The propagation of an vibrationThe propagation of an vibrationIn an elastic mediumIn an elastic medium

Wave: Any disturbance propagates with time from one region of space to another.

1. Mechanical waves: earthquake waves, sound wave, water wave, …

The types of waves

2. Electromagnetic waves: light, microwaves, x-rays...

3. Gravity waves:

4. Matter waves: electron, atom, molecule…….

Ogle :

§15-1 Generation & Propagation of a Mechanical Wave 机械波的产生和传播

§15-3 Wave Function of a Plane SHW 平面简谐波的函数

§15-4 Energy Energy Flow and Wave Intensity 波的能量 波动强度

§15-2 Wave Speed & Elasticity of the Medium 机械波的传播速度 媒质的弹性

§15-5 Huygen’s Principle 惠更斯原理

§15-8 The Doppler Effect 多普勒效应

§15-6 Principle of Superposition of Waves Interference of Waves 波的叠加原理 波的干射

§15-7 Standing Waves 驻波

I. Mechanical waveI. Mechanical waveThe propagation of a mechanical vibrationThe propagation of a mechanical vibrationin an elastic medium.in an elastic medium.

§15-1 Generation & Propagation of a mechanical Wave

The conditions of generation and propagation of a M-waves:

1. There must be a vibrating center called source of wave.

2. There must be an elastic an elastic medium propagating wave.

Transverse waveTransverse wave ：： the individual particles the individual particles of medium vibrate in the direction that is of medium vibrate in the direction that is perpendicular to the direction of wave perpendicular to the direction of wave propagating.propagating.

Longitudinal waveLongitudinal wave ：： the individual particles the individual particles of medium vibrate in the direction that is of medium vibrate in the direction that is parallel to the direction of wave propagating.parallel to the direction of wave propagating.

II. ClassificationII. Classification

Continuous waveContinuous wave

Pulse wavePulse wave

Moving waveMoving wave: The vibration states “travel” in : The vibration states “travel” in some direction.some direction.

Standing waveStanding wavepropagatepropagate

III.The mathematical description of moving III.The mathematical description of moving wave:---wave:---Wave functionWave function ：：

Choose Choose o o –source of wave –source of wave ，， xx----the direction the direction of wave propagatingof wave propagating ，， vv----wave speedwave speed

oo xx

yy

Assume, at time Assume, at time tt, that the displacement of the , that the displacement of the particle locating in particle locating in o o isis

tfy 0

vvyy00

vv

yyPP

xx

oo xx

yy vvyy00

vv

yyPP

xx

Then , at time Then , at time tt, the displacement of the particle, the displacement of the particle

locating in locating in PP

v

xt

= = the displacement of the particle locating in the displacement of the particle locating in o o

at time at time , ,

u

xtfy --Wave function propagating --Wave function propagating

in in ++x-directionx-directioni.e.i.e.

oo xx

yyuuyy00

uu

yy

Wave function propagating in Wave function propagating in - - x-direction x-direction isis

u

xtfy

Spherical waveSpherical wave

波 面波 面 波 线 波 面波 面 波 线

Plane wavePlane wave

IV.The geometric description of moving wave: IV.The geometric description of moving wave: wave front wave front andand ray ray

§15-2 Wave Speed & Elasticity of the Medium

tension or compressiontension or compression

Basic deformationBasic deformationshearshear

I. Deformation of elastic objectsI. Deformation of elastic objects

1.Elastic modulus1.Elastic modulus

（（ 11 ）） Bulk Bulk

modulusmodulus

体 变体 变

An object has volume An object has volume VV00 at the pressure at the pressure pp . .

Let Let 0V

V

--bulk modulus--bulk modulusDefinitionDefinition0V

Vp

B

The volume =The volume =VV00+ + V V at pressure at pressure

p+p+ p p..

pp

VV00VV00+ + VV

p+ p+ p p

----Volumetric strainVolumetric strain (( 体应变体应变 ))

（（ 22 ）） Tensile elastic modulus Tensile elastic modulus lS

ffll

The bar is pulled by The bar is pulled by ff,,

Stress(Stress( 应力应力 )) S

f

l

lTensile strain(Tensile strain( 线应变线应变 ))

长 变长 变

llS

f

Y

--Young’s modulus--Young’s modulusDefinition

s

f

N ----shear modulus----shear modulus

（（ 33 ）） Shear modulusShear modulus f

f

Shear stressShear stress(( 剪切应力剪切应力 ))S

f

D

dShear strainShear strain(( 切应变切应变 ))

A shearing force is exerted A shearing force is exerted on the faces of the object.on the faces of the object.

切 变切 变

Definition

DD

dd SS

NoteNoteliquid and gas produce bulk strain only. liquid and gas produce bulk strain only.

Solid can produce bulk, tensile, and shear strain.Solid can produce bulk, tensile, and shear strain.

2.2.Wave speed in elastic mediumWave speed in elastic medium

v v depends only on the depends only on the densitydensity and and elasticityelasticity of the medium in which wave propagates.of the medium in which wave propagates.

Wave speed(phase speedWave speed(phase speed 相速相速 )) vv ：：

-- the propagating speed of the phases of a vibration.

In some kind of uniform gas or liquid, longiIn some kind of uniform gas or liquid, longitudinal wave can be propagated only.tudinal wave can be propagated only.

Bv

In some kind of uniform solid, In some kind of uniform solid,

Transverse wave:Transverse wave: Nv

Longitudinal wave:Longitudinal wave: Yv

oror Bv

Longitudinal wave:Longitudinal wave:

In ideal gas,In ideal gas,

pv molMRT

::The ratio of molar capacityThe ratio of molar capacity

In a stretched string ,In a stretched string ,

Tv

T ： the tension in string ： the mass per unit length of the string

Transverse wave:Transverse wave:

Harmonic waveHarmonic wave ：： The propagation of simple harmonic motion The propagation of simple harmonic motion

in elastic mediumin elastic medium..

Wave functionWave function ：： The moving function of any The moving function of any particle of medium when there is a wave in the particle of medium when there is a wave in the medium.medium.

The source of wave and every segment of The source of wave and every segment of medium are moving in harmonic motion with medium are moving in harmonic motion with same frequency.same frequency.

§15§15-3 Wave Function of a Plane SHM-3 Wave Function of a Plane SHM

I. Wave function of plane harmonic waveI. Wave function of plane harmonic wave

Assume that the source of point Assume that the source of point OO is moving in is moving in SHMSHM

tAy cos0

v

That the state ( phase ) of point That the state ( phase ) of point OO

is propagated by the wave from O is propagated by the wave from O

to P needs time :to P needs time :v

x

x x

y

O

Pi.e.i.e.

)v

xt(y)t,x(y 0

)

v

xt(cosA

-- Wave function of plane harmonic waveWave function of plane harmonic wave

u

x x

y

O

P

SoSo the displacement of the displacement of point point PP at time at time tt

==the displacement of the displacement of point point OO at time at time

v

xt

i.e.i.e.

)()(v

tvxtt

v

xt

)(v

xxtt

),(),( ttxxytxy

--the vibrating state propagates the distance x along the wave ray during the time interval t

Wave function describes the propagating of the Wave function describes the propagating of the vibrating state.vibrating state.

DiscussionDiscussion

x

If If x x ==xx = = a constanta constant, then, then

)(ty])'

(cos[ u

xtAy

ux''Let Let

t

y

O

vibrating curve

thenthen

)'cos( tAy

--the motion of the particle at point --the motion of the particle at point xx is SHM is SHM

If If t t ==tt = = a constanta constant, then, then

])'(cos[ u

xtAy )(xy

Let Let '' t

)'cos(

u

xAy

Then Then

----the displacement of each particle on the the displacement of each particle on the wave ray at any timewave ray at any time t’t’

the wave pattern at time t

x

y

O

Other forms of Other forms of the wave function of plane hthe wave function of plane harmonic wavearmonic wave

22 T

v x t A ycos

x Tt A2 cos

x t A2 cosTv

Wave function of plane harmonic wave traveWave function of plane harmonic wave traveling in theling in the – –xx direction direction

])(cos[ vxtAy

Wave function of a spherical harmonic wave Wave function of a spherical harmonic wave

])(cos[),( vrtr

Atry

Wave function of a cylindrical harmonic wave Wave function of a cylindrical harmonic wave

])(cos[),( vrtr

Atry

II.The dynamic equation of mechanical wavesII.The dynamic equation of mechanical waves

2

2

2

2

y

t

y

Assume a general wave function is Assume a general wave function is )(v

xtfy

t

y

t

y

v

xt

x

y

x

y

andand

Let Let

y

Then Then

2

2

22

2 1

y

vx

y

y

v

1

2

2

22

2 1

t

y

vx

y

-- -- The dynamic equation of The dynamic equation of mechanical wavesmechanical waves

2

2

22

2 1

t

y

vx

y

can describe all kinds of waves.can describe all kinds of waves.

[[ExampleExample] A plane harmonic wave is traveling] A plane harmonic wave is traveling inin + +xx direction. direction. AmplitudeAmplitude==AA ，， T=18sT=18s , , =36=36mm. The particle at origin . The particle at origin OO is at its is at its equilibrium position and is moving toward the equilibrium position and is moving toward the +y direction+y direction when when t=t=00. Find . Find the wave the wave function, function, the oscillation equation at the oscillation equation at xx=9=9mm, , the wave pattern equation and the coordinates the wave pattern equation and the coordinates of all crests at of all crests at t t =3=3ss..

Solution Solution Assume the wave function is Assume the wave function is

])(cos[ vxtAythen the vibration equation of the particle then the vibration equation of the particle at point at point OO is is ：：

)cos()(0 tAty

According to initial conditions:According to initial conditions:

cos)0(0 Ay 0 sin)0(0 Av 0

We getWe get2

9

2 T T

v

m]2

)2

(9

cos[

xtAy

sm2

At At xx=9m=9m ， ， the oscillation equation isthe oscillation equation is

)9

cos( tAy (m)

9cos tA

At At t t =3s=3s ， ， the wave pattern equation isthe wave pattern equation is

(m))]13

(6

cos[ x

Ay

The coordinates The coordinates xx of crests should of crests should satisfy satisfy

2,1,0)112(3 kkx

We getWe get

1)]13

(6

cos[ x

kx

2136

　　

Kinetic energies Potential energies

Energies propagatingEnergies propagating

§15-4 Energy, Energy flow and §15-4 Energy, Energy flow and Wave intensityWave intensity

Wave propagating

The medium deformation

The particles in medium vibrating

I. Energy of waveI. Energy of wave Take Take a transverse wave generating and proa transverse wave generating and pro

pagating along a streched stringpagating along a streched string as an exa as an example.mple.

Take any segment Take any segment ABAB as a particle. as a particle.

Assume the wave function is Assume the wave function is

)(cosv

xtAy

A B

A

B

x

x

yl

Y

x

vv 22 yxl

The vibrating speed of the particle (The vibrating speed of the particle (ABAB) is) is

t

y

)(sinu

xtA

If the mass of If the mass of ABAB is is mm==xx ，， its vibrating its vibrating K-energyK-energy is is

2

2

1

t

ymEk

)(sin2

1 222

v

xtxA

Kinetic energyKinetic energy

xlTEP

When When AB AB AA B B, it possesses elastic P-energy, , it possesses elastic P-energy,

Potential energyPotential energy

22

2

1

x

yxv

xyx 22

the tension in string T=T=vv22

11

2

2

x

yx

1

2

11

2

2

x

yx

The total energy of The total energy of ABAB

pk EEE )(sin)( 222

v

xtAx

When wave propagates in a volume medium,When wave propagates in a volume medium,

，， x x VV

)(sin)( 222

v

xtAVE

)(sinv

xtA

vx

y

pE )(sin2

1 222

v

xtAx

i.e.i.e. pk EE

Remarks Remarks TheThe EEkk, , EEp p andand E E change with same phase fchange with same phase f

or any particle of the medium. or any particle of the medium.

The EEkk, , EEp p andand E E hashas maximum maximum when the pawhen the pa

rticle locate in its equilibrium position. rticle locate in its equilibrium position. EEkk==EEp p

== EE= = 0 0 when the particle locate in its maximum when the particle locate in its maximum displacement.displacement.

EE is not conservative for any particle. A particle absorbs energy continuously from its contiguous particle in the front of it and gives off the energy to the particle behind it. ----energy pro----energy propagationpagation

II. Energy density of waveII. Energy density of wave

Energy densityEnergy density ：： the wave energy stored the wave energy stored in a unit volume.in a unit volume.

V

Ew

)(sin222

v

xtA

The average energy density:The average energy density: the average the average value over one period.value over one period.

T

wdtT

w0

1 22

2

1 A

IIIIII. Energy flow density of wave. Energy flow density of wave (wave intensity)(wave intensity)

Energy flow: Energy flow: the amount of the amount of energy passing a given area energy passing a given area in unit time.in unit time.

Average energy flow(Average energy flow( 平均能流平均能流 ))

svwPdtT

PT

0

1

S

v

v

vswP

Energy flow density of wave (wave Energy flow density of wave (wave

intensity) intensity) ：：

S

PI 22

2

1Avvw

22 , IAI

IV. Sound wave

1. SW is a mechanical longitudinal wave

Frequency

range:<20HZ-----infrasonic wave

>20000HZ-----supersonic wave

Acoustics ： study the generation, propagation, receiving of SW and the reactions between SW and matters..

20~~20000HZ ， -----(audible)sound

3. sound intensity level 3. sound intensity level (( 声强级声强级 ) ) ::

2 2 12/ 1 / 10m W m W

212

0 /10 mWI --reference intensity

The range of audible wave ：

Let Let

Then the sound intensity level of I is

)(log0

BelI

IL

)(log100

dbI

ILOr Or

22

2

1AvI 2. sound intensity ：

Decibel (Decibel ( 分贝分贝 ))

4. Characters and applications of sound wave4. Characters and applications of sound wave

supersonic ： larger , smaller , less diffraction, good direction--sonar

infrasonic ： less decline ， longer propagating distance

audible ： music noise

Easily reflected by the boundary surfaces--B 超

Larger energy, easily focused --weld ( 焊接 ), drill( 钻孔 )

--study the motions of earth, ocean…

　　I. Huygens’ principle (hypothesis)I. Huygens’ principle (hypothesis) All points on a wave front serve as point All points on a wave front serve as point

sources of spherical secondary wavelets. After sources of spherical secondary wavelets. After a time a time tt, the envelope of these secondary , the envelope of these secondary wavelets will be the new wave front.wavelets will be the new wave front.

§§15-5 Huygens’ Principle15-5 Huygens’ Principle

It provides a geometrical method to find the new wave front at later instant from known shape of a portion of a wave front at some instant.

It can be used to discuss various problems concerning the propagation of waves.

tv

spherical wavespherical wave plane waveplane wave

tv

II. Diffraction, reflection and refraction of waveII. Diffraction, reflection and refraction of wave

1.diffraction1.diffraction

--The bending of wave into the shadow region --The bending of wave into the shadow region of obstacle during propagating.of obstacle during propagating.

2. reflection2. reflection

A

1A2A

B C

'A

'ii i 'iCAAA 2'

CAAACA '2

CAAACA 2'

'ii

3. refraction3. refraction

A

1A2A

B C

i1v

2v'A'B

iACCA sin2 tv 1sin' ACAA tv 2

212

1

sin

sinn

v

vi

21 vv AssumeAssume

nn2121 ：： ------the relative refractive index of the relative refractive index of

medium medium 22 versus versus 11

I. Propagating law of waveI. Propagating law of wave IndependenceIndependence ：： the propagation of waves is the propagation of waves is

independent when they propagate in same independent when they propagate in same areas of a medium simultaneously.areas of a medium simultaneously.

Principle of superpositionPrinciple of superposition ： ： When two or When two or more waves combine, the resultant more waves combine, the resultant displacement of each particle of medium is the displacement of each particle of medium is the vector sum of the individual displacement vector sum of the individual displacement produced by each wave.produced by each wave.

§15-6 Principle of superposition of wave §15-6 Principle of superposition of wave Interference of waveInterference of wave

II. Interference of wavesII. Interference of waves

Coherent wave source:Coherent wave source:

Two wave sources have Two wave sources have equal frequency, the same equal frequency, the same direction of vibration and direction of vibration and constant phase difference.constant phase difference.

Coherent wavesCoherent waves

ConstructivConstructive e interferenceinterference

destructive destructive interferenceinterference

Assume two coherent sources vibrate as followsAssume two coherent sources vibrate as follows

)cos( 11010 tAy

)cos( 22020 tAy1S

2S

P1r

2rThey cause vibrations at point They cause vibrations at point P:P:

)2cos( 1111 rtAy

)2cos( 2222 rtAy

The resultant vibration at The resultant vibration at P P ::

21 yyy )cos( tA

Here Here )2(cos2 12

122122

21

rrAAAAA

)2

cos()2

cos(

)2

sin()2

sin(

222

111

222

111

r

Ar

A

rA

rA

tg

DiscussionsDiscussions at any point of the medium,at any point of the medium,

12

12 2rr

A A is a constant at any fixed point.is a constant at any fixed point.

--do not change with time--do not change with time

AA is not same when is different. is not same when is different.

AsAs 2AI

The intensity of resultant wave The intensity of resultant wave

cos2 2121 IIIII

--the intensity I at each point of the medium is determined by and does not change with time t.

2,1,02 kk WhenWhen

21 AAA We haveWe have

2121 2 IIIII

Largest intensityLargest intensity

maxA

Constructive interference

The resultant amplitude has maximum value.The resultant amplitude has maximum value.

2,1,0)12( kk WhenWhen

21 AAA

We haveWe have

2121 2 IIIII

--Smallest intensity--Smallest intensity

minA

Destructive interference

--Smallest resultant amplitude --Smallest resultant amplitude

IfIf , , thenthen21

k

2

12 k

,2,1,0 k

12 rr

maxAA

minAA

--integral number of wavelength--integral number of wavelength

We haveWe have

--even number of wavelength--even number of wavelength

We haveWe have

Path difference:

122rr

[[ExampleExample] Two wave sources ] Two wave sources SS11 and and SS22have distahave dista

nce nce d=30md=30m. They generate two plane coherent wa. They generate two plane coherent waves propagating along ves propagating along x x axis. axis. SS11 locates in the ori locates in the ori

gin gin OO. Two points locating in . Two points locating in xx11=9m=9m and and xx22=12m =12m

respectively, which are contiguously still points brespectively, which are contiguously still points because of interference. ecause of interference. Find the wavelength of thFind the wavelength of the waves and the minimum phase difference of twe waves and the minimum phase difference of two wave sources.o wave sources.

x0 1x 2x1S 2S

SolutionSolution ：：

The point at The point at xx11 is still because of interference, is still because of interference,

)12(]

2[]

)(2[ 1

11

2

kxxd

)12()2(2 1

12

kxd

Similarly, at point Similarly, at point xx22 , we can get , we can get

)32()2(2 2

12

kxd

Assume that the initial phases of Assume that the initial phases of SS11, , SS22 are are 11 , ,22 respectively. respectively.

thenthen

i.e.i.e. ……

……

Eq.Eq. - - Eq.Eq., we get, we get

2

)(4 12 xx

)(2 12 xx )912(2 m6

)2(2

)32( 212

xdk

)52( k

We can get the minimum phase difference:We can get the minimum phase difference:

12

Substituting this result into Eq. Substituting this result into Eq. ,,

for for kk=-2.=-2.

I. Standing waveI. Standing wave

is formed by the two coherent waves with same is formed by the two coherent waves with same amplitude and opposite propagating direction.amplitude and opposite propagating direction.

vv0t

0 x

§15-7 Standing wave§15-7 Standing wave

4

Tt

2

Tt

always at rest--node

0 x

0x

always has maximum amplitude--antinode

II. Equation of standing waveII. Equation of standing wave

)](cos[1 v

xtAy

)](cos[2 v

xtAy

Two coherent waves traveling in contrary directionTwo coherent waves traveling in contrary direction

Resultant wave:Resultant wave:

21 yyy tv

xA

cos)cos(2

Antinode (loop) Antinode (loop) ：：

1cos v

x

,2,1,0 nnv

x

v

nx 2

n

The distance between two The distance between two adjacent antinodes is adjacent antinodes is nn xxx 1

2

The amplitude of a particle at The amplitude of a particle at xx is is

v

xA

cos2

Notes Notes

nodenode ：： 0cos v

x

2,1,02

2 nnx

4)12(

nx

The distance between two adjacent nodes isThe distance between two adjacent nodes is

2

, too., too.

All particles between two adjacent nodes have All particles between two adjacent nodes have same phase. same phase.

0 x

The energy of the wave transfers between aThe energy of the wave transfers between antinodes and nodes alternately ntinodes and nodes alternately – no propag– no propagation of energy.ation of energy.

Same phaseSame phase

The particles locate in two side of a node have The particles locate in two side of a node have opposite phase (phase difference is opposite phase (phase difference is ). ).

Opposite phaseOpposite phase

v

v

III. Half-wave loss of reflected waveIII. Half-wave loss of reflected waveSmall refractive indexSmall refractive index ： ： is smalleris smallervLarge refractive index Large refractive index ： ： is largeris largerv

Smaller Smaller Larger Larger : the reflecting po : the reflecting point is always int is always nodenode..

22v11v 1 2

The phase of the reflected wave changes The phase of the reflected wave changes 18018000(() with respect to incident wave at the ) with respect to incident wave at the reflecting point.reflecting point. ---half-wave loss---half-wave loss

v

Larger Larger Smaller Smaller : : the reflecting poithe reflecting point is always nt is always antiantinodenode..

22v11v

1 2

The phase of the reflected wave is theThe phase of the reflected wave is the same same with incident wave at the reflecting point.with incident wave at the reflecting point.

v

--No half-wave loss at reflecting point.--No half-wave loss at reflecting point.

SolutionSolution

)(),( 001 vxttx

[[ExampleExample] As shown in Fig., the incident wave ] As shown in Fig., the incident wave function isfunction is

)(cos1 vxtAy Find the function Find the function

of reflected wave.of reflected wave.

0xincident

reflectedx

y0

Small Small

large large

The phase of incident wave at point The phase of incident wave at point xx00 at time at time t t ::

The reflected wave has same The reflected wave has same AA, , , , vv with incident with incident wave.wave.

])(cos[2 vxtAyAssume the function of reflected wave isAssume the function of reflected wave is

Its phase at point Its phase at point xx00 is is

)(),( 002 v

xttx

),(),( 0102 txtx

vx02

])2

(cos[ 02

c

x

c

xtAy

As half-wave loss,As half-wave loss,

=?=?

When the wave source When the wave source (or observer, or both) is (or observer, or both) is moving in the medium, moving in the medium, the frequency received the frequency received by observer may differ by observer may differ from the vibrating from the vibrating frequency of the source. frequency of the source.

Let Let vvss------the speed of source the speed of source

vvoo------the speed of observer

f f --- ---the frequency of sourcethe frequency of source v---v---the propagating speed of wave in mediumthe propagating speed of wave in medium

§15-8 Doppler effect§15-8 Doppler effect

呜呜

1. 1. S S at rest, at rest, O O movingmoving

O O moving towardmoving toward SS

s Oovv

The speed of wave The speed of wave with respect to with respect to OO is is

ovv

ovv

f

'fv

vv o

fv

vo )1( ---the received frequency is ---the received frequency is higher than the frequency of higher than the frequency of source.source.

The frequency received by The frequency received by OO is is

O O moving away frommoving away from SS

fv

vf o )1('

----the received frequency is lower than the received frequency is lower than the frequency of source.the frequency of source.

s Oovv

2. 2. O O is at rest , is at rest , SS moving moving

The source of wave is moving to the left in water.

's

s O

O

sv

S S moving towardmoving toward OO

Tvs

'

Tvs' TvvT s

Tvv s )(

''v

f Tvv

v

s )( f

vv

v

s

----the received frequency is higher than the the received frequency is higher than the frequency of source.frequency of source.

S S moving away frommoving away from OO

fvv

vf

s' --- f--- f < f < f

3. 3. S S and and OO both moving both moving

SS and and O O moving toward each other, moving toward each other,

ovv The wavelength in mediumThe wavelength in medium ：： Tvv s )('

''

ovv

f

Tvv

vv

s

o

)(

fvv

vv

s

o

SS and and O O moving away from each other,moving away from each other,

fvv

vvf

s

o

'

The speed of wave with respect to The speed of wave with respect to OO ：：

RemarksRemarks

All conclusions above are used for All conclusions above are used for vvss , , vvoo << vv

The The shock waveshock wave is formed as is formed as vvs s >> vv

The bullet is moving in the air with supersonic speed.

3. 3. Doppler effect of electromagnetic wave

Source, observer are moving toward each other

f

cv

cv

f

1

1

Source, observer are moving away from each other

f

cv

cv

f

1

1

vv: relative speed between : relative speed between source and observersource and observer

cc: the speed of : the speed of electromagnetic waveelectromagnetic wave

[[ExampleExample]As shown in Fig., the source of wave ]As shown in Fig., the source of wave SS is fixed. An object moves toward the observer is fixed. An object moves toward the observer O O with speed with speed uu=0.2m/s=0.2m/s. The beat frequency . The beat frequency f f = 4= 4Hz Hz is receivedis received by by OO. Find the frequency of source . Find the frequency of source ffs . s . (the sound speed in the air is (the sound speed in the air is 340m/s340m/s))

uO S

SolutionSolution

sff 1 objectobject

O O can receives two waves:can receives two waves:

one comes from one comes from SS directly, directly,

another comes from reflection another comes from reflection of the object. of the object.

Beat frequency is Beat frequency is

12 fff sfuv

uv)1(

fu

uvf s

2

Hz3398

The frequency of the wave The frequency of the wave reflected by the object isreflected by the object is

sfuv

uvf

2

uO S

objectobject