12-1

Chapter 12 Sound Waves

We study the properties and detection of a particular type of wave – sound waves.

A speaker generates sound. The density of the air changes as the wave propagates.

Notice that the displacement maxima and minima occur where the pressure variation is zero and

the pressure variation maxima and minima occur where the displacement is zero.

The range of frequencies that can be heard by humans is typically taken to be between 20 Hz and

20,000 Hz. Most people struggle to hear the highest frequencies and that ability lessens with

age.

Speed of Sound

Recall

Inertia

ForceRestoringv

In fluids

12-2

Bv

The speed of the wave in a fluid (especially air) depends on temperature. In solids

Yv

Amplitude and Intensity of Sound Waves

A sound wave can be described either by talking about pressure or displacement. Since the

displacement creates the pressure change, there is a relationship between the amplitude of the

pressure p0 and the amplitude of the displacement s0. For a harmonic sound wave the relation is

00 svp

A larger amplitude wave appears louder, but the relation between amplitude and loudness is very

complicated. Loudness is subjective and depends on the response of the ear and the brain.

Usually the intensity and not the amplitude is used for loudness. Again for a harmonic wave

v

pI

2

2

0

“The most important thing to remember is that intensity is proportional to the amplitude

squared, which is true for all waves, not just sound.” (p 425)

12-3

Decibel Scale

The perception of hearing is roughly proportional to the logarithm of the intensity. The lowest

intensity of sound that can be heard by most people is

212

0 W/m100.1 I

I0 is called the threshold of hearing. It is used as the reference level for measuring sound

intensity. The sound intensity level in decibels is defined as

0

10log)dB10(I

I

(Be sure to practice with the decibel scale. Logarithms can be tricky.) An intensity level of 0 dB

corresponds to the threshold of hearing.

For incoherent sound waves with intensities I1 and I2, the total intensity is

21 III

If the sound waves are coherent, the waves can interfere and the intensity is between |I1 – I2| and

I1 + I2, depending on the phase relationship between the two waves.

Decibels can be used in a relative sense. The difference in two dB readings

12-4

1

210

0

110

0

21012

log)dB10(

log)dB10(log)dB10(

I

I

I

I

I

I

is related to the ratio of the intensities.

Standing Sound Waves

Recall that a standing wave is the superposition of two traveling waves. The wave reflects at the

boundary of the wave.

Pipe open at Both Ends

The boundary conditions are the same at both ends. Since the end is open to the atmosphere, the

pressure at the ends can not deviate much from atmospheric pressure. The ends are pressure

nodes. Pressure nodes are displacement antinodes.

From the diagram, the wavelengths satisfy

n

Ln

2

The frequencies

12

nfL

vn

vf

n

n

The index n is an integer and it can vary from 1, 2, etc.

12-5

Pipe Open at One End

The situation is different from the pipe opened at both ends. The closed end is a pressure

antinode. The air at the closed end is isolated from the atmosphere and the pressure can deviate

far from atmospheric. The air at the closed end is a displacement node since the rigid wall

prevents the air from moving.

From the diagram, the wavelengths satisfy

n

Ln

4

The frequencies

14

nfL

vn

vf

n

n

This time n has odd values only (1, 3, 5, etc.)

12-6

Problem 27 Two tuning forks A and B, excite the next-to-lowest resonant frequencies in two air

columns of the same length, but A’s column is closed at one end and B’s column is open at both

ends. What is the ratio of A’s frequency to B’s frequency.

Since A excites the pipe open at one end, only the odd harmonics are possible

14

nfL

vn

vf

n

n

Where n = 1, 3, 5, etc. Next to lowest resonant frequency refers to the second frequency. Here

that mean n = 3 and

143

4 L

v

L

vnf

A

For B, all the harmonics are possible since it is exciting a pipe open at both ends.

12

nfL

vn

vf

n

n

n = 1, 2, 3, etc. Next to lowest in this sequence corresponds to n = 2,

L

v

L

vnf

B2

22

Forming a ratio

4

3

2

2

4

3

22

43

v

L

L

v

L

vL

v

f

f

B

A

Timbre

In general, a musical instrument will produce sounds that are made of combinations of the

available frequencies. The lowest frequency in a complex sound wave is called the

fundamental and the other frequencies are called overtones. Since all the overtones are integral

multiples of the fundamental, they are also called harmonics.

The complex shape of the sound wave means that different instruments playing the same note

will have a different tone quality. You can recognize your favorite singer by the timbre of the

singer’s voice.

12-7

A complex periodic signal can be created by adding together a set of harmonic waves. The wave

having three frequencies 110, 165, and 220 Hz repeats at 55 Hz since each of these frequencies

are harmonics of 55 Hz.

In principle, any complex waveform can be decomposed into a series of harmonic waves. This is

called Fourier (or spectral) analysis. Limiting our study to harmonic waves actually includes all

waveforms.

12-8

Human Ear

The physiology of the ear is detailed in the text. Please read it.

A brief overview of the ear: http://www.youtube.com/watch?v=p3Oy4lodZU4

The perception of loudness depends on frequency.

Pitch is the perception of frequency. Higher pitch means higher frequency (and shorter

wavelengths).

Beats occurs when two sound waves are close in frequency. It is very useful for tuning

instruments. The beat frequency is the difference in the two frequencies

12-9

21 fffbeat

The Doppler Effect

Probably best explained in http://www.youtube.com/watch?v=Y5KaeCZ_AaY

Or not! Maybe this will work? http://www.youtube.com/watch?v=yWIMWqkcRDU

The Doppler effect is the change in observed frequency (pitch) resulting from the motion of the

sound source and/or sound observer. It can be used to measure the speed of a moving car (or

baseball).

In front of the moving source, the wave crests are closer together and the frequency is higher.

Behind the moving source, the wave crests are further apart and the frequency is lower.

S

S

O fvv

f

/1

1

Important: vS > 0 for a source moving in the direction of the wave.

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An observer moving relative to a stationary source will experience a different frequency. An

observer moving towards the source will experience a higher frequency and an observer moving

away from the source will experience a lower frequency.

SOO fvvf )/1(

Important: vO > 0 for an observer moving in the direction of the wave.

If both the source and observer move

S

S

OO f

vv

vvf

/1

/1

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The sign convention is given for the individual cases. You must get the signs right to get the

correct answer!!

Shock Waves

When the source moves faster than the speed of sound, the wave crests pile on top of each other

and a large amplitude wave occurs. For airplanes, this is called a sonic boom.

Shock waves: http://www.youtube.com/watch?v=-d9A2oq1N38

Echolocation and Medical Imaging

Sound and echoes are used to locate objects. Here is an animation explaining sonar:

http://www.youtube.com/watch?v=w_q2dqUdi8U

Ultrasound can be used to image structures inside the body.

12-12

High frequency sound is used since higher frequencies will have shorter wavelengths. Short

wavelengths diffract less around small obstacles.

Time for the end of the semester music festival.