Chapter 13 Wave Motion and Sound
Chapter 13&14
Wave Motion and Sound
1,Simple Harmonic Waves
2,Wave Equation
3,Energy and Power of Waves
4,Interference of Waves
5,Standing Waves
6,The Doppler Effect
Chapter 13 Wave Motion and Sound
Key words:
Mechanical wave 机械波
Electromagnetic wave 电磁波
Transverse wave（ 横波）
Longitudinal wave（纵波）
crests (波峰 ) and troughs (波谷 )
Sinusoidal wave（正弦波）
Traveling (moving,progressive) wave（行波）
Standing (stationary) wave（驻波）
Stretched string（拉紧的线、弦线）
Angular wave number（角波数）
Interference of waves（波的干涉）
Chapter 13 Wave Motion and Sound
一定的扰动的传播称为波动，简称为波。
波动是振动的传播过程；振动是激发波动的波源。
机械波电磁波波动机械振动在弹性介质中的传播,
交变电磁场在空间的传播,
两类波的不同之处
机械波的传播需有传播振动的介质 ;
电磁波的传播可不需介质,可在真空中传播。
具有传播速度
能量传播
反射
折射
干涉
衍射两类波的共同特征几个基本概念
Chapter 13 Wave Motion and Sound
Classification According to Natures of Matter
§ 13-2 Wave Types p327-331
1,Mechanical Waves
机械波,机械振动以一定速度在弹性介质中由近及远地传播出去,就形成机械波 。 Such as water waves,
sound waves,All these waves have certain central
features,they are governed by Newton’s laws,and
they can exist only within a material medium（ 媒质,
介质 ）,such as water,air,and rock.
Chapter 13 Wave Motion and Sound
2,Electromagnetic Waves
电磁波,Such as light,radio and television waves,
X rays,These waves require no material
medium to exist,Its speed through the vacuum
is 299792458m/s.
Examples,Visible and ultraviolet light,radio
and television waves,microwaves,x rays,and
radar waves.
Chapter 13 Wave Motion and Sound
3,Matter Waves 物质波
The most remarkable property of the matter
waves is that wave functions of matter waves are
referred to as probability amplitudes of waves.
Much of what we discuss in this chapter applies
to waves of all kinds,However,for specific
examples we shall refer to mechanical waves.
these waves are associated with electrons,
protons （ 质子 ）,and other fundamental
particles.
Chapter 13 Wave Motion and Sound
Classification According to Oscillation Types(P327)
1,Transverse Waves 横波
The direction of oscillation of medium elements
is perpendicular to the direction of travel of the
wave.
Compressions and expansions correspond to the
crests (波峰 ) and troughs (波谷 ) of a transverse
waves.
横波,质点振动方向与波的传播方向相 垂直 的波,
（仅在固体中传播 ）
Chapter 13 Wave Motion and Sound
From the figure,the wave has the shape of a
sine curve or a cosine curve,A typical string
element moves up and down continuously as
the wave passes.
Chapter 13 Wave Motion and Sound
特征 1：具有交替出现的波峰 crest和波谷 trough.
特征 2,各质点振动方向与波的传播方向垂直。
如绳波为横波。
传播方向振动方向
Chapter 13 Wave Motion and Sound
2,Longitudinal Waves 纵波 p328
The direction of oscillation of medium elements
is parallel to the direction of the wave’s travel,
the motion is said to be longitudinal,and the
wave is said to be a longitudinal wave.
纵波,质点振动方向与波的传播方向互相 平行 的波,
（可在固体、液体和气体中传播）
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
特征 1：具有交替出现的密部和疏部,
Expansions and compressions
特征 2,各质点振动方向与波的传播方向平行 。
纵波是靠介质疏密部变化传播的，如声波，
弹簧波为纵波。
传播方向振动方向
Chapter 13 Wave Motion and Sound
1 2 3 4 5 6 7 8 9 10 11 12 13t=0
4 5 6 7 8 9 10 11 12 13
1 2 3
t=T/4
7 8 9 10 11 12 13
65
43
2
1t=T/2
6
5432
1 137
8
10 11
12
t=T
Both a transverse wave and a longitudinal wave
are said to be traveling waves because they both
travel from one point to another.
Chapter 13 Wave Motion and Sound
13-1 The Characteristic Quantities of wave,p327
波长,沿波的传播方向，两个相邻的、相位差为 的振动质点之间的距离，即一个完整波形的长度,
The wavelength? of a wave is the distance
between two successive crests.
π2
O
yA
A-
u
x
Chapter 13 Wave Motion and Sound
周期,波前进一个波长的距离所需要的时间,Period is the time required for one complete
oscillation or one complete cycle of the wave to pass
a given point along the line of travel.
T
Tf 1?
频率 f,周期的倒数,即单位时间内波动所传播的完整波的数目,The frequency is the number
of crests—or complete cycles-that pass a given
point per unit time.
振幅 Amplitude Dm,it is the maximum
displacement of the elements from their equilibrium
positions as the wave passes through them,The
height of crest or the depth of a trough.
Chapter 13 Wave Motion and Sound
Wave velocity often referred to as the phase
velocity (波速 v 即相速 ),The distance of an
oscillatory state propagating in a unit-time,in
one T,is the wave moves a distance of? ——
wavelength,so
fTv /
波速 v,波动过程中，某一振动状态（即振动相位）单位时间内所传播的距离（相速）,
Wave velocity,is the velocity at which wave
crests (or any other part of the waveform) move.
Chapter 13 Wave Motion and Sound
①,周期、频率与介质无关，与波源的相同。
波长、波速与介质有关。
③,波在不同介质中频率不变。
②,不同频率的同一类波在同一介质中波速相同。
注意周期或频率只决定于波源的振动 !
波速只决定于媒质的性质！
Chapter 13 Wave Motion and Sound
例 在室温下，已知空气中的声速 为 340 m/s，
水中的声速 为 1450 m/s,求频率为 200 Hz和 2000 Hz
的声波在空气中和水中的波长各为多少？
1u
2u
m7.1
Hz200
sm340 1
1
1
1?
-
f
u? m17.0
2
1
2 f
u?
m25.7Hz20 0 sm14 50
1
1
2
1?
-
f
u? m7 2 5.0
2
2
2 f
u?
在水中的波长解 由,频率为 200 Hz和 2000 Hz 的声波在
fu
空气中的波长
Chapter 13 Wave Motion and Sound
13-4Math’s Expression of Traveling waves (P332-335)
1,The wave function 简谐波的波函数简谐波在介质中传播时,各质元都在做简谐运动,
他们的位移随时间变化 。 由于各质元开始振动的时刻不同,各质元的简谐运动并不同步,即在同一时刻各质元的位移随它们位置的不同而不同 。 介质中任一质点 （ 坐标为 x） 相对其平衡位置的位移 （ 坐标为 D） 随时间的变化关系,即 称为波函数 。),( txD
),( txDD?
各质点相对平衡位置的 位移波线上各质点平衡 位置
Chapter 13 Wave Motion and Sound
以速度 u 沿 x 轴正向传播的简谐波,令原点 O 的初相为零，其振动方程,
tDD MO?c o s?
由于波沿 x 轴正向传播，所以在 x>0的个质点将依次较晚开始振动。
以 v表示沿 x 轴正向传播的简谐波的速度。
位于原点的质元的振动方程
在时刻 t位于 x处的质元的振动方程
Chapter 13 Wave Motion and Sound
点 O 的振动状态
tDD MO?c o s?
点 P
v
x
t
t 时刻点 P 的运动t-x/v时刻点 O 的运动
)(c o s
v
xtDD
MP -
点 P 振动方程时间推迟方法
P
x
*
y
x
u?A
A-
O
tDD Mo?co s?
点 O 振动方程
0,0x
Chapter 13 Wave Motion and Sound
0,0x
])(c o s [),( vxtDtxD Mp
v沿 x 轴 负 向
)c o s ( tDD MO点 O 振动方程波函数
v沿 x轴 正 向 ])(c o s [),(-?
v
xtDtxD
Mp
y
x
uA
A-
O
如果原点的初相位 不 为零
Chapter 13 Wave Motion and Sound
Angular wave number（角波数） k,
2?k
Using relationships:
vTf
T?
22 an
d v
k 2
波数：单位长度内含的波长数目（波长倒数）
角波数,2?长度内含的波长数目（简称波数）
如果把横波中相接的一峰一谷算做一个,完整波,,
波数的含义为,波数等于在 2?长度内含有,完整波,
的数目。
Chapter 13 Wave Motion and Sound
Neglecting the subscript p,we have equivalent
expressions for SH wave equation:
])(c o s [),( oM vxtDtxD-?
])(2c os [),( oM xTtDtxD-?
])(2c o s [),( oM xtfDtxD-?
)c o s (),( oM kxtDtxD-?
Chapter 13 Wave Motion and Sound
The consistent forms with our book should be:
])(s i n [),( otvxAtxy-?
])(2s i n [),( oTtxAtxy-?
)s i n (),( otkxAtxy-?
])(2s i n [),( otfxAtxy-?
In general,wave function of an arbitrary shape
of traveling wave has a form:
)s i n (),( tkxDtxD M
all waves in which the variables x and t enter in
combination are traveling waves,tkx
Chapter 13 Wave Motion and Sound
2,The Physical Meaning of Wave Equation:
波 函数 具有 时间周期性 （ T )o t
A
y
T
(1) Fixed x,corresponding to the oscillating curve
of medium element at position x,i.e,y(t,xo).
(2) Fixed t,corresponding to the y-x curve
(波形图 ) at to.
波 函数 具有 空间周期性 （? )o
y
x
2x1xA
Chapter 13 Wave Motion and Sound
波线上各点的简谐运动图
Chapter 13 Wave Motion and Sound
振动方程与波函数的区别波函数是波程 x 和时间 t的函数，描写 某一时刻 任意位置处质点 振动位移 。
)(tfx?
),( txfy?
振动方程是 时间 t的函数 o
x
t
o
y
x
)c o s ( tAx
-
u
xtAD c os
Chapter 13 Wave Motion and Sound
例 1 已知波动方程如下，求波长、周期和波速,
].)cm01.0()2,5 0 s[(πc o s)cm5( -1-1 xty -?
解,方法一（比较系数法）,
)(π2c o s?xTtAy -?
])cm201.0()s22,5 0[(π2c o s)cm5( 1-1- xty -?
把题中波动方程改写成
s8.0s5.2 2T cm20001.0 cm2 1scm250 -
T
u?
比较得
Chapter 13 Wave Motion and Sound
解,方法二（由各物理量的定义解之）,
--- txt )2,5 0 s[(π])cm01.0()2,5 0 s[(π -11-1-1
π2])cm01.0( 2-1?x
cm20012?-? xx?
])cm01.0()2,5 0 s[(π])cm01.0()2,5 0 s[(π 2-12-11-11-1 xtxt -?-
s8.012?-? ttT
1
12
12 scm2 5 0 -
-
-?
tt
xxu
周期 为相位传播一个波长所需的时间波长 是指同一时刻,波线上相位差为 的两点间的距离,
π2t
cm20012-?xx
Chapter 13 Wave Motion and Sound
])(π2c o s [?
-? x
T
tAy
1） 波动方程
2
π-
例 2 一平面简谐波沿 O x 轴正方向传播,已知振幅,,,在 时坐标原点处的质点位于平衡位置沿 O y 轴正方向运动,求
0?tm0.2m0.1?A s0.2?T
0,0 tyy v
00 xt
解 写出波动方程的标准式
y
A?
O
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
Chapter 13 Wave Motion and Sound
2） 求 波形图,
x)ms i n ( πm)0.1( 1-?
s0.1?t
])m( π2πco s [m)0.1( 1 xy --?
波形方程
s0.1?t
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
o
m/y
m/x2.0
1.0
-1.0
时刻波形图s0.1?t
Chapter 13 Wave Motion and Sound
3） 处质点的振动规律并做图,m5.0?x
]π)sc o s [ ( πm)0.1( 1 -? - ty
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
处质点的振动方程m5.0?x
0
m/y
1.0
-1.0
s/t2.0
O
y
1
2
3
4
*
*
* *
*
*
1
2
3
4
处质点的振动曲线m5.0?x
1.0
Chapter 13 Wave Motion and Sound
例 3 一平面简谐波以速度 沿直线传播,波线上点 A 的简谐运动方程,
s/m20?u
ty A )sπ4c o s ()m103( 12 --
1） 以 A 为坐标原点，写出波动方程
m10 uT?m103 2-A s5.0?T 0
)
m10s5.0
(π2c o s)m103( 2 xty - -
])(π2c o s [?
-? x
T
tAy
u
ABC D
5m 9m
xo
8m
Chapter 13 Wave Motion and Sound
ABAB xx --?- π2 10 5π2 --? π?
π?B? ]π)sπ4c o s [ ()m103( 12 -- ty B
]π)m10s5.0(π2c o s [)m103( 2?- - xty
2） 以 B 为坐标原点，写出波动方程
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
Chapter 13 Wave Motion and Sound
3） 写出传播方向上点 C,点 D 的简谐运动方程
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
点 C 的相位比点 A 超前
]π2)sπ4co s [()m103( 12?ACty C --
]π513)sπ4co s [()m103( 12 -- t
点 D 的相位落后于点 A ]π2)sπ4co s [()m103(
12
ADty D - --
]π59)sπ4co s [()m103( 12 - -- t
m10
Chapter 13 Wave Motion and Sound
4） 分别求出 BC,CD 两点间的相位差
π4.4
10
22π2π2?--?--?-
DCDC xx
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
π6.110 8π2π2 -?-?--?- CBCB xx
m10
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
§ 13-3 Energy,Power and Intensity (波强）
of Traveling Wave P331-332
当机械波在媒质中传播时，媒质中各质点均在其平衡位置附近振动，因而具有振动动能,
同时，介质发生弹性形变，因而具有弹性势能,
每个质元振动所具有的动能每个质元形变所具有的势能波动的过程实际是能量传递的过程。这是波动过程的一个重要特征。
之和机械波的能量
Chapter 13 Wave Motion and Sound
以固体棒中传播的纵波为例分析波动能量的传播,
弹性介质中取一体积元 dV，质元振动速度为 u，
质量 (m is the mass of a particle
or small value of the medium)
dVdm
dV
u波函数 )/(c o s uxtAy -
质元振动速度 t
yv
)/(s i n uxtA --
动能
1,Energy
Chapter 13 Wave Motion and Sound
2
2
1 vdmdE
k?
)/(si n)(21 222 uxtAdV -
Chapter 13 Wave Motion and Sound
dVxyGW 2P )dd(21d?
弹性势能
x
y
)/(s i n uxt
u
A --
22 )(
2
1
2
1
D
dGGw
p
由
dV
x
tA
dV
x
tA
u
G
)(s i n
2
1
)(s i n
2
1
222
222
2
u
u
-?
-?
Gu?2
Chapter 13 Wave Motion and Sound
体积元的总机械能
)(s i ndddd 222pk
u
xtVAWWW -
)(s i nd21dd 222pk uxtVAWW -
For Elastic Potential Energy,it has the same
expression as K,
Both K & U of element dm varies periodically
with time,同时达到最大值,又同时达到最小值,体积元的机械能不守恒 。
Chapter 13 Wave Motion and Sound
讨 论
1） 在波动传播的媒质中,任一体积元的 动能,势能,
总机械能 均随 作 周期性变化,且变化是 同相位 的,tx,
2） 任一体积元都在不断地接收和放出能量,即不断地传播能量,任一体积元的机械能不守恒,波动是能量传递的一种方式,An important feature of wave
motion is the transfer of energy,
3） 波动的能量与振动能量的区别
振动能量中 Ek,EP相互交换,系统总机械能守恒 。
Chapter 13 Wave Motion and Sound
2,Density of Energy 能量密度
能量密度,介质 单位体积 中的波动能量,
)(s i ndd 222 uxtAVWw -
平均 能量密度能量密度在一个周期内的平均值,
介质中 x处在时刻 t的能量密度为：
Chapter 13 Wave Motion and Sound
22222
0
2
2
1d1 fAAtw
T
w
T
2Aw?普适结论 2fw?
The average power,which is average rate at which
energy of both kinds is transmitted by the wave,is:
Chapter 13 Wave Motion and Sound
3,Wave Intensity 波的能 流 密度（强度）
随着振动在介质中的传播，能量也从介质的一端传到另一端，波动是能量传递的一种形式。用一个物理量来描述传播能量的本领。
能流密度 ( 波的强度 ),
平均在 单位时间 内通过垂直于波传播方向的 单位面积 的平均能流,
I
udt dS
u?
The intensity I of a wave,or density of energy flow,
at a surface is average rate per unit area at which
energy is transferred by the wave through or onto
the surface.
Chapter 13 Wave Motion and Sound
udt dS
u?取垂直于波的传播方向的一个小面积 ds,平均在 dt时间内通过此面积后方体积为 udtds
的立方体的平均总能量为：
uw
d t d S
dWI
u d t d SwdW?
以 I表示波的强度，有
22
2
1 Aw
Chapter 13 Wave Motion and Sound
where? is the density of medium in volume
and v is wave speed.
uAI 2221
The SI unit of I is the watt per square meter.
任意谐波 2AI?
Chapter 13 Wave Motion and Sound
4,平面波 (planar wave) 和球面波 (spherical wave)的振幅 p331-332
1) 平面波
1S 2S
u?
21 WW?
21 AA?
由得
uSAuSwSIW 2211111 21
uSAuSwSIW 2222222 21
这表明平面波在媒质不吸收的情况下,振幅不变。
Chapter 13 Wave Motion and Sound
2)球面波
2
22
21
22
1 2
1
2
1 uSAuSA由
1S
2S
1r
2r
22222121 π4 π4 rArA
2211 rArA?
得球面波振幅与离点波源的距离成反比。球面波的振幅在媒质不吸收的情况下,随 r增大而减小,
22222121 π4 π4 rArA
1
2
2
1
2
2
I
I
r
r?
The intensity is proportional to the square of the amplitude.
Chapter 13 Wave Motion and Sound
13-6 The principle of superposition for
waves（波的叠加原理 P337）
Chapter 13 Wave Motion and Sound
几列波相遇之后,仍然保持它们各自原有的特征
（频 率,波长、振幅、振动方向等）不变，并按照原来的方向继续前进，好象没有遇到过其他波一样,
在相遇区域内任一点的振动，为各列波单独存在时在该点所引起的振动位移的矢量和,
叠加性,When two or more waves traverse the
same medium,the displacement of any particle
of the medium is the sum of the displacements
that the individual waves would give it.
独立性,But the directions of individual waves
can not change,as if the other were not present.
Chapter 13 Wave Motion and Sound
§ 13-8 Interference (干涉 ) of Waves (P339-340)
干涉现象相干波相干条件 频率相同、振动方向相同、相位差恒定。
一般情况下，各个波的振动方向和频率均不同，相位关系不确定，叠加的合成波较为复杂。
当两列 （ 或多列 ） 相干波叠加的结果,其合振幅 A
和合强度 I 将在空间形成一种稳定的分布,即某些点上的振动始终加强,某些点上的振动始终减弱 。
—— 波的干涉
相干波源满足相干条件的波产生相干波的波源?
Chapter 13 Wave Motion and Sound
频率相同、
振动方向平行、
相位相同或相位差恒定的两列波相遇时，使某些地方振动始终加强，而使另一些地方振动始终减弱的现象，称为波的干涉现象,
Chapter 13 Wave Motion and Sound
1,Interference of Waves(波的干涉 ):
There is a stable distribution of oscillatory
constructiveness and destructiveness in the
region of overlapping waves.
Chapter 13 Wave Motion and Sound
Interference is one of unique characteristics of
waves (干涉现象是波动形式所 独具 的重要特征之一 ).
)c o s (21 tDDDD M
The resultant oscillation in point P is also SHM:
Based on the superposition of oscillation,(See
Chapter12 Oscillation2,slides 34)
c o s2 212 22 1 MMMMM DDDDD
2211
2211
c o sc o s
s i ns i nt a n
MM
MM
DD
DD
Chapter 13 Wave Motion and Sound
The in expression A is the phase difference
arisen by two waves at point P,It does not vary
with time,so does not amplitude at every points.
1s
2s
P
*1r
2r
12
12 π2
rr --- 常量
Chapter 13 Wave Motion and Sound
(1) when (k=0,1,2,…)?
k
rr 22 12
12
---
21 MMM DDD
The greatest possible DM —– Fully
constructive interference (相干加强 ),
(2) When (k=0,1,2,…), )12( k
MMM DDD 21 -?
The lest possible A —– Fully
destructive interference (相干减弱 ).
(3) When (k=0,1,2,…), )12(2 kk
Then DM is between —–
intermediate interference (neither fully
constructive nor fully destructive).
2121 MMMM DDandDD -?
Chapter 13 Wave Motion and Sound
Special case,(i) 21
MM
MMM
DDD
DDD
kk
k
rrL
21
21
21,.,,)2,1,0(
2
)12( -?
-
两列相干波源同相位时,其相干加强和减弱情况仅取决于波程差。
(ii) When,
021 MMM DDD
)c o s1(2c o s2 0212 22 1 MMMMMM DDDDDD
2c o s2 0
MM DD
x 2
Chapter 13 Wave Motion and Sound
§ 13-9 Standing Waves (驻波 ),(P341-344)
1 the produce of standing waves 驻波的产生振幅、频率、传播速度都相同的两列相干波，在同一直线上沿 相反 方向传播时叠加而形成的一种特殊的干涉现象,
If two sinusoidal waves of the same amplitude
and wavelength travel in opposite directions
along a stretched string,their interference with
each other produces a standing wave.
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
驻 波 的 形 成
Chapter 13 Wave Motion and Sound
2,the equation of standing waves 驻波方程
To analyze a standing wave,we represent two
waves with:
)s i n (),(1 tkxDtxD m?-?
)s i n (),(2 tkxDtxD m
For combined waves,from superposition-principle,
)s i n ()s i n (),(' tkxDtkxDtxD mm-?
tkxDtxD m?co s]s i n2[),('?
驻波的振幅与位置有关各质点都在作同频率的简谐运动
Chapter 13 Wave Motion and Sound
It does not describe a traveling wave because
it is not of the form of,Instead,it
describe a Standing Wave.
tkx
tkxDtxD m?co s]s i n2[),('?
Chapter 13 Wave Motion and Sound
The amplitude in a SW varies with position.
(i) The zero amplitude is for values of kx that
give |sinkx|=0,Those values are
)(,...2,1,0, nnkx or 2 nx –— notes
The stationary position of zero displacement in
SW is called nodes (波节 ),The interval between
pairs of nodes is given by
21
-
nn xx
Adjacent nodes are separated by half a wavelength.
3.The Distribution of Amplitude ( ):
kxD m s in2
Chapter 13 Wave Motion and Sound
(ii) The amplitude of the SW has a maximum
value 2Dm (or 2A),which occurs where |sinkx|=1.
The positions satisfied these condition are
called antinodes (波腹 ),The interval between
pairs of two antinodes is also?/2.
),2,1,0(,)12(21 nnkx?
2)2
1( nxor –— antinodes
(iii) The amplitude of the rest positions except
nodes and antinodes are between 0 and 2DM.
Look at the ―Example 13-7 of P343
Chapter 13 Wave Motion and Sound
The phase of particles between adjacent nodes
are same and out of phase at two sides of nodes
(相邻波节之间的相位相同,而波节的两边相位相反 ),
The Phase Shift:
tkxDtxD m?co s]s i n2[),(?
相邻两波节之间质点振动同相位，任一波节两侧振动相位相反，在 波节 处产生 的 相位跃变,（与行波不同，无相位的传播）,
π
Chapter 13 Wave Motion and Sound
txAy π2co sπ2co s2?
xπ2c o s,44,0
-? x txAy?
π2co sπ2co s2?
)ππ2c o s (π2c o s2 txAy,
4
3
4,0
x
xπ2c o s
x
y
o
2?2?-
4
x 为 波节例
Chapter 13 Wave Motion and Sound
3,Reflections at a Boundary:
In a fixed (or ―hard‖ ) reflection,there must be a
node at the boundary —– reflected and incident
pulses must be out of phase at the point (在反射点处 反 射 波 有?相位 的突变,称为 半 波 损 失 (Half-
wavelength loss );
P338
Medium with larger?v is dense medium (波 密 媒质 );the one with smaller?v is Thin medium (波疏 媒质 ).
Chapter 13 Wave Motion and Sound
当波从波疏介质垂直入射到波密介质,被反射到波疏介质时形成 波节,入射波与反射波在此处的相位时时 相反,即反射波在 分界处 产生 的相位 跃变,
相当于出现了半个波长的波程差，称 半波损失,
π
波密介质
u? 较大波疏介质较小
u?
Thin medium?dense medium:Half-wavelength loss !
Chapter 13 Wave Motion and Sound
当波从波密介质垂直入射到波疏介质,被反射到波密介质时形成 波腹,入射波与反射波在此处的相位时时 相同,即反射波在分界处 不 产生相位 跃变,
波密?波疏,No half-? loss !
Chapter 13 Wave Motion and Sound
4 Standing wave patterns 振动的简正模式 p341-343
Nature frequencies or resonant frequencies:
The frequencies at which standing waves are
produced are the natural frequencies.
Chapter 13 Wave Motion and Sound
应满足,由此频率两端 固定 的弦线形成 驻 波时，波长 和弦线长
2
nnl
,2,12 nlunn?
n?
l
决定的各种振动方式称为弦线振动的 简正模式,
The lowest frequency is called the fundamental
frequency; the other natural frequencies are
called overtones; when they are integral
multiples of the fundamental they are called
harmonics,with the fundamental being referred
to as the first harmonic,The next mode after the
fundamental has two loops and is called the
second harmonic.
Chapter 13 Wave Motion and Sound
,2,12 nnl n?
两端 固定 的弦振动的简正模式一端 固定 一端 自由的弦振动的简正模式
,2,12)21(?-? nnl n?
2
1l
2
2 2l
2
3 3l
4
1l
4
3 2l
4
5 3l
Chapter 13 Wave Motion and Sound
13-7 Reflection and transmission p338-339
13-10 Refraction p344-345
13-11 Diffraction p345-346
Study by yourselves
Chapter 13 Wave Motion and Sound
Summary for Chapter Twelve
See Page 346
Chapter 13 Wave Motion and Sound
Homework,
p349:21,24
P351:61
Chapter 13&14
Wave Motion and Sound
1,Simple Harmonic Waves
2,Wave Equation
3,Energy and Power of Waves
4,Interference of Waves
5,Standing Waves
6,The Doppler Effect
Chapter 13 Wave Motion and Sound
Key words:
Mechanical wave 机械波
Electromagnetic wave 电磁波
Transverse wave（ 横波）
Longitudinal wave（纵波）
crests (波峰 ) and troughs (波谷 )
Sinusoidal wave（正弦波）
Traveling (moving,progressive) wave（行波）
Standing (stationary) wave（驻波）
Stretched string（拉紧的线、弦线）
Angular wave number（角波数）
Interference of waves（波的干涉）
Chapter 13 Wave Motion and Sound
一定的扰动的传播称为波动，简称为波。
波动是振动的传播过程；振动是激发波动的波源。
机械波电磁波波动机械振动在弹性介质中的传播,
交变电磁场在空间的传播,
两类波的不同之处
机械波的传播需有传播振动的介质 ;
电磁波的传播可不需介质,可在真空中传播。
具有传播速度
能量传播
反射
折射
干涉
衍射两类波的共同特征几个基本概念
Chapter 13 Wave Motion and Sound
Classification According to Natures of Matter
§ 13-2 Wave Types p327-331
1,Mechanical Waves
机械波,机械振动以一定速度在弹性介质中由近及远地传播出去,就形成机械波 。 Such as water waves,
sound waves,All these waves have certain central
features,they are governed by Newton’s laws,and
they can exist only within a material medium（ 媒质,
介质 ）,such as water,air,and rock.
Chapter 13 Wave Motion and Sound
2,Electromagnetic Waves
电磁波,Such as light,radio and television waves,
X rays,These waves require no material
medium to exist,Its speed through the vacuum
is 299792458m/s.
Examples,Visible and ultraviolet light,radio
and television waves,microwaves,x rays,and
radar waves.
Chapter 13 Wave Motion and Sound
3,Matter Waves 物质波
The most remarkable property of the matter
waves is that wave functions of matter waves are
referred to as probability amplitudes of waves.
Much of what we discuss in this chapter applies
to waves of all kinds,However,for specific
examples we shall refer to mechanical waves.
these waves are associated with electrons,
protons （ 质子 ）,and other fundamental
particles.
Chapter 13 Wave Motion and Sound
Classification According to Oscillation Types(P327)
1,Transverse Waves 横波
The direction of oscillation of medium elements
is perpendicular to the direction of travel of the
wave.
Compressions and expansions correspond to the
crests (波峰 ) and troughs (波谷 ) of a transverse
waves.
横波,质点振动方向与波的传播方向相 垂直 的波,
（仅在固体中传播 ）
Chapter 13 Wave Motion and Sound
From the figure,the wave has the shape of a
sine curve or a cosine curve,A typical string
element moves up and down continuously as
the wave passes.
Chapter 13 Wave Motion and Sound
特征 1：具有交替出现的波峰 crest和波谷 trough.
特征 2,各质点振动方向与波的传播方向垂直。
如绳波为横波。
传播方向振动方向
Chapter 13 Wave Motion and Sound
2,Longitudinal Waves 纵波 p328
The direction of oscillation of medium elements
is parallel to the direction of the wave’s travel,
the motion is said to be longitudinal,and the
wave is said to be a longitudinal wave.
纵波,质点振动方向与波的传播方向互相 平行 的波,
（可在固体、液体和气体中传播）
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
特征 1：具有交替出现的密部和疏部,
Expansions and compressions
特征 2,各质点振动方向与波的传播方向平行 。
纵波是靠介质疏密部变化传播的，如声波，
弹簧波为纵波。
传播方向振动方向
Chapter 13 Wave Motion and Sound
1 2 3 4 5 6 7 8 9 10 11 12 13t=0
4 5 6 7 8 9 10 11 12 13
1 2 3
t=T/4
7 8 9 10 11 12 13
65
43
2
1t=T/2
6
5432
1 137
8
10 11
12
t=T
Both a transverse wave and a longitudinal wave
are said to be traveling waves because they both
travel from one point to another.
Chapter 13 Wave Motion and Sound
13-1 The Characteristic Quantities of wave,p327
波长,沿波的传播方向，两个相邻的、相位差为 的振动质点之间的距离，即一个完整波形的长度,
The wavelength? of a wave is the distance
between two successive crests.
π2
O
yA
A-
u
x
Chapter 13 Wave Motion and Sound
周期,波前进一个波长的距离所需要的时间,Period is the time required for one complete
oscillation or one complete cycle of the wave to pass
a given point along the line of travel.
T
Tf 1?
频率 f,周期的倒数,即单位时间内波动所传播的完整波的数目,The frequency is the number
of crests—or complete cycles-that pass a given
point per unit time.
振幅 Amplitude Dm,it is the maximum
displacement of the elements from their equilibrium
positions as the wave passes through them,The
height of crest or the depth of a trough.
Chapter 13 Wave Motion and Sound
Wave velocity often referred to as the phase
velocity (波速 v 即相速 ),The distance of an
oscillatory state propagating in a unit-time,in
one T,is the wave moves a distance of? ——
wavelength,so
fTv /
波速 v,波动过程中，某一振动状态（即振动相位）单位时间内所传播的距离（相速）,
Wave velocity,is the velocity at which wave
crests (or any other part of the waveform) move.
Chapter 13 Wave Motion and Sound
①,周期、频率与介质无关，与波源的相同。
波长、波速与介质有关。
③,波在不同介质中频率不变。
②,不同频率的同一类波在同一介质中波速相同。
注意周期或频率只决定于波源的振动 !
波速只决定于媒质的性质！
Chapter 13 Wave Motion and Sound
例 在室温下，已知空气中的声速 为 340 m/s，
水中的声速 为 1450 m/s,求频率为 200 Hz和 2000 Hz
的声波在空气中和水中的波长各为多少？
1u
2u
m7.1
Hz200
sm340 1
1
1
1?
-
f
u? m17.0
2
1
2 f
u?
m25.7Hz20 0 sm14 50
1
1
2
1?
-
f
u? m7 2 5.0
2
2
2 f
u?
在水中的波长解 由,频率为 200 Hz和 2000 Hz 的声波在
fu
空气中的波长
Chapter 13 Wave Motion and Sound
13-4Math’s Expression of Traveling waves (P332-335)
1,The wave function 简谐波的波函数简谐波在介质中传播时,各质元都在做简谐运动,
他们的位移随时间变化 。 由于各质元开始振动的时刻不同,各质元的简谐运动并不同步,即在同一时刻各质元的位移随它们位置的不同而不同 。 介质中任一质点 （ 坐标为 x） 相对其平衡位置的位移 （ 坐标为 D） 随时间的变化关系,即 称为波函数 。),( txD
),( txDD?
各质点相对平衡位置的 位移波线上各质点平衡 位置
Chapter 13 Wave Motion and Sound
以速度 u 沿 x 轴正向传播的简谐波,令原点 O 的初相为零，其振动方程,
tDD MO?c o s?
由于波沿 x 轴正向传播，所以在 x>0的个质点将依次较晚开始振动。
以 v表示沿 x 轴正向传播的简谐波的速度。
位于原点的质元的振动方程
在时刻 t位于 x处的质元的振动方程
Chapter 13 Wave Motion and Sound
点 O 的振动状态
tDD MO?c o s?
点 P
v
x
t
t 时刻点 P 的运动t-x/v时刻点 O 的运动
)(c o s
v
xtDD
MP -
点 P 振动方程时间推迟方法
P
x
*
y
x
u?A
A-
O
tDD Mo?co s?
点 O 振动方程
0,0x
Chapter 13 Wave Motion and Sound
0,0x
])(c o s [),( vxtDtxD Mp
v沿 x 轴 负 向
)c o s ( tDD MO点 O 振动方程波函数
v沿 x轴 正 向 ])(c o s [),(-?
v
xtDtxD
Mp
y
x
uA
A-
O
如果原点的初相位 不 为零
Chapter 13 Wave Motion and Sound
Angular wave number（角波数） k,
2?k
Using relationships:
vTf
T?
22 an
d v
k 2
波数：单位长度内含的波长数目（波长倒数）
角波数,2?长度内含的波长数目（简称波数）
如果把横波中相接的一峰一谷算做一个,完整波,,
波数的含义为,波数等于在 2?长度内含有,完整波,
的数目。
Chapter 13 Wave Motion and Sound
Neglecting the subscript p,we have equivalent
expressions for SH wave equation:
])(c o s [),( oM vxtDtxD-?
])(2c os [),( oM xTtDtxD-?
])(2c o s [),( oM xtfDtxD-?
)c o s (),( oM kxtDtxD-?
Chapter 13 Wave Motion and Sound
The consistent forms with our book should be:
])(s i n [),( otvxAtxy-?
])(2s i n [),( oTtxAtxy-?
)s i n (),( otkxAtxy-?
])(2s i n [),( otfxAtxy-?
In general,wave function of an arbitrary shape
of traveling wave has a form:
)s i n (),( tkxDtxD M
all waves in which the variables x and t enter in
combination are traveling waves,tkx
Chapter 13 Wave Motion and Sound
2,The Physical Meaning of Wave Equation:
波 函数 具有 时间周期性 （ T )o t
A
y
T
(1) Fixed x,corresponding to the oscillating curve
of medium element at position x,i.e,y(t,xo).
(2) Fixed t,corresponding to the y-x curve
(波形图 ) at to.
波 函数 具有 空间周期性 （? )o
y
x
2x1xA
Chapter 13 Wave Motion and Sound
波线上各点的简谐运动图
Chapter 13 Wave Motion and Sound
振动方程与波函数的区别波函数是波程 x 和时间 t的函数，描写 某一时刻 任意位置处质点 振动位移 。
)(tfx?
),( txfy?
振动方程是 时间 t的函数 o
x
t
o
y
x
)c o s ( tAx
-
u
xtAD c os
Chapter 13 Wave Motion and Sound
例 1 已知波动方程如下，求波长、周期和波速,
].)cm01.0()2,5 0 s[(πc o s)cm5( -1-1 xty -?
解,方法一（比较系数法）,
)(π2c o s?xTtAy -?
])cm201.0()s22,5 0[(π2c o s)cm5( 1-1- xty -?
把题中波动方程改写成
s8.0s5.2 2T cm20001.0 cm2 1scm250 -
T
u?
比较得
Chapter 13 Wave Motion and Sound
解,方法二（由各物理量的定义解之）,
--- txt )2,5 0 s[(π])cm01.0()2,5 0 s[(π -11-1-1
π2])cm01.0( 2-1?x
cm20012?-? xx?
])cm01.0()2,5 0 s[(π])cm01.0()2,5 0 s[(π 2-12-11-11-1 xtxt -?-
s8.012?-? ttT
1
12
12 scm2 5 0 -
-
-?
tt
xxu
周期 为相位传播一个波长所需的时间波长 是指同一时刻,波线上相位差为 的两点间的距离,
π2t
cm20012-?xx
Chapter 13 Wave Motion and Sound
])(π2c o s [?
-? x
T
tAy
1） 波动方程
2
π-
例 2 一平面简谐波沿 O x 轴正方向传播,已知振幅,,,在 时坐标原点处的质点位于平衡位置沿 O y 轴正方向运动,求
0?tm0.2m0.1?A s0.2?T
0,0 tyy v
00 xt
解 写出波动方程的标准式
y
A?
O
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
Chapter 13 Wave Motion and Sound
2） 求 波形图,
x)ms i n ( πm)0.1( 1-?
s0.1?t
])m( π2πco s [m)0.1( 1 xy --?
波形方程
s0.1?t
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
o
m/y
m/x2.0
1.0
-1.0
时刻波形图s0.1?t
Chapter 13 Wave Motion and Sound
3） 处质点的振动规律并做图,m5.0?x
]π)sc o s [ ( πm)0.1( 1 -? - ty
]2π)m0.2s0.2(π2co s [m)0.1( --? xty
处质点的振动方程m5.0?x
0
m/y
1.0
-1.0
s/t2.0
O
y
1
2
3
4
*
*
* *
*
*
1
2
3
4
处质点的振动曲线m5.0?x
1.0
Chapter 13 Wave Motion and Sound
例 3 一平面简谐波以速度 沿直线传播,波线上点 A 的简谐运动方程,
s/m20?u
ty A )sπ4c o s ()m103( 12 --
1） 以 A 为坐标原点，写出波动方程
m10 uT?m103 2-A s5.0?T 0
)
m10s5.0
(π2c o s)m103( 2 xty - -
])(π2c o s [?
-? x
T
tAy
u
ABC D
5m 9m
xo
8m
Chapter 13 Wave Motion and Sound
ABAB xx --?- π2 10 5π2 --? π?
π?B? ]π)sπ4c o s [ ()m103( 12 -- ty B
]π)m10s5.0(π2c o s [)m103( 2?- - xty
2） 以 B 为坐标原点，写出波动方程
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
Chapter 13 Wave Motion and Sound
3） 写出传播方向上点 C,点 D 的简谐运动方程
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
点 C 的相位比点 A 超前
]π2)sπ4co s [()m103( 12?ACty C --
]π513)sπ4co s [()m103( 12 -- t
点 D 的相位落后于点 A ]π2)sπ4co s [()m103(
12
ADty D - --
]π59)sπ4co s [()m103( 12 - -- t
m10
Chapter 13 Wave Motion and Sound
4） 分别求出 BC,CD 两点间的相位差
π4.4
10
22π2π2?--?--?-
DCDC xx
u
ABC D
5m 9m
xo
8m
ty A )sπ4c o s ()m103( 12 --
π6.110 8π2π2 -?-?--?- CBCB xx
m10
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
§ 13-3 Energy,Power and Intensity (波强）
of Traveling Wave P331-332
当机械波在媒质中传播时，媒质中各质点均在其平衡位置附近振动，因而具有振动动能,
同时，介质发生弹性形变，因而具有弹性势能,
每个质元振动所具有的动能每个质元形变所具有的势能波动的过程实际是能量传递的过程。这是波动过程的一个重要特征。
之和机械波的能量
Chapter 13 Wave Motion and Sound
以固体棒中传播的纵波为例分析波动能量的传播,
弹性介质中取一体积元 dV，质元振动速度为 u，
质量 (m is the mass of a particle
or small value of the medium)
dVdm
dV
u波函数 )/(c o s uxtAy -
质元振动速度 t
yv
)/(s i n uxtA --
动能
1,Energy
Chapter 13 Wave Motion and Sound
2
2
1 vdmdE
k?
)/(si n)(21 222 uxtAdV -
Chapter 13 Wave Motion and Sound
dVxyGW 2P )dd(21d?
弹性势能
x
y
)/(s i n uxt
u
A --
22 )(
2
1
2
1
D
dGGw
p
由
dV
x
tA
dV
x
tA
u
G
)(s i n
2
1
)(s i n
2
1
222
222
2
u
u
-?
-?
Gu?2
Chapter 13 Wave Motion and Sound
体积元的总机械能
)(s i ndddd 222pk
u
xtVAWWW -
)(s i nd21dd 222pk uxtVAWW -
For Elastic Potential Energy,it has the same
expression as K,
Both K & U of element dm varies periodically
with time,同时达到最大值,又同时达到最小值,体积元的机械能不守恒 。
Chapter 13 Wave Motion and Sound
讨 论
1） 在波动传播的媒质中,任一体积元的 动能,势能,
总机械能 均随 作 周期性变化,且变化是 同相位 的,tx,
2） 任一体积元都在不断地接收和放出能量,即不断地传播能量,任一体积元的机械能不守恒,波动是能量传递的一种方式,An important feature of wave
motion is the transfer of energy,
3） 波动的能量与振动能量的区别
振动能量中 Ek,EP相互交换,系统总机械能守恒 。
Chapter 13 Wave Motion and Sound
2,Density of Energy 能量密度
能量密度,介质 单位体积 中的波动能量,
)(s i ndd 222 uxtAVWw -
平均 能量密度能量密度在一个周期内的平均值,
介质中 x处在时刻 t的能量密度为：
Chapter 13 Wave Motion and Sound
22222
0
2
2
1d1 fAAtw
T
w
T
2Aw?普适结论 2fw?
The average power,which is average rate at which
energy of both kinds is transmitted by the wave,is:
Chapter 13 Wave Motion and Sound
3,Wave Intensity 波的能 流 密度（强度）
随着振动在介质中的传播，能量也从介质的一端传到另一端，波动是能量传递的一种形式。用一个物理量来描述传播能量的本领。
能流密度 ( 波的强度 ),
平均在 单位时间 内通过垂直于波传播方向的 单位面积 的平均能流,
I
udt dS
u?
The intensity I of a wave,or density of energy flow,
at a surface is average rate per unit area at which
energy is transferred by the wave through or onto
the surface.
Chapter 13 Wave Motion and Sound
udt dS
u?取垂直于波的传播方向的一个小面积 ds,平均在 dt时间内通过此面积后方体积为 udtds
的立方体的平均总能量为：
uw
d t d S
dWI
u d t d SwdW?
以 I表示波的强度，有
22
2
1 Aw
Chapter 13 Wave Motion and Sound
where? is the density of medium in volume
and v is wave speed.
uAI 2221
The SI unit of I is the watt per square meter.
任意谐波 2AI?
Chapter 13 Wave Motion and Sound
4,平面波 (planar wave) 和球面波 (spherical wave)的振幅 p331-332
1) 平面波
1S 2S
u?
21 WW?
21 AA?
由得
uSAuSwSIW 2211111 21
uSAuSwSIW 2222222 21
这表明平面波在媒质不吸收的情况下,振幅不变。
Chapter 13 Wave Motion and Sound
2)球面波
2
22
21
22
1 2
1
2
1 uSAuSA由
1S
2S
1r
2r
22222121 π4 π4 rArA
2211 rArA?
得球面波振幅与离点波源的距离成反比。球面波的振幅在媒质不吸收的情况下,随 r增大而减小,
22222121 π4 π4 rArA
1
2
2
1
2
2
I
I
r
r?
The intensity is proportional to the square of the amplitude.
Chapter 13 Wave Motion and Sound
13-6 The principle of superposition for
waves（波的叠加原理 P337）
Chapter 13 Wave Motion and Sound
几列波相遇之后,仍然保持它们各自原有的特征
（频 率,波长、振幅、振动方向等）不变，并按照原来的方向继续前进，好象没有遇到过其他波一样,
在相遇区域内任一点的振动，为各列波单独存在时在该点所引起的振动位移的矢量和,
叠加性,When two or more waves traverse the
same medium,the displacement of any particle
of the medium is the sum of the displacements
that the individual waves would give it.
独立性,But the directions of individual waves
can not change,as if the other were not present.
Chapter 13 Wave Motion and Sound
§ 13-8 Interference (干涉 ) of Waves (P339-340)
干涉现象相干波相干条件 频率相同、振动方向相同、相位差恒定。
一般情况下，各个波的振动方向和频率均不同，相位关系不确定，叠加的合成波较为复杂。
当两列 （ 或多列 ） 相干波叠加的结果,其合振幅 A
和合强度 I 将在空间形成一种稳定的分布,即某些点上的振动始终加强,某些点上的振动始终减弱 。
—— 波的干涉
相干波源满足相干条件的波产生相干波的波源?
Chapter 13 Wave Motion and Sound
频率相同、
振动方向平行、
相位相同或相位差恒定的两列波相遇时，使某些地方振动始终加强，而使另一些地方振动始终减弱的现象，称为波的干涉现象,
Chapter 13 Wave Motion and Sound
1,Interference of Waves(波的干涉 ):
There is a stable distribution of oscillatory
constructiveness and destructiveness in the
region of overlapping waves.
Chapter 13 Wave Motion and Sound
Interference is one of unique characteristics of
waves (干涉现象是波动形式所 独具 的重要特征之一 ).
)c o s (21 tDDDD M
The resultant oscillation in point P is also SHM:
Based on the superposition of oscillation,(See
Chapter12 Oscillation2,slides 34)
c o s2 212 22 1 MMMMM DDDDD
2211
2211
c o sc o s
s i ns i nt a n
MM
MM
DD
DD
Chapter 13 Wave Motion and Sound
The in expression A is the phase difference
arisen by two waves at point P,It does not vary
with time,so does not amplitude at every points.
1s
2s
P
*1r
2r
12
12 π2
rr --- 常量
Chapter 13 Wave Motion and Sound
(1) when (k=0,1,2,…)?
k
rr 22 12
12
---
21 MMM DDD
The greatest possible DM —– Fully
constructive interference (相干加强 ),
(2) When (k=0,1,2,…), )12( k
MMM DDD 21 -?
The lest possible A —– Fully
destructive interference (相干减弱 ).
(3) When (k=0,1,2,…), )12(2 kk
Then DM is between —–
intermediate interference (neither fully
constructive nor fully destructive).
2121 MMMM DDandDD -?
Chapter 13 Wave Motion and Sound
Special case,(i) 21
MM
MMM
DDD
DDD
kk
k
rrL
21
21
21,.,,)2,1,0(
2
)12( -?
-
两列相干波源同相位时,其相干加强和减弱情况仅取决于波程差。
(ii) When,
021 MMM DDD
)c o s1(2c o s2 0212 22 1 MMMMMM DDDDDD
2c o s2 0
MM DD
x 2
Chapter 13 Wave Motion and Sound
§ 13-9 Standing Waves (驻波 ),(P341-344)
1 the produce of standing waves 驻波的产生振幅、频率、传播速度都相同的两列相干波，在同一直线上沿 相反 方向传播时叠加而形成的一种特殊的干涉现象,
If two sinusoidal waves of the same amplitude
and wavelength travel in opposite directions
along a stretched string,their interference with
each other produces a standing wave.
Chapter 13 Wave Motion and Sound
Chapter 13 Wave Motion and Sound
驻 波 的 形 成
Chapter 13 Wave Motion and Sound
2,the equation of standing waves 驻波方程
To analyze a standing wave,we represent two
waves with:
)s i n (),(1 tkxDtxD m?-?
)s i n (),(2 tkxDtxD m
For combined waves,from superposition-principle,
)s i n ()s i n (),(' tkxDtkxDtxD mm-?
tkxDtxD m?co s]s i n2[),('?
驻波的振幅与位置有关各质点都在作同频率的简谐运动
Chapter 13 Wave Motion and Sound
It does not describe a traveling wave because
it is not of the form of,Instead,it
describe a Standing Wave.
tkx
tkxDtxD m?co s]s i n2[),('?
Chapter 13 Wave Motion and Sound
The amplitude in a SW varies with position.
(i) The zero amplitude is for values of kx that
give |sinkx|=0,Those values are
)(,...2,1,0, nnkx or 2 nx –— notes
The stationary position of zero displacement in
SW is called nodes (波节 ),The interval between
pairs of nodes is given by
21
-
nn xx
Adjacent nodes are separated by half a wavelength.
3.The Distribution of Amplitude ( ):
kxD m s in2
Chapter 13 Wave Motion and Sound
(ii) The amplitude of the SW has a maximum
value 2Dm (or 2A),which occurs where |sinkx|=1.
The positions satisfied these condition are
called antinodes (波腹 ),The interval between
pairs of two antinodes is also?/2.
),2,1,0(,)12(21 nnkx?
2)2
1( nxor –— antinodes
(iii) The amplitude of the rest positions except
nodes and antinodes are between 0 and 2DM.
Look at the ―Example 13-7 of P343
Chapter 13 Wave Motion and Sound
The phase of particles between adjacent nodes
are same and out of phase at two sides of nodes
(相邻波节之间的相位相同,而波节的两边相位相反 ),
The Phase Shift:
tkxDtxD m?co s]s i n2[),(?
相邻两波节之间质点振动同相位，任一波节两侧振动相位相反，在 波节 处产生 的 相位跃变,（与行波不同，无相位的传播）,
π
Chapter 13 Wave Motion and Sound
txAy π2co sπ2co s2?
xπ2c o s,44,0
-? x txAy?
π2co sπ2co s2?
)ππ2c o s (π2c o s2 txAy,
4
3
4,0
x
xπ2c o s
x
y
o
2?2?-
4
x 为 波节例
Chapter 13 Wave Motion and Sound
3,Reflections at a Boundary:
In a fixed (or ―hard‖ ) reflection,there must be a
node at the boundary —– reflected and incident
pulses must be out of phase at the point (在反射点处 反 射 波 有?相位 的突变,称为 半 波 损 失 (Half-
wavelength loss );
P338
Medium with larger?v is dense medium (波 密 媒质 );the one with smaller?v is Thin medium (波疏 媒质 ).
Chapter 13 Wave Motion and Sound
当波从波疏介质垂直入射到波密介质,被反射到波疏介质时形成 波节,入射波与反射波在此处的相位时时 相反,即反射波在 分界处 产生 的相位 跃变,
相当于出现了半个波长的波程差，称 半波损失,
π
波密介质
u? 较大波疏介质较小
u?
Thin medium?dense medium:Half-wavelength loss !
Chapter 13 Wave Motion and Sound
当波从波密介质垂直入射到波疏介质,被反射到波密介质时形成 波腹,入射波与反射波在此处的相位时时 相同,即反射波在分界处 不 产生相位 跃变,
波密?波疏,No half-? loss !
Chapter 13 Wave Motion and Sound
4 Standing wave patterns 振动的简正模式 p341-343
Nature frequencies or resonant frequencies:
The frequencies at which standing waves are
produced are the natural frequencies.
Chapter 13 Wave Motion and Sound
应满足,由此频率两端 固定 的弦线形成 驻 波时，波长 和弦线长
2
nnl
,2,12 nlunn?
n?
l
决定的各种振动方式称为弦线振动的 简正模式,
The lowest frequency is called the fundamental
frequency; the other natural frequencies are
called overtones; when they are integral
multiples of the fundamental they are called
harmonics,with the fundamental being referred
to as the first harmonic,The next mode after the
fundamental has two loops and is called the
second harmonic.
Chapter 13 Wave Motion and Sound
,2,12 nnl n?
两端 固定 的弦振动的简正模式一端 固定 一端 自由的弦振动的简正模式
,2,12)21(?-? nnl n?
2
1l
2
2 2l
2
3 3l
4
1l
4
3 2l
4
5 3l
Chapter 13 Wave Motion and Sound
13-7 Reflection and transmission p338-339
13-10 Refraction p344-345
13-11 Diffraction p345-346
Study by yourselves
Chapter 13 Wave Motion and Sound
Summary for Chapter Twelve
See Page 346
Chapter 13 Wave Motion and Sound
Homework,
p349:21,24
P351:61
