mp v_s}$ . Then pick the signs that make $f^{'}$ larger for approach.

The Doppler effect (Bozeman Science). Paul Andersen explains how the frequency you hear depends on the motion of both the source and the observer; compare his explanation with Figure 4.3.

✦

Practice

Two tuning forks of 256 Hz and 260 Hz sound together. State the beat frequency and the frequency of the note heard.
A guitar string sounded with a 330 Hz fork gives 5 beats per second. When the string is loosened slightly, the beat frequency increases. Find the original frequency of the string.
A train approaches a station at $34\,\mathrm{m\,s^{-1}}$ , sounding a 400 Hz whistle. Find the frequency heard on the platform before and after it passes ( $v = 340\,\mathrm{m\,s^{-1}}$ ).
A motorcyclist rides at $20\,\mathrm{m\,s^{-1}}$ towards a stationary alarm emitting 1000 Hz. What frequency does she hear ( $v = 340\,\mathrm{m\,s^{-1}}$ )?
A siren is heard at 520 Hz as a vehicle approaches and 480 Hz after it passes. Find the speed of the vehicle ( $v = 340\,\mathrm{m\,s^{-1}}$ ).
Explain, using a diagram of wavefronts, why an observer BEHIND a moving source hears a lower frequency.
A student says: "An observer moving at 30 m s⁻¹ towards a source hears exactly the same frequency as a source moving at 30 m s⁻¹ towards the observer." Evaluate this statement with a calculation for a 600 Hz source ( $v = 340\,\mathrm{m\,s^{-1}}$ ).

Show answersHide answers

4 Hz; the note heard is 258 Hz.
Possible values 325 Hz or 335 Hz. Loosening lowers the frequency; the beats increased, so the string moved further from 330 Hz: it was 325 Hz.
Approaching: $400 \times \frac{340}{306} = 444\,\mathrm{Hz}$ ; receding: $400 \times \frac{340}{374} = 364\,\mathrm{Hz}$ .
$1000 \times \frac{340 + 20}{340} = 1059\,\mathrm{Hz} \approx 1.06\,\mathrm{kHz}$
$v_s = 340 \times \frac{520 - 480}{520 + 480} = 13.6\,\mathrm{m\,s^{-1}}$
Each wavefront spreads from where the source was when it was emitted. The source moves away from B between emissions, so the fronts reaching B are spaced $\frac{v + v_s}{f}$ apart, a longer wavelength; they pass B at speed $v$ , so B receives fewer per second.
Not exact. Observer moving: $600 \times \frac{370}{340} = 653\,\mathrm{Hz}$ . Source moving: $600 \times \frac{340}{310} = 658\,\mathrm{Hz}$ . Close at low speeds, but different: the formula is not symmetric for sound.

Key points

Beats: $f_{\text{beat}} = |f_1 - f_2|$ ; the note heard has frequency $\frac{f_1 + f_2}{2}$ .
Doppler: $f' = f\,\frac{v \pm v_o}{v \mp v_s}$ , upper signs for approach.
A moving source changes the wavelength; a moving observer changes the rate at which wavefronts are met.

Sources and credits

17.7: Beats and 17.8: The Doppler Effect, from University Physics Volume 1 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
17.4: Doppler Effect and Sonic Booms, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Doppler Effect, Bozeman Science, embedded from YouTube.

The beats experiment, the three diagrams, the worked examples and the practice exercises are written by Ganzaa.

1The note that drops as a moto passes

2Experiment 4.1: hearing beats

3Explaining beats

4The Doppler effect