Acoustic phenomena

Everything you hear — the pitch of a temple bell, the roar of a bus on the Ring Road, the rising whine of an ambulance as it passes — is a pressure wave arriving at your eardrum. This unit turns the everyday words loud, shrill and sweet into measurable physical quantities, and then works out what happens to a note when the source or the listener is moving.

9.1 Sound waves: pressure amplitude

A sound wave can be described in two equivalent ways: by the displacement of the air particles, or by the excess pressure — the small amount by which the pressure rises above or falls below the steady atmospheric value.

Let a plane progressive wave travel along x with displacement

Consider a thin layer of air of original thickness \Delta x. When the wave passes, its two faces move by different amounts, so its new thickness is \Delta x + (\partial y/\partial x)\Delta x. The volume strain is therefore \partial y/\partial x, and by the definition of the bulk modulus B the excess pressure is

Two consequences matter. First, \Delta P contains \cos where y contains \sin: the pressure wave is a quarter of a wavelength out of step with the displacement wave. Where the displacement is zero the pressure change is greatest, and vice versa. Second, for a given displacement amplitude the pressure amplitude grows in proportion to the frequency.

The ear is astonishingly sensitive. At 1 kHz the faintest audible sound has a root-mean-square pressure of only about 2\times10^{-5}\ \text{Pa} — one five-thousand-millionth of atmospheric pressure — which corresponds to a displacement amplitude of roughly 10^{-11}\ \text{m}, smaller than the diameter of an atom.

9.2 Characteristics of sound: intensity, loudness, quality and pitch

A musical note has three characteristics you can hear — loudness, pitch and quality — and each is governed by a measurable physical quantity.

Intensity

For a plane progressive wave the intensity works out to

So intensity depends on the square of the amplitude and the square of the frequency, and on the medium through \rho v. For a point source radiating power P uniformly in all directions, the energy at distance r is spread over a sphere of area 4\pi r^{2}:

This inverse-square law is why you must stand close to hear a soft speaker.

Intensity level and the decibel

The ear can cope with intensities spanning a factor of 10^{12}, so a logarithmic scale is used. Taking the standard reference I_0 = 10^{-12}\ \text{W m}^{-2} (roughly the threshold of hearing at 1 kHz), the intensity level is

Sound	I (W m⁻²)	\beta (dB)
Threshold of hearing	10^{-12}	0
Rustling leaves, soft whisper	10^{-10}	20
Quiet room at night	10^{-8}	40
Normal conversation	10^{-6}	60
Busy Kathmandu street	10^{-4}	80
Motorcycle horn at 1 m	10^{-1}	110
Threshold of pain	1	120

Loudness

Loudness is the sensation produced in the ear. It grows with intensity, but not in proportion to it: by the Weber–Fechner law the sensation is proportional to the logarithm of the stimulus,

Loudness also depends on frequency — the human ear is most sensitive between about 1 kHz and 5 kHz, so a 100 Hz note and a 3 kHz note of the same intensity do not sound equally loud. Loudness is measured in phon or sone, intensity level in decibels; they are not the same thing.

Pitch

Pitch is the sensation of how high or low a note is, and it is decided by frequency: the higher the frequency, the higher the pitch. A normal human ear responds from about 20 Hz to 20 000 Hz. Below 20 Hz is infrasonic (earthquake waves, elephant calls); above 20 kHz is ultrasonic (bat and dolphin echolocation, medical scanning).

Quality (timbre)

Two instruments playing the same note at the same loudness still sound different. A real instrument produces the fundamental plus a set of overtones, and the number of overtones and their relative intensities decide the shape of the resulting waveform. That shape is the quality or timbre.

Characteristic	Physical quantity	Depends on
Loudness	intensity	square of amplitude, 1/r^{2}, sensitivity of ear
Pitch	frequency	frequency of the fundamental
Quality	waveform	number and relative intensity of overtones

9.3 Doppler's effect

Situation	Apparent frequency
Source approaches, observer at rest	f' = \dfrac{v}{v-v_s}f
Source recedes, observer at rest	f' = \dfrac{v}{v+v_s}f
Observer approaches, source at rest	f' = \dfrac{v+v_o}{v}f
Observer recedes, source at rest	f' = \dfrac{v-v_o}{v}f
Both approach each other	f' = \dfrac{v+v_o}{v-v_s}f
Both recede from each other	f' = \dfrac{v-v_o}{v+v_s}f

When there is no Doppler effect. If the source and observer move with the same velocity in the same direction, their separation is constant and f' = f. If the motion is perpendicular to the line joining them — the instant a train passes you broadside — there is no shift either. And a steady wind produces no change for a stationary source and observer, because it increases the speed of sound and the wavelength in exactly the same proportion.

Applications. Traffic police speed guns and weather radar; SONAR for measuring the speed of a submarine or shoal of fish; Doppler ultrasound (echocardiography) to measure the speed of blood in an artery; and, in astronomy, the red shift of light from distant galaxies, which is the evidence that the universe is expanding.

Chapter summary

• Excess pressure \Delta P = -B\,\partial y/\partial x; the pressure amplitude is P_0 = Bak = \rho v\omega a = 2\pi\rho v f a.
• The pressure wave is \lambda/4 out of phase with the displacement wave: a displacement node is a pressure antinode.
• Intensity I = \frac{1}{2}\rho v\omega^{2}a^{2} = P_0^{2}/2\rho v, in W m⁻²; for a point source I = P/4\pi r^{2}, so I\propto 1/r^{2}.
• Intensity level \beta = 10\log_{10}(I/I_0) dB with I_0 = 10^{-12}\ \text{W m}^{-2}. Doubling I adds 3 dB; doubling the distance subtracts 6 dB.
• Loudness (Weber–Fechner, L = k\log I) is the sensation; pitch is set by frequency; quality is set by the overtones present, i.e. by the waveform.
• Doppler effect: f' = \dfrac{v+v_o}{v-v_s}f, with v_o and v_s counted positive when the motion brings source and observer closer.
• There is no Doppler shift for motion perpendicular to the line of sight, for equal velocities in the same direction, or for a steady wind.

Practice questions

Group A — Multiple choice (1 mark each)

1. The pressure amplitude of a sound wave of displacement amplitude a and angular frequency \omega is [1]a\rho v ab\rho v\omega ac\rho v\omega^{2}ad\rho\omega a
2. The displacement wave and the pressure wave in air differ in phase by [1]a0b\pi/4c\pi/2d\pi
3. If the intensity of a sound is doubled, its intensity level increases by about [1]a2 dBb3 dBc6 dBd10 dB
4. The quality of a musical note depends on [1]afrequencybamplitudecnumber and intensity of overtonesdspeed of sound
5. A source and an observer move with the same velocity in the same direction. The apparent frequency is [1]agreater than fbless than fcequal to fdzero

Answers to Group A

1. (b) — P_0 = Bak and B=\rho v^{2}, k=\omega/v, giving \rho v\omega a.

2. (c) — \Delta P \propto \cos where y\propto\sin, a quarter cycle apart.

3. (b) — 10\log_{10}2 = 3.0\ \text{dB}.

4. (c) — the harmonic mixture fixes the waveform, and the waveform is the quality.

5. (c) — the separation does not change, so no wavefronts are gained or lost.

Group B — Short answer (5 marks each)

1. Show that the pressure amplitude of a sound wave is P_0 = \rho v\omega a and explain why the pressure wave leads the displacement wave by a quarter of a wavelength. [5]
2. Distinguish between intensity, intensity level and loudness of sound, giving the unit of each. [5]
3. The intensity level of a sound is 70 dB. Find its intensity, and the new intensity level if the intensity is increased 100 times. [5]
4. Explain how the pitch and the quality of a note are related to the physical properties of the sound wave. Why do a bansuri and a sarangi sounding the same note sound different? [5]
5. A whistle of frequency 500 Hz is sounded on a car moving away from a stationary listener at 25\ \text{m s}^{-1}. Taking v = 340\ \text{m s}^{-1}, find the frequency heard. [5]

Answers to Group B

1. With y = a\sin(\omega t-kx), \Delta P = -B\,\partial y/\partial x = Bak\cos(\omega t-kx), so P_0 = Bak. Using B=\rho v^{2} and k=\omega/v gives P_0 = \rho v\omega a. The cosine leads the sine by \pi/2, which in space is a quarter wavelength.

3. \beta = 10\log_{10}(I/I_0) = 70 gives I/I_0 = 10^{7}, so I = 10^{7}\times10^{-12} = 1.0\times10^{-5}\ \text{W m}^{-2}. Multiplying the intensity by 100 = 10^{2} adds 10\log_{10}10^{2} = 20\ \text{dB}, giving \beta' = 90\ \text{dB}.

5. The source recedes, so v_s = -25\ \text{m s}^{-1}:

f' = \frac{v}{v+25}f = \frac{340}{365}\times500 = 465.8\ \text{Hz}

The whistle sounds about 34 Hz flatter than it really is.

Group C — Long answer (8 marks each)

1. (a) State the Doppler effect and derive an expression for the apparent frequency when both the source and the observer are in motion along the line joining them. [5](b) Give two situations in which there is relative motion but no Doppler shift, and state two practical applications of the effect. [3]
2. A point source emits sound uniformly with a power of 0.50 W.(a) Find the intensity and the intensity level at a distance of 5.0 m. [4](b) At what distance does the intensity level fall to 60 dB? [4] (Take I_0 = 10^{-12}\ \text{W m}^{-2}.)

Answer to Group C question 2

(a)

I = \frac{P}{4\pi r^{2}} = \frac{0.50}{4\pi(5.0)^{2}} = \frac{0.50}{314.16} = 1.59\times10^{-3}\ \text{W m}^{-2}

\beta = 10\log_{10}\frac{1.59\times10^{-3}}{10^{-12}} = 10\log_{10}(1.59\times10^{9}) = 92.0\ \text{dB}

(b) A level of 60 dB means I' = 10^{6}I_0 = 1.0\times10^{-6}\ \text{W m}^{-2}. From I' = P/4\pi r'^{2},

r'^{2} = \frac{P}{4\pi I'} = \frac{0.50}{4\pi\times1.0\times10^{-6}} = 3.979\times10^{4}\ \text{m}^{2}

r' = 199\ \text{m}

So the level drops from 92 dB to 60 dB — a factor of about 1600 in intensity — between 5 m and about 200 m, exactly as the inverse-square law requires.