Chapter 10 Chapter 10

Wave Motion Wave Motion

Chapter 10 Wave MotionChapter 10 Wave Motion

§1. Several Concepts of Mechanical Wave

§2. Wave Function of Simple Harmonic Wave

§3. Energy in Wave Motion, Energy Flux Density

§4. Huygens Principle, Diffraction and Interference of Waves

§5. Standing Waves

§6. Doppler Effect

§7. Plane Electromagnetic Waves

§1. Several Concepts of Mechanical Wave§1. Several Concepts of Mechanical Wave

1. The formation of mechanical wave

2. Transverse wave and longitudinal wave

3. Wavelength, wave period and frequency, wave speed

4. Wave line, wave surface, wave front

Elastic medium which can propagate mechanical oscillation

2 Medium

An object which is oscillating mechanically

1 Wave source

What are propagated is the oscillation states, whi

le the mass points do not flow away.

Notice

1. The formation of mechanical wave1. The formation of mechanical wave

1 Transverse wave

2. Transverse wave and longitudinal wave2. Transverse wave and longitudinal wave

Characteristics: The oscillation directions of mass poi

nts are perpendicular to the direction of travel of the wa

ve.

2 Longitudinal wave

Characteristics: The oscillation directions of mass poi

nts are parallel to the direction of travel of the wave.

3 Complex wave

Characteristics: Any complex wave motions can be vi

ewed as a superposition of transverse waves or longitudi

nal waves.

O

yA

A

u

x

1 Wavelength

3. Wavelength, wave period and frequency, wave speed3. Wavelength, wave period and frequency, wave speed

2 Period T

uT

3 Frequency

The period (or frequency) of wave is equal to the oscillation period (or frequency) of the wave source.

T1

The magnitude of the wave velocity depends on the nature of media.

4 Wave velocity u

Tu

In solid

In liquid and gas

K

u

(transverse wave)

(longitudinal wave)E

u

G

u

(longitudinal wave)

The curved surface by connecting the points with the

same phase on the different wave lines

1 Wave line

2 Wave surface

The lines drawn with arrows along the direction

of wave propagation

At one instant, the curved surface connected by every po

int with the original state of wave source

4. Wave line, wave surface, wave front4. Wave line, wave surface, wave front

Wave front

In isotropic medium, wave line is perpendicular to wave surface.

Classification

(2) Spherical wave

(1) Plane wave

1. Wave function of simple harmonic wave

2. The physical meaning of wave function

§2. Wave Function of §2. Wave Function of SimpleSimple Harmonic Wave Harmonic Wave

In the homogeneous and non-absorbing medium

as the wave source is in simple harmonic motion,

the wave formed is called plane simple harmonic

wave.

ttxxytxy ,,

1. Wave function of simple harmonic wave1. Wave function of simple harmonic wave

A mass point is in simple harmonic motion at

origin O. Its motion equation is:

tAyO cos

y

x

uA

AO

P

x

φttωAttyy OP Δcos)Δ(

u

xtAcos

At time t, the displacement of point P is:

This is the function of plane simple harmonic

wave spreading along the positive direction of Ox

axis, and it is also called the wave equation of plane

simple harmonic wave.

The equation can be written in the following three commonly used forms:

xtA

x

T

tA

u

xtAy

π2cos

π2cos

cos

tAy costhen

xπ2

if

xtAy

π2cos

O

y

t

1 When x is fixed

The equation gives the displacement of the

mass point, which is x away from origin O, at

different time.

2. The physical meaning of wave function2. The physical meaning of wave function

The curve of displacement versus time for a simple harmonic motion of every mass point on wave line

x

Ayπ2

costhen

xtAy

π2cos2 When t is fixed

The equation represents the distribution of displ

acement of every mass point at the given time.

2112

2x

Ct if

y

o xx1 x2

The equation expresses the overall situation of d

isplacement varying with time of all mass points.

3 When both x and t are in variation

y

x

u

O

waveform at time twaveform at time t+ t

x x

y

x

uA

AO

Px

4 If the plane simple harmonic wave travels along

the negative x-direction:

])(cos[)( u

xtAttyy o

§3. Energy in Wave Motion, Energy Flux Density§3. Energy in Wave Motion, Energy Flux Density

1. The propagation of wave energy

2. Energy flux and energy flux density

1 Wave energy

1. The propagation of wave energy1. The propagation of wave energy

Take the longitudinal wave in a rod as an example:

22k d

2

1d

2

1d vv VmW

Kinetic energy of oscillation:

)(sind2

1 222

ux

tVA

xxO xd

xO y yy d

)(sind2

1 222

ux

tVA

22 )d

d(d

2

1

x

yVu 22

p )d

d(d

2

1d

2

1d

x

yxESykW

Elastic potential energy:

Total energy of this volume element:

)(sindddd 222pk u

xtVAWWW

)(sind2

1dd 222

u

xtVAWW pk

They all reach the maximum at the equilibrium

position, whereas they are all zero at the maximum

displacement.

2) The mechanical energy in each volume element is not constant.

Discussion

1) have the same phase. WWW pk d andd,d

3) Wave motion is a mode of dissemination of energy.

Energy density:

)(sindd 222

ux

tAVW

w

Average density of energy :

22

0 21

d1

AtwT

wT

xxO xd

xO y yy d

Energy flux:

SuwP

udtS

u

SuwP

2. Energy flux and energy flux density2. Energy flux and energy flux density

Average energy flux:

uwSP

I

Energy flux density ：

uAI 22

21

udtS

u

uSwP

§4. Huygens Principle, Diffraction and Interference §4. Huygens Principle, Diffraction and Interference of Wavesof Waves

1. Huygens principle

2. The diffraction of waves

3. The interference of waves

sph

erical wave

plan

e wave

The every point of a wave front in the medium may

be considered the sources of emitting secondary wavel

ets that spread out in all directions, and at any later ti

me the envelope of these secondary wavelets is the new

wave front.

O

1R

2R

tu

1. Huygens principle1. Huygens principle

wave d

iffraction

diffraction

ph

enom

ena

formed

by w

ater

When wave strikes a barrier in the process of spreading, it

can round the edge of the barrier and go on spreading in the

shade area of the barrier.

2. The diffraction of waves2. The diffraction of waves

1 Superposition principle of waves

The waves will keep their own properties without any change after they meet, and keep traveling in their original directions as if they had never met each other.

The oscillation at any point in the area where the waves meet is the vector sum of their separate oscillation displacements produced by every wave existing at the same point independently.

3. The interference of waves3. The interference of waves

If there are two waves wi

th the same frequency, the

parallel oscillation directio

n, the same phase or the inv

ariant phase difference, wh

en they meet the oscillation

s of some areas are always s

trengthened and the oscillat

ions of some other areas are

always weakened.

2 The wave’s interference

Oscillations of

wave sources:

)cos( 111 tAy

)cos( 222 tAy

)π2cos( 1111

rtAy P

)π2cos( 2222

rtAy P

Oscillations at point P:

The conditions for constructive and destructive interference

1s

2s

P*1r

2r

)cos(21 tAyyy PPP

)π2

cos()π2

cos(

)π2

sin()π2

sin(tan

122

111

222

111

r

Ar

A

rA

rA

cos2 2122

21 AAAAA

12

12 π2rr

constant

1s

2s

P*1r

2r

cos2 212

22

1 AAAAA

when ...3,2,1,0π2 kk

21max AAA

21min AAA

when 时π12 k

“Phase Difference” conditions for interference

Discussion

Phase difference )(π2

1212 rr

The difference of wave paths 21 rr

π2π221 rrthen

if 12

π)12(

π2π2π221 k

krr

constructive

destructive

when krr 21

21max AAA

when2

)12(21

krr

21min AAA

“Wave Path Difference” conditions for interference

§5. Standing Waves§5. Standing Waves

1. Formation of standing waves

2. Equation of standing waves

3. Phase jump

4. Energy in standing waves

5. Normal modes of oscillation

1 Phenomena

1. Formation of standing waves1. Formation of standing waves

2 Conditions

3 Formation of standing waves

The standing wave is a particular interference phenomenon that produced by two coherent waves with the same amplitude, frequency and wave speed traveling in the opposite direction along the same straight line.

tx

A

π2cosπ2cos2

)(π2cos1 xtAy the positive

x-direction

)(π2cos2 xtAy the negative

x-direction

21 yyy

)(π2cos)(π2cos

xtA

xtA

2. Equation of standing waves2. Equation of standing waves

tx

Ay

π2cosπ2cos2Equation of standing waves

x

π2cos,2,1,0ππ2 kk

x

,2,1,0π)

2

1(π2 kk

x

1

0

(1) Amplitude, , only depends on xx

A π2cos2

Discussion

1π2

cos x

AA 2 antinodesb when

42

kx )2,1,0( ，k

0π2

cos x

a when 0A nodes

)2,1,0( ，k4

)12(

kx

Distance between two adjacent node and antinode

Distance between two adjacent nodes 24

Conclusions

4

x

y

2

node

antinode

4

34

54

some points remain still all the time; while the amplitudes of some other points are the maximum.

(2) Phase

txAy

π2cos)π2

cos2(

Conclusion 1 Between two adjacent nodes, the phase of every point is the same.

0π2

cos),4

,4

( xx

txAy

2cos)π2

cos2(

x

y

4

4

34

54

Conclusion 2 The phases of both sides of one node are opposite.

0π2

cos),4

3,

4( xx

π)π2cos()π2

cos2(

π2cos)π2

cos2(

txA

txAy

x

y

4

4

34

54

At any time, the standing wave has a certain waveform, but it does not appear to be moving in either direction along string. Every point oscillates in the vicinity of its own equilibrium position with the certain amplitude.

x

y

4

4

34

54

den

ser med

ium

thin

ner m

ediu

m

thinner medium denser medium

3. Phase jump3. Phase jump

denser medium thinner medium

When the wave shoots from the denser medium to the

thinner medium, the wave node forms at the reflected end.

This indicates the incident and reflected waves are exactly

out of phase with each other all the time. It means the wave

path difference with half of the wavelength is produced,

which is called the phase jumping or half-wave loss.π

2k )(d

t

yW

2p )(d

x

yW

A B C

node

antinode

x

x

maximum displacement

equilibrium position

4. Energy in standing waves4. Energy in standing waves

The standing wave does not spread energy.

——The normal mode of the string oscillation

For a string of which both ends are fixed, the wavelength

and the string length should satisfy the following relatio

nship:

n

2nnl

,2,1

2 n

lu

nn,

l

5. Normal modes of oscillation5. Normal modes of oscillation

The normal mode of oscillation on a string with two

fixed ends:

21l

2

2 2l

2

3 3l

,2,14

2 nnl n

The normal mode of oscillation of an air column in a

glass tube with one opening end and one closed end:

,2,14

)12( nnl n

41l

4

3 2l

4

5 3l

§6. Doppler Effect§6. Doppler Effect

1. Observer moving with velocity v0

relative to medium while wave source is at rest

2. Wave source moving with velocity vs relative to mediu

m while observer is at rest

3.

Wave source and observer moving simultaneously relat

ive to medium

Frequency of wave source ——the number

of complete oscillations of wave source occurrin

g per unit time

Frequency received by observer ——the n

umber of oscillations that observer receives per

unit time

Frequency of wave —— the number of o

scillations of mass point in medium per unit tim

e

b

P

1. Observer moving with velocity 1. Observer moving with velocity vv0 0 relative to relative to

medium while wave source is at restmedium while wave source is at rest

Frequency received

by observer u

u 0'v

Observer moving towards wave source

Observer moving away from wave source

u

u 0'v

2. Wave source moving with velocity 2. Wave source moving with velocity vvs s relative to relative to

medium while observer is at restmedium while observer is at rest

b

As's

Tsv

uT

Tsb v

svu

ub'

s

b

v

uu

s

'v

uu

s

'v

uu

b

As 's

Tsv

Frequency received

by observer

Wave source moving towards observer

Wave source moving away from observer

s

0'vv

uu

0v observer moving towards wave source +

away from -

wave source moving towards observer –

away from +

sv

As long as the two approach each other, the received frequency is higher than that of original wave source; and if the two are apart from each other, the received frequency is lower than that of original wave source.

3. Wave source and observer moving 3. Wave source and observer moving simultaneously relative to mediumsimultaneously relative to medium

If wave source and observer do not move down

their connection line:

ov

sv o'v

s'v

s

0

'

''

vv

uu

§7. Plane Electromagnetic Waves§7. Plane Electromagnetic Waves

1. Generation and propagation of

electromagnetic waves

2. Characteristics of plane electromagnetic

waves

3. Energy in electromagnetic waves

4. The electromagnetic spectrum

0Q+

0Q

CL

Electromagnetic waves are formed by the propagation

of alternating electromagnetic fields in space.

LCT π2 LCπ2

1

-

+

oscillation dipolar

+

-

1. Generation and propagation of 1. Generation and propagation of electromagnetic waveselectromagnetic waves

Electric filed for different moments in

the vicinity of oscillating electric

dipole

tpp cos0

++

++

++

E

B

E

c c

c c

+

-

B

Electric and magnetic fields in the vicinity of oscillating electric dipole

)(cosπ4

sin),(

20

u

rt

r

ptrE

)(cosπ4

sin),(

20

u

rt

r

ptrH

1u

0p

pole axispropagation direction

r

E

H

uEH

)(cos0 u

xtEE

)(cos0 u

xtHH

Plane electromagnetic waves

uE

H xo

)cos()(cos 00 kxtEu

xtEE

)cos()(cos 00 kxtHu

xtHH

π2

k

2. Characteristics of plane electromagnetic waves2. Characteristics of plane electromagnetic waves

(1) Electromagnetic wave is transverse wave: ,

E u

H u

(2) and are in phase. E

H

E

H u

(3) Values of and are in proportion:E

H

EH

(4) The propagation speed of electromagnetic wave in

vacuum equals the speed of light in vacuum:

00/1 u

The energy propagating in the form of electromagnetic waves is called the radiant energy.

Energy flux density of electromagnetic waves wuS

)(2

1 22me HEwww

Energy density of electromagnetic field

Vector of the energy flux density of electromagnetic wave (Poynting’s vector) HES

3. Energy in electromagnetic waves3. Energy in electromagnetic waves

)(2

22 HEu

S EH

and /1u EH

Average of energy flux density of

the plane electromagnetic wave

002

1HES

Average radiation power of

oscillating dipole

442

0

π12

up

pH

E

S

760 nm 400 nmvisible light

Electromagnetic Spectrum

infrared ultraviolet -rayγ

X-ray

long-wavelength radio

610

1010

1410

1810

2210

210

410

810

1210

1610

2010

2410

010

frequency / Hz

1610810

wavelength / m

410 410010 810 1210

Short-wavelength radio

4. The electromagnetic spectrum4. The electromagnetic spectrum

radio waves cm1.0~m103 4

nm400~nm760visible light

infrared ray

nm5~nm400ultraviolet ray

nm0.04~nm5 X-rays

nm04.0γ -rays

nm760~nm106 5