Chapter 19 Sound waves19-1 Properties of Sound waves19-2 Traveling sound waves

19-4 Power and intensity of sound waves 19-3* The speed of sound

19-5 Interference of sound waves19-6* Standing longitudinal waves19-7* Vibrating system and sources of sound19-8 Beats19-9 The Doppler effect

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.

19-1 Properites of sound waves

Fig19-2

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.

x

x

x

x

vmu

0P

0

mS

m

mp

x),(),( txtx 0

),(),( txPPtxP 0

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

Under certain conditions

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

)sin( tkxm

)sin( tkxPP m

)( vvP

B

v

m

B

ρP

B

Pρ

v

vρ

v

v

v

mv

v

mρ 0

2Δ

Δ)

Δ(

ΔΔΔ

19-2 Traveling sound waves

P

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

BPmm

0 (19-3)

x=0

x

s(x,t)

s(x+ ,t)xx'

xx x’ x’’

xA

m

0

The undisturbed density of isx

A is the corss-sectional area.

])([])([ tx,sxxtx,xsxx'-x'x'x'

]),(),(

[1δx

txstxxsx

.1),1(1' 0

0

xs/ifxs/-ρxs/

ρ

xA

δmρ

xs00 / (19-6)

or

]s

[1δx

x

(19-8)

kB

P

ks mm

m

0

)cos(),( tkxstxs m

)(sin)( m ωtkxut

sx,tux

B

Pvu m

m

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

)sin(),(

tkxtx

x

s

0

m

0

(19-9)

v= /k is the velocity of sound wave.

is the velocity of oscillation of an element in fluids.)( tx,μx

(19-4)0

Bv

As in the case of the transverse mechanical wave, the speed of a sound wave depends on the ratio of an elastic property of the medium and an inertial property. For a 3D fluid,

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

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

( )xcmxext MaF ,,

19-3* The speed of sound

Csmvair20,/343

0

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 ,P

)sin( tkxPAPAF mx

The power delivered by the element is:

)(sin 2 tkxuPAFuP mmxx

v

PAP m

av 2

)( 2 (19-18)

Average over any number of full cycles.

)(sin)( m ωtkxux,tux

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 )).

0

log10I

ISL

0I212 /10 mw

v

P

A

PI mav

2

)( 2

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.

0I

2/1 mw

声源 声强W/m2 声强级 dB 响度引起痛觉的声音 1 120

钻岩机或铆钉机 10-2 100 震耳交通繁忙的街道 10-5 70 响通常的谈话 10-6 60 正常耳语 10-10 20 轻

树叶的沙沙声 10-11 10 极轻引起听觉的最弱声音 10-12 0

几种声音近似的声强、声强级和响度几种声音近似的声强、声强级和响度

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:

222

/32.0)5.2(4

25

4mw

m

w

r

PI

dBmw

mw

I

ISL 115

/10

/32.0log10log10

212

2

0

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 . Thetotal pressure disturbance at point P is .

sourceP1r1s

2s2r

2P1P

21 PPP 1s

2s

Fig 19-6

19-5 Interference of sound waves

The type of interference that occurs at point P depends on the phase difference between the waves.

L

2

,2

|||| 2121 Lrrkkrkr

(19-22)

)sin( 1 tkrP

The phase difference:

21 rrL

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.

mL

)2

1( mL

)sin()]2/cos(2[

),(),(),('

21

tkxy

txytxytxy

m

We assume a train of

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

1) If the end is open,

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

2) If the end is closed, a pressure antinode (波腹 ) will form at the end.

(a)

(b)

(c)

(d)

close end

open end

Fig 19-7

19-6* Standing longitudinal waves

n

Ln

2 , n=1,2,3…

n

Ln

4 , 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 frequency matches one of the natural frequency of the system, which are determined by the length of the tube (L).

180

0P

Notes:

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.

(a)The vibrating system has a large number of

19-7* Vibrating system and sources of sound

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:

321 fff

1f

3f2f

1nffn

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.

(b) Why do some vibrating systems produce pleasant sounds while others produce harsh ( 刺耳的 ) or discordant ( 不和谐 ) sounds?

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.

tPtP m 11 sin)( tPtP m 22 sin)(

)~( 21

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

The resultant pressure is

(19-32)

Set (19-33)

221

av

221

amp

)sin()cos(2)( ttPtP avampm

ttPtPtPtP m 2

)(sin]

2

)(cos2[)()()( 2121

21

(19-34)

(19-35)

In Fig19-13, the ear would perceive a toneat a frequency .

Since ~ , the amplitude frequency is small. The amplitude fluctuates slowly.

amp

1 2

21 av

av

(a)

(b)

Fig 19-13

)(2 tP

)(1 tP

t

t

t

)sin()cos(2)( ttPtP avampm 21 ~

21 ~~ av

|)cos(2| tP ampm

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)

)cos( tamp

212 ampbeat

)sin()cos(2)( ttPtP avampm

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 beats per 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 beats per second in the fundamental mode increases. What was the original frequency of the fundamental?

Sample Problem 19-5

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.

1.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 S at rest and an observer O moving toward the source at a speed .

*S

Fig 19-140

0v

0v

0

sv

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.

tv0

vt

2. Moving source, observer at rest

In this case, the wavelength is shortened from to .

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

v

vvf 0'

fv

vvvv

t

tvvtf 00

0'

(19-39)

(19-38)

Tvs '

The frequency that is actually heard is 'f

1 23 4 5 6

……S1

sv

00

v

Fig 19-15

'S7

See 动画库 \ 波动与光学夹 \2-20 多普勒效应 3

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)fvv

vf

s

)('

fvv

v

t

vtf

s

)(/ '

'

Tvs '

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

fvv

vvf

s

)( 0'

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 theobject. This is the basic principle of radar monitors,

and it is also used to track satellites.

tsv

1P2P

vt

1 23 4 56

sv.

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

sv

Wavefront when .vvs

Wavefront when .vvs

See 动画库 \ 波动与光学夹 \2-30 冲击波

Doppler cooling in Bose-Einstein Condensate (BEC)

•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.

...Created in 1995，Won Nobel Prize in 2001 Predicted in 1924...

A. Einstein S. N. Bose E. A. Cornell W. Ketterle C. E. Wieman

•Doppler cooling (Laser cooling) •Evaporative cooling

10-9 K obtained 10-6 K obtained

What’s the conditions to observe BEC?Extremely low temperature;The atoms are still in gas state

Sample problem 19-6The 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

Solution: Using Eq(9-44)

(a) Here

(b)

00 v smvs /29Hzff

vv

vf

s

1229)29343

343('

0sv smv /290

HzHzfv

vvf 1220)1125(

343

293430'

(c)

(d)

smvvs /5.140

Hzfvv

vvf

s

122411255.14343

5.143430'

smvs /38smv /90

Hzff 123238343

9343'