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## Chapter 19 Sound waves

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### Chapter 19 Sound waves

19-1 Properties of Sound waves

19-2 Traveling sound waves

19-3* The speed of sound

19-4 Power and intensity of sound waves

19-5 Interference of sound waves

19-6* Standing longitudinal waves

19-7* Vibrating system and sources of sound

19-8 Beats

19-9 The Doppler effect

19-1 Properites of sound waves

When we discuss sound waves, we normally mean longitudinal wave in the frequency range 20 Hz to 20,000 Hz, the normal range of human hearing.

For simplification, we will consider the sound wave in 1D case.

x

x

v

x

In Fig19-2, as the piston (活塞) moves back and forth, it alternately compresses and expands (使稀薄)) the air next to it.

This disturbance travels

down the tube as a sound wave.

Under certain conditions

It is convenient to use density and pressure to describe the properties of fluids.

Fig19-2

1) Let us assume that the pistol is driven so that the density and pressure of air in the tube will vary as a sine function.

(19-1)

(19-2)

2) What’s the relationship between and ?

From the definitions of bulk modulus(体模量/膨胀系数) (Eq(15-5)) and density , when m is fixed, we have

s(x+ ,t)

x

x

x’

x’’

x=0

or

(19-3)

3) How to find the displacement of an element of gas inside the tube?

The undisturbed density of is

A is the corss-sectional area.

(19-6)

Combine Eqs. (19-1) and (19-6), we have:

(19-8)

(19-9)

isthe velocity ofoscillation of an element in fluids.

v= /k isthe velocity of sound wave.

As in the case of thetransverse mechanical wave, the speed of a sound wave depends onthe ratio ofanelastic propertyof the medium and aninertial property. For a 3D fluid,

(19-4)

Note:1) B is the bulk modulus, is the mass density.

2) Use Newton’s law for a system of particles.

( )

19-4 Power and intensity of sound waves

As the wave travels, each fluid element exerts a force on the fluid element ahead of it. If the pressure increase in the fluid element is ,

The power delivered bythe element is:

(19-18)

Average over any number of full cycles.

Intensity I: (19-19)

The response of the ear to sound of increasing

intensity is approximately logarithmic.

One can define a logarithmic scale of intensity called the “sound level SL”

(19-20)

Where is a reference intensity, which is

chosen to be (a typical value for the

threshold of human hearing(听觉阈)).

The unit of the sound level is “decibels” (dB).

A sound of intensity (听觉阈)has a sound level of 0 dB.

The sound at the upper range of human hearing, called the threshold of pain (痛觉阈) has an intensity of and a SL of 120 dB.

Sample problem 19-2

Spherical sound waves are emitted uniformly in all

directions from a point source, the radiated power

P being 25 w. What are the intensity and the sound

level of the sound wave at a distance r=2.5m from the source?

Solution:

Fig19-6 shows two loudspeakers

driven from a common source.

At point P the pressure variation

due only to speaker is and

that due to alone is . The

total pressure disturbance at

point P is .

19-5 Interference of sound waves

Fig 19-6

P

source

The type of interference that occurs at point P

depends on the phase difference between the

waves.

When ( m=0,1,2,…...) (19-23),

The intensity reaches a maximum value, forming constructive interference.

When destructive interference occurs. The intensity has a minimum value.

(19-22)

19-6* Standing longitudinal waves

Fig 19-7

, n=1,2,3…

We assume a train of

sine waves travels down a tube( Fig19-7).

1) If theend is open,

the wave at the end will behave as a pressure node(波节);

2)If theend is closed, a pressureantinode(波腹) will form at the end.

(a)

open end

(b)

close end

(c)

(d)

, n=1,3,5...

See动画库\波动与光学夹\2-16纵驻波

a).For open end, the longitudinal pressure wave is reflected with a phase change of , because the pressure at the open end must at the value , same as the environment’s.

In this case, it likes the string fixed at both ends.

b).For the closed end, the pressure can vary freely.

c).The superposition of the original and reflected

waves gives a pattern of standing waves.

d).Resonance can happen, when the driving frequencymatches one of the natural frequency of the system, which are determined by the length of the tube (L).

19-7* Vibrating system and sources of sound

We have already studied the propagation of the sound wave, and now to understand the nature of the sound we must study the vibration system that produces it.

We can classify musical instruments into three categories: those based on vibration string; those based on vibration column of air, and more complex system including plates, rods, and membranes.

- The vibrating system has a large number of

natural vibrational frequencies. We write these in

ascending order, so that .

The lowest frequency, is called the “fundamental

frequency(基频)”, and the corresponding mode of

oscillation is called the “fundamental mode”. The

higher frequencies are called “overtones(泛音)”, with being the first overtone, the second overtone, and so on. In some systems:

(b) Why do some vibrating systems produce

pleasant sounds while others produce harsh (刺耳的) or discordant (不和谐) sounds?

When several frequencies are heard simultaneously, a pleasant sensation results if the frequencies are in the ratio of small whole numbers(整数), such as 3:2 or 5:4.

19-8 Beats (节拍)

- Previously, we have considered the “interference in space”.
- Now we shall discuss “interference in time”.
- We consider two waves which have nearly the same frequency.

We have chosen the phase constants to be zero, and same amplitudes.

In Fig19-13, the ear

would perceive a tone

at a frequency .

Since ~ , the amplitude frequency

is small. The amplitude

fluctuates slowly.

t

(a)

t

t

(b)

Fig 19-13

A beat--- that is, a maximum intensity—occurs,

whenever equals +1 or -1 ,since the

intensity depends on the square of the amplitude.

Each of these values occurs once in each cycle of

the envelope, thus

(19-36)

A violin string that should be tuned to concert A (440Hz) is slightly mistuned. When the violin string is played in its fundamental mode along with a concert A tuning fork, 3 beatsper second are heard. (a) What are the possible values of the fundamental frequency of the string? (b)Suppose the string were played in its first overtone simultaneously with a tuning fork with 880Hz. How many beats per second would be heard? (c) When the tension of the string is increased slightly, the number of beatsper second in the fundamental mode increases. What was the original frequency of the fundamental?

19-9 The Doppler effect

In a paper written in 1842, Doppler (1803~1853)

called attention to the fact that the color of a

luminous body must be changed by relative motion

of the body and the observer. This “Doppler effect”

as it is called, applies to waves in general.

- Moving observer, source at rest

Suppose the source and observer move along the line joining them.

See动画库\波动与光学夹\2-21Doppler Effect A.exe

Let us adopt a reference frame at rest in the medium through which the sound travels.

Fig19-14 shows a source of sound Sat rest and an observer Omoving toward the source at a speed .

*

S

0

Fig 19-14

If the frequency of wave is f, what is the actually one f ’ heard by the ear?

An observer at rest in the medium would receive waves in time t, where v is the speed of sound in the medium and is the wavelength.

Because of the motion toward the source, the observer receives additional waves in the same time t.

The frequency that is actually heard is

(19-38)

2. Moving source, observer at rest

In this case, the wavelength is shortenedfrom to .

* When the observer is in motion away from the source,

(19-39)

The frequency of the sound heard by the observer

Is given by

(19-40)

* If the source moves away form the observer, the

frequency heard is

(19-41)

3. If both source and observer move through the

transmitting medium,

(19-44)

Where the upper signs (+ numerator, -denominator)

correspond to the source and observer moving toward the other and the lower signs in the direction away from the other.

4. If a source of sound is moved away from an

observer and toward a wall, the observer hears

two notes (音符) of different frequency. The note heard directly from the receding source is lowered in pitch by the motion.The other note is due to the

waves reflected from the wall, and this is raised in

pitch. The superposition of these two wave trains

produces beats.

A similar effect occurs if a wave from a stationary

source is reflected from a moving object. The beat

frequency can be used to deduce the speed of the

object. This is the basic principle of radar monitors,

and it is also used to track satellites.

How about the wavefront if is larger than v, sound speed?

.

6

5

4

3

2

1

Wavefront when .

Wavefront when .

See动画库\波动与光学夹\2-30冲击波

Doppler cooling in Bose-Einstein Condensate (BEC)

Predicted in 1924...

E. A. Cornell

W. Ketterle

C. E. Wieman

A. Einstein S. N. Bose

- BEC: A given number of particles approach each other sufficiently closely and move sufficiently slowly they will together convert to the lowest energy quantum state.

What’s the conditions to observe BEC?

Extremely low temperature;

The atoms are still in gas state

- Doppler cooling (Laser cooling)

- Evaporative cooling

10-9 K obtained

10-6 K obtained

Sample problem 19-6

The siren （警报器）of a police car emits a pure tone at a

frequency of 1125 Hz. Find the frequency that you

would perceive in your car.

(a) your car at rest, police car moving toward you at 29 m/s;

(b) police car at rest your moving toward it at 29 m/s

(c) you and police car moving toward one another at 14.5 m/s

(d) you moving at 9 m/s, police car chasing behind

you at 38 m/s

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