Sound Waves

Sound WavesSound is a longitudinal wave in a mediumAudible range of human ear is 20 to 20000Hz

0

uF ma pA cA t

t

P1 P2

0

uF ma pA cA t

t

0

V A t u uV A t u

V A tc c

2 20 0

0

u Vp c c

c V

0 1 1

0 1

V A tc V A tc

V V V A t u

20

0 0 0

V V Bp c B c

V V

2 20 0

0

u Vp c c

c V

0

Vp B

V

Bulk

modulus

Please estimate the speed of sound in water, the bulk modulus of fresh water at 20oC is about 2.28×106 Pa.

A.150 B. 340 C. 710

D. 1480 E. 3000 m/s

63

0

2.18 101.48 10 /

1000

Bc m s

VdV

dpB pB

V

p

dV

dpVdppdV isothermal ,0

M

RTpv

pBV

p

dV

dpdVpVVdp

constpV

a

,0

.

1

空氣的 bulk modulus 若為等溫過程， pV=nRT ，

用此結果代入所得之聲速較實際值小。在聲音的傳遞，速度很快，空氣和周圍來不及做熱交換，要考慮絕熱過程，即

。

max( , ) cos( )y x t y kx t

max0

sin( )V A y

p B B Bky kx tV A x

Characteristic impedance or specific acoustic resistance

Energy in Wave Motion

( , ) ( , ) ( , )

( , ) ( , ) ( , )( , ) ( , )

P x t F x t v x t

P x t F x t v x tI p x t v x t

A A

The instantaneous rate of doing work at the point x and time t is

For sinusoidal wave with wave function given as

max( , ) cos( )y x t y kx t

The speed of the displacement would be

max( , ) / sin( )v x t dy dt y kx t

Energy in Wave Motion

( , ) ( , )I p x t v x t

The instantaneous rate of doing work at the point x and time t is

2 2max

2 2 20 max

sin ( )

sin ( )

I Bk y kx t

c y kx t

0

Bc

k

max( , ) sin( )v x t y kx t

max sin( )p Bky kx t

2 2 2max 0 max

1 1

2 2I Bkωy cω y

intensity of a sinusoidal sound wave

B

p

v

pI

22

2max

2max

0

logdB10I

Iβ

definition of sound intensity level

B

v max maxp Bky

Sound Velocity and Gauge Pressure

The maximum sound wave pressure human can tolerate is 28 Pa above the atmospheric pressure. Find the max displacement if frequency is 1000Hz. Given speed of sound is 344m/s

max

3

6

2

28

(343 / )(1.21 / )(2000 )

1.1 10 11

p py

v v f

Pa

m s kg m

m m

A.11 nm B. 0.11μm C. 11 μm

D. 0.11 mm E. 1.1 mm

Sound Attenuation

Why distant thunder is heard as a low rumble while nearby one is a higher

frequency crash.High or low frequency is better for

long distance communication?

Which thunder is heard as a higher frequency crash?

A. distant one B. nearby one C. same

Sound Attenuation

max sin( )p Bky kx t

Which material will have largest displacement under same sound pressure variation?

A. Steel B. Alumina C. water D. Air

Sources of Musical Sound

What factors would affect the frequency of sound generated from a guitar string?

The Doppler effect

Moving listener and stationary source

1 LL s

vf f

v

SS

LL f

vv

vvf

Moving source and moving listener

Shack Wave (Sonic Boom)

http://en.wikipedia.org/wiki/Sound_barrier

H.W. Estimate the speed of the fighter.

ttss

ttsttssss

tsstss

m

mm

mm

coscos2

)(),(

)(cos)(cos2coscos

cos ,cos

2121

2121

2121

2121

2121

2211

21beat

21beat 2

fff

2smcos't

拍音可利用來做樂器的調音。人類可感覺之拍音約 <7Hz

Beats Beats 拍音拍音

People used Doppler flow meter to measure the blood flow rates. The speed of sound in blood is known to be 1570 m/s and a typical red blood cell has a radius of 5μm. Please estimate what would be the frequency for the flow meter?

If the frequency shift measured by the Doppler flow meter is 100Hz (the meter was operated at 107Hz), please find the average velocity of blood flow.

A.3.5 mm/s B.1.5 cm/s C. 23 cm/s

D. 65 cm/s E. 120cm/s

G

v

橫波 (transverse wave) 彈性體所產生之形變為剪應變 (shear strain)

Y

v

縱波 (longitudinal wave) 彈性體所產生之形變為拉伸應變 (tensile strain) 恢復力為拉伸應力 (tensile stress)

m/s 3000kg/m1083.7

Pa103.833

10

G

v

(a) 鋼之密度為 7.83×103 kg/m3 ， G=8.3×1010 Pa ，橫波在鋼中傳遞之速率為

(b) 鑽石的橫波聲速高達 12800 m/s ，密度為 3.51×103 kg/m3 ，他的 shear modulus 為

2vG =5.75×1011 Pa

Can bat tells what’s its prey?

The echo is modulated by the beating wing of prey, as a result, the motion of wings impose a “warble” to the echo.

Insect has its own characteristic flapping frequency

In air or water, which is easier to detect via Doppler shift?

The otolith organs are the primary means by which we sense linear acceleration of the head and the orientation of the head with respect to Earth's gravity. Each of these otolith organs contains a small sensory area known as a macula. Each macula contains several thousand vestibular hair cells. The cilia are embedded in a gelatinous matrix called the otolithic membrane. This membrane contains small piles of calcium carbonate crystals (CaCO3), called otoliths, a word which literally means "ear stones."

Hearing

Hearing

There is a problem for terrestrial animals.

In most animal, a change in pressure is heard

Energy propagation at interfaces.

Newton’s 3rd law at the interfaces

i r tp p p

i r tp p p

i r t

i t r

A A A

A A A

0 sin( )p c A kx t

sin sin

sini a a r a a

t w w

A c t A c t

A c t

i a a r a a t w wA c A c A c

w wr t i

a a

cA A A

c

2 a ai t

a a w w

cA A

c c

The amplitude of sound transmitted into water is only about 1/1800th that in air

i r t

i t r

A A A

A A A

w wr t i

a a

cA A A

c

2w w w wi t t i i t t

a a a a

c cA A A A A A A

c c

2

4 a a w wt i

a a w w

c cI I

c c

2 2 20 sin ( )I c A kx t 2 a a

i ta a w w

cA A

c c

2 2 2 22

2 22 2

4

4 4

a a a at w w t w w i

a a w w

a a w w a a w wa a i i

a a w w a a w w

c cI c A c A

c c

c c c cc A I

c c c c

2

4 a a w wt i

a a w w

c cI I

c c

The intensity of sound transmitted from air into water is about ?

A.50% B. 20% C. 1%

D. 0.1% E. 0.01%

How can terrestrial animal hear?

The Anatomy of Human Ear

A eardrum = 43mm2 A stirrup = 3.2 mm2 V fluid =1500m/s

PaB fluid91018.2 PaP earinner 4.0)max( mAinner

11104.4

http://www.sfu.ca/sca/Manuals/ZAAPf/e/equal_loud_cont.html

Sound Amplifier

When the size of an air bubble in water is smaller than sound wavelength, it can function as a sound amplifier

More realistic model is to treat the bubble as a damped oscillator system

/ 3600i t a a a w w w

a W

p p c A c A

A A