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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 nonabsorbing 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 xdirection:
])(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
xdirection
)(π2cos2 xtAy the negative
xdirection
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 halfwave 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
Xray
longwavelength radio
610
1010
1410
1810
2210
210
410
810
1210
1610
2010
2410
010
frequency / Hz
1610810
wavelength / m
410 410010 810 1210
Shortwavelength 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 Xrays
nm04.0γ rays
nm760~nm106 5