1
Chapter 12 Waves
Wave motion is the most universal physical
phenomenon, we discover the wave motion
all around us.
Mechanical waves—acoustic wave, water wave
Electromagnetic waves—radio, light wave
Matter waves—electron wave
We will concentrate to the mechanical waves.
Wave is a propagating disturbance that
transfers energy, momentum and
information as its own characteristic speed
from one region of space to another.
2
§ 12.1 mechanical waves
1. What is mechanical waves?
Source of wave—a disturbance(oscillation)
Medium—for propagation of waves
Longitudinal waves:
waves whose
oscillation or jiggling
is along the line the
wave propagates.
According to the direction of particle motion
2. Classification of mechanical waves
Plane of polarization-- the
plane in which the jiggling
of a transverse waves
disturbance occurs.
Pulse wave:
Periodic train of waves:
§ 12.1 mechanical waves
Transverse waves: the
waves whose oscillation or
jiggling is perpendicular to
the line along which the
waves propagates.
3
Wavefronts—the surfaces on them all the
points are in the same state of motion.
Wave ray—the direction which are
perpendicular to the wavefronts or parallel to
the velocity of the waves
§ 12.1 mechanical waves
According to the shape of the wavefronts
Plane waves: the
wavefronts are plane
Cylindrical wave: the
wavefronts are
cylindrical
Spherical wave: the
wavefronts are
spherical
§ 12.1 mechanical waves
4
Seismic waves—
earthquakes
produce both
longitudinal and
transverse
waves
Sand scorpion—
located its prey
by using
longitudinal and
transverse
waves.
§ 12.1 mechanical waves
m/s150=
m/s50=
td
v
d
v
d
t
tl
?=??=? 75
§ 12.2 one-dimensional waves and the equation
of waves
1. Wavefunction of one-dimensional waves
moving at constant velocity
x
y
o
x(x′)
yy′
oo′
x
??
vt
x′
P
P
)()0,( xΨxy =
)(
)(),(
vtxΨ
xΨtxy
?=
′
=
vv ?→
)(
)(),(
vtxΨ
xΨtxy
+=
′
=
5
x
y
o
x(x′)
yy′
oo′
x
??
vt
x′
P
P
),()0,( txyxy
PP
=
=?vtx
constant
phase
t
x
vv
t
x
d
d
0
d
d
=?=?
Phase speed
§ 12.2 one-dimensional waves and the equation
of waves
2. The classical wave equation
vtxu
uΨvtxΨtxy
?=
=?= )()(),(
2
2
2
2
)(
u
Ψ
x
u
u
Ψ
ux
Ψ
u
Ψ
x
u
u
Ψ
x
Ψ
?
?
=
?
?
?
?
?
?
=
?
?
?
?
=
?
?
?
?
=
?
?
2
2
2
2
2
)(
u
Ψ
v
t
u
u
Ψ
v
ut
Ψ
u
Ψ
v
t
u
u
Ψ
t
Ψ
?
?
=
?
?
?
?
?
?
?
=
?
?
?
?
?=
?
?
?
?
=
?
?
2
2
2
2
2
x
Ψ
v
t
Ψ
?
?
=
?
?
0
1
2
2
22
2
=
?
?
?
?
?
t
Ψ
vx
Ψ
or
§ 12.2 one-dimensional waves and the equation
of waves
6
3. Periodic waves
x
),(
0
txΨ
λ
o
λ
(a) Space period
A snapshot at some instant t
0
t
o
),(
0
txΨ
T
T
(b) Time period
An oscillation at any given position x
0
§ 12.2 one-dimensional waves and the equation
of waves
In one time period, some oscillatory state
propagate one space period.
Phase speed: νλ
λ
?==
T
v
λ
T
§ 12.2 one-dimensional waves and the equation
of waves
7
x
),( txΨ v
r
o
vT=λ
tv?
(c) Traveling wave
νλλν === vvT
T
1
4. Sinusoidal (harmonic) waves
)](cos[),( vtxkAtxΨ ?=
x
),( txΨ v
r
o
λ
§ 12.2 one-dimensional waves and the equation
of waves
4. The physical meaning of the harmonic waves
Model: One-dimensional harmonic waves
x
v
r
o
If the oscillatory equation at point o is
)cos(
0
φω += tAΨ
?),( == txΨΨ
--harmonic motion
§ 12.2 one-dimensional waves and the equation
of waves
8
Method 1
if )cos(
0
φω
′
+= tAΨ
v
OP(x)
x
The propagating time of the
state from point o to point P(x)
v
x
t =?
Choose a ray of wave as x axis, construct a
one-dimensional coordinate system. O is the
origin, v is the velocity along the +x direction.
The phase of point P(x) at instant t is the same
as the point o at instant t-?t.
§ 12.2 one-dimensional waves and the equation
of waves
])(cos[),( φω +?= t
v
x
AtxΨ
(1)
),0(),(
0
ttΨtxΨ
p
??= ])(cos[ φω
′
+?=
v
x
tA
therefore
u
OP(x)
x
Method 2:
The phase at point P lag
relative to point o
π
λ
2?
x
The phase difference between two points of
space interval of λ is 2π.
§ 12.2 one-dimensional waves and the equation
of waves
9
Equations (1) and (2) is identical.
)2cos(),( π
λ
φω ??
′
+=
x
tAtxΨ
p
)2cos(),( φωπ
λ
+??= t
x
AtxΨ
or
(2)
ω
π
λ
2
vvT ==
Due to
§ 12.2 one-dimensional waves and the equation
of waves
(1)
])(cos[),( φω +?= t
v
x
AtxΨ
λ
ππ
ωλ
22
=== k
T
vT
where
§ 12.2 one-dimensional waves and the equation
of waves
])(cos[),( φω +?= t
v
x
AtxΨ
)2cos( φω
λ
π +?= t
x
A
])(2cos[ φ
λ
π +?=
T
tx
A
LL=
+?= ])(cos[ φvtxkA
)cos( φω +?= tkxA
10
§ 12.2 one-dimensional waves and the equation
of waves
6. The comparison of graphs of oscillation and
waveform
T, A, φ, the moving direction of point
(a) The graphs of oscillation
A, λ, determine φ from the graph of waveform
at t=0, the moving direction of point
(b) The graphs of waveform
??
a
b
)m(x
04.0
)0,(xΨ
o
2.0
P
m/s08.0=v
Example: figure depicts the waveform of a
traveling sinusoidal wave at instants t=0 s,
find (a) the oscillatory equation of point o; (b)
the wavefunction ;(c) the oscillatory
equation of point P; (d) the moving directions
of points a and b.
),( txΨ
§ 12.2 one-dimensional waves and the equation
of waves
11
(a)
)srad(4.02
)m(4.0)(04.0
1?
?==
==
π
λ
πω
λ
v
mA
?
?
a
b
)m(x
04.0
),( txΨ
o
2.0
P
m/s08.0=v
Solution:
)
2
4.0cos(04.0),0(
π
π += ttΨ
2
0
π
? =
The initial phase angle of point o:
?
?
?
?
?
?
+?=
2
)
08.0
(4.0cos04.0),(
π
π
x
ttxΨ
(b) the wavefunction
§ 12.2 one-dimensional waves and the equation
of waves
Ψ
a
b
x
04.0
),( txΨ
o
2.0
p
??
?
?
?
?
?
?
+=
?
?
?
?
?
?
+?=
2
4.0cos04.0
2
)
08.0
4.0
(4.0cos04.0),(
π
π
π
π
t
ttxΨ
P
(c) the oscillatory equation of point P
(d) the moving directions of points a and b.
§ 12.2 one-dimensional waves and the equation
of waves
12
§ 12.3 Energy transport via mechanical waves
1. The velocity of wave on a string
μ
μ
θμ
θθ
/
)/(
/,
)2/sin(2)/(
2
2
Tv
r
l
Trvl
rllm
TTrvm
=
?
=?∴
?=?=?
≈=?Q
The wave is viewed from a reference frame
moving with the wave so it is seen to be at
rest. The disturbance is motionless but the
string is observed to move to the left at a
constant speed v.
x
y
T
r
T′
r
θ
2/θ
l?
r
C
2/θ
§ 12.3 Energy transport via mechanical waves
The general relationship for the speed of
mechanical waves in material media:
2/1
factormass
factorforceaofmagnitude
?
?
?
?
?
?
?
?
∝v
2/1
solid
?
?
?
?
?
?
?
?
=
ρ
E
v
2/1
liquid
?
?
?
?
?
?
?
?
=
ρ
B
v
Sound wave in solid:
Sound wave in liquid:
young′s
modulus
bulk
modulus
Generalize to the other waves:
13
PEKEE ddd +=
For a string wave:
xxd
y
ldyd
xm
yxl
dd
)d()d(d
22
μ=
+=
§ 12.3 Energy transport via mechanical waves
2. The mechanical energy of a small string
element of medium
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?+≈
?
?
?
?
?
?
?+=?+=
22
222
)(
2
1
d1)
d
d
(
2
1
1d
1)
d
d
(1dd)d()d(
x
y
x
x
y
x
x
y
xxyxlδ
x
x
y
TPE d)(
2
1
d
2
?
?
=
then
x
t
y
t
y
mKE d)(
2
1
)(d
2
1
d
22
?
?
=
?
?
= μKinetic energy:
)dd(d xlTlTPE ?== δ
22
)d()d(d yxl +=
Potential energy:
§ 12.3 Energy transport via mechanical waves
14
The mechanical energy:
x
x
y
Tx
t
y
PEKEE
d)(
2
1
d)(
2
1
ddd
22
?
?
+
?
?
=
+=
μ
For a harmonic wave
[]
[]φω
ω
φω
φωωφω
+??=
?
?
?
?
?
?
+?
?
?
=
?
?
+?=
?
?
?
?
?
?
+?
?
?
=
?
?
)(sin)(cos
)(sin)(cos
tkxA
v
t
v
x
A
xx
y
tkxAt
v
x
A
tt
y
§ 12.3 Energy transport via mechanical waves
[]xtkxA
v
TE d)(sin)(
2
1
d
2222
φω
ω
μω +?
?
?
?
?
?
?
+=
μ
T
v =due to
[ ] VtkxAE d)(sind
222
φωρω +?=
generalize
[ ] xtkxAE d)(sind
222
φωμω +?=
we have
§ 12.3 Energy transport via mechanical waves
15
An non-isolated differential medium
Comparison:
An isolated harmonic oscillator
The mechanical energy conserved
KE and PE is changed with out of phase
The mechanical energy is not conserved
KE and PE is changed in phase
§ 12.3 Energy transport via mechanical waves
Equilibrium position:
Transverse
waves:
x
xd
y
ld
yd
with maximum distortion,
maxmax
d,)( PE
x
y
?
?
oscillating speed
maxmax
d, KEu
Position of the maximum displacement:
with minimum distortion,
minmin
d,)( PE
x
y
?
?
oscillating speed
minmin
d, KEu
§ 12.3 Energy transport via mechanical waves
16
Longitudinal waves:
Equilibrium position(the center of compression
and rarefaction):
with maximum distortion,
maxmax
d,)( PE
x
y
?
?
oscillating speed
maxmax
d, KEu
Position of the maximum displacement:
with minimum distortion,
minmin
d,)( PE
x
y
?
?
oscillating speed
minmin
d, KEu
§ 12.3 Energy transport via mechanical waves
3. The power of wave
[ ] xtkxAE d)(sind
222
φωμω +?=
[]φωωμ +?== )(sin
d
d
222
tkxAv
t
E
P
[]zytkxAv
t
E
P dd)(sin
d
d
222
φωωρ +?==
[ ] VtkxAE d)(sind
222
φωρω +?=or
§ 12.3 Energy transport via mechanical waves
17
VtkxAPEKEE d])[(sinddd
222
?+?=+= φωρω
4. The density of energy
])[(sin
d
d
222
φωωρ +?== tkxA
V
E
w
the density of energy:
22
2
1
ωρA=
∫
+?=
T
ttkxA
T
w
0
222
av
d])[(sin
1
φωωρ
The average density of energy:
§ 12.3 Energy transport via mechanical waves
vAI
r
r
22
2
1
ωρ= --wave intensity
vA
vw
s
P
I
av
22
av
av
2
1
ωρ
?
=
=
=
5. Wave intensity
[]zytkxAv
t
E
P dd)(sin
d
d
222
φωωρ +?==
The direction of energy transportation is
same as that of wave
§ 12.3 Energy transport via mechanical waves
v
tv?
18
The power through the area
s
1
and s
2
is same
2
22
21
22
1
2
1
2
1
svAsvA ?=? ωρωρ
1
2
1
2
1
2
2
1
2
2
r
r
hr
hr
s
s
A
A
===
π
π
1
2
2
2
2
1
2
1
r
r
A
A
I
I
==
Example: a cylindrical plane wave propagates
in a homogeneous medium, find the relationship
of wave intensity with respect to the amplitude
and the distance to the wave source.
Solution:
§ 12.3 Energy transport via mechanical waves
The wave function of cylindrical plane waves:
])cos[(
0
φω +?= tkr
r
A
Ψ
])cos[(
0
φω +?= tkr
r
A
Ψ
Prove the wave function of spherical plane
waves is
§ 12.3 Energy transport via mechanical waves
19
§ 12.4 Reflection and transmission of waves
1. Tie one end of the rope to a
rigid wall
The pulse experiences a
change of phase of π.
2. the one end of the rope is free
There is no phase change
on reflection.
3. Two different ropes with
different mass densities
A pulse is transmitted to the
second rope and a pulse is
reflected. P546
§ 12.5 Sound wave and the acoustic Doppler effect
1. What is a sound wave?
Sound waves in air
or other gases
represent such
rapid oscillations in
density that the
inertial properties
of the air make it
quite elastic.
20
2. Sound intensity and sound level
Threshold for hearing: ~10
-12
W/m
2
at ~1 kHz
Cause deafening pain: ~1 W/m
2
12log10log
12
=? αα
The sound level:
Let
212
0
W/m10
?
≡I
0
log)dB10(
I
I
≡β
Unit: decibels
Each factor of 10 increase in sound intensity I
produces an additive change in sound level of
10dB.(P551 table 12.1)
§ 12.5 Sound wave and the acoustic Doppler effect
3. The acoustic Doppler effect
νν =
′
§ 12.5 Sound wave and the acoustic Doppler effect
21
1motion of the observer with the source and
medium at rest
)1()1(
obsobsobs
v
vvv
±=±=±=
′
ν
λν
ν
λ
νν
§ 12.5 Sound wave and the acoustic Doppler effect
2motion of the sound source with the observer
and medium at rest
)(
/
sss s
s
vv
v
v
v
v
v
Tv
v
Tv
mmm
m
ν
λν
ν
νλλ
ν
λλ
===
?
=
′
=
′
§ 12.5 Sound wave and the acoustic Doppler effect
22
3the motion of both observer and sound source
)(
obs
s
vv
vv
m
±
=
′
νν
4motion of the medium with the source and
observer at rest
νν =
′
§ 12.5 Sound wave and the acoustic Doppler effect
)(
smed
obsmed
vvv
vvv
m±
±±
=
′
νν
5motion of the observer and/or source with a
wind
4. Shock waves
)(
s
vv
v
?
=
′
νν
Mach cone:
)(sin
s
s
vv
v
v
>=φ
Mach number:
v
v
s
numberMach =
§ 12.5 Sound wave and the acoustic Doppler effect
23
§ 12.6 the superposition of waves
1. The linear principle of superposition
When two waves encounter
together, if the amplitudes
is not too large, the total
wave distance at any point
x and time t is the sum of
the individual wave
disturbances.
L+++= ),(),(),(),(
321total
txΨtxΨtxΨtxΨ
)cos(),(
11
φω +?= tkxAtxΨ
)cos(),(
22
φω +?= tkxAtxΨ
)cos()cos(
),(),(),(
21
21total
φωφω +?++?=
+=
tkxAtkxA
txΨtxΨtxΨ
2. Interference of waves
)
2
cos()
2
cos(2
]
2
)()(
cos[
]
2
)()(
cos[2),(
2112
21
21
total
φφ
ω
φφ
φωφω
φωφω
+
+?
?
=
+??+?
?
+?++?
=
tkxA
tkxtkx
tkxtkx
AtxΨ
§ 12.6 the superposition of waves
24
)
2
cos()
2
cos(2),(
2112
total
φφ
ω
φφ +
+?
?
= tkxAtxΨ
),2,1,0(0,)12(
),2,1,0(2,2
total
total
L
L
==+=?
===?
nAn
nAAn
πφ
πφ constructive
destructive
if
21
AA ≠
then
),2,1,0(,)12(
),2,1,0(,2
21total
21total
L
L
=?=+=?
=+==?
nAAAn
nAAAn
πφ
πφ
§ 12.6 the superposition of waves
0=?φ
πφ =? 32πφ =?
),(
and
),(
2
1
txΨ
txΨ
),(
1
txΨ ),(
2
txΨ ),(
1
txΨ ),(
2
txΨ
),( txΨ
),( txΨ
),( txΨ
Ψ
Ψ
Ψ
Ψ
ΨΨ
Interference of waves
§ 12.6 the superposition of waves
25
)cos(),(
11
φω +?= tkxAtxΨ
)cos(),(
22
φω ++= tkxAtxΨ
)cos()cos(
),(),(),(
21
21total
φωφω ++++?=
+=
tkxAtkxA
txΨtxΨtxΨ
)
2
cos()
2
cos(2
]
2
)()(
cos[
]
2
)()(
cos[2),(
1221
21
21
total
φφ
ω
φφ
φωφω
φωφω
?
+
+
+=
++?+?
?
++++?
=
tkxA
tkxtkx
tkxtkx
AtxΨ
3. Standing waves
§ 12.6 the superposition of waves
)
2
cos()
2
cos(2
1212
total
φφ
ω
φφ ?
+
+
+= tkxAΨ
Special case:
0
21
==φφ
tkxAΨ ωcoscos2
total
=
§ 12.6 the superposition of waves
Standing wave is a special phenomenon of
interference.
26
§ 12.6 the superposition of waves
The characteristic of the standing waves:
Some points keep zero disturbance: a,c,e,g,…
Some points have maximum amplitudes: o,b d f, …
1
20 AA<<
The other points
0)
2
cos(
12
=
+
+
φφ
kx
)(
44
)12(
12
φφ
π
λλ
+?+= kx
2
)12(
2
12
πφφ
+=
+
+ kkx
),2,1,0( L±±=k
The position of nodes:
§ 12.6 the superposition of waves
27
)(
42
12
φφ
π
λλ
+?= kx
1)
2
cos(
12
=
+
+
φφ
kx
π
φφ
kkx =
+
+
2
12
),2,1,0( L±±=k
The position of antinodes:
2
λ
? =x
The distance between the
successive nodes or antinodes:
§ 12.6 the superposition of waves
0)
2
1
(
2
1
22
1
22
1
=?+= vAvAI
rr
r
ωρωρ
The wave intensity of standing waves:
The energy is
switched
between the
nodes and the
antinodes.
§ 12.6 the superposition of waves
28
The characteristic of energy for standing wave
:2/,0 Tt =
PEE =
Concentrate
near the nodes.
:4/3,4/ TTt =
KEE =
Concentrate near
the antinodes.
§ 12.6 the superposition of waves
4. The musical instruments with strings
μ
ν
μ
νλ
λλ
T
L
T
T
vL
2
1
2
1
1
=
====
--fundamental frequency
§ 12.6 the superposition of waves
μ
ν
T
L
n
n
2
1
=
--eigenfrequency
2
n
nL
λ
=
29
§ 12.6 the superposition of waves
1. Wave groups and beats
)cos(
)cos(
),(),(),(
)cos(),(
)cos(),(
22
11
21
222
111
txkA
txkA
txΨtxΨtxΨ
txkAtxΨ
txkAtxΨ
ω
ω
ω
ω
?+
?=
+=
?=
?=
§ 12.7 Wave groups and beats
]
22
cos[
]
22
cos[2),(
2121
2121
tx
kk
tx
kk
Atx
ωω
ωω
Ψ
?
?
?
?
+
?
+
=
30
]
22
cos[]cos[2),( tx
k
tkxAtx
ω??
ωΨ ??=
)]
2
()
2
cos[(2 tx
k
A
ω??
?The other factor
if
21
or2/ ωωω? <
This wave represents a slowly varying amplitude
factor for superposition, which travels at its own
group speed.
§ 12.7 Wave groups and beats
1
1
1
21
21
v
kkkk
v =≈
+
+
==
ωωωω
if
, then the phase speed
21
ωω ≈
Let x=0
)
2
cos(cos2 ttAΨ
ω
ω
?
=
k
v
kvkv
kk
v
k
v
k
v
k
v
d
d
)(
d
d
d
d
,
d
d
phase
phasephasegroup
phasegroup
group
+===
==
?
?
=
ω
ωω
ω
dispersion
The superposition as a function of time:
§ 12.7 Wave groups and beats
31
)
2
cos(cos2 ttAΨ
ω
ω
?
=
§ 12.7 Wave groups and beats
2. Fourier analysis
§ 12.7 Wave groups and beats
From sinusoidal
waves, any manner
of more complex
periodic waves can
be built.
)7cos
7
1
5cos
5
1
3cos
3
1
(cos
4
???+?+?= tttt
A
x ωωωω
π
For instance:
A square wave
32
tAtAxxx ωω 3coscos
2121
+=+=
1
x
2
x
t
t
O
1
T
O
2
T
tO
x
1
T
1
x
2
x
t
t
t
O
O
1
T
O
2
T
x
1
T
)3cos(
3
1
cos
21
πωω ++=+= ttxxx
§ 12.7 Wave groups and beats
§ 12.7 Wave groups and beats
A flute
An oboe
A saxophone