第 15章 机械波
Chapter 15 Mechanical Wave
§ 15-1 Formation & 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 波动方程
§ 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(自学)
§ 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
§ 15-5 Huygen’s Principle
惠更斯原理
§ 15-7 Standing Waves 驻波
教学要求
1、确切理解描述波动的物理量的物理意义,并能熟练
地确定这些物理量;
2、深刻理解平面简谐波波动方程的物理意义,并会建
立波动方程,运用它来讨论与分析波动现象;
3、理解波的能量 能流密度
4、熟练掌握波的干涉原理和干涉强弱的条件;
5、理解驻波形成条件和干涉强弱条件;
6、理解多勒效应。
Introduction
message
what method?
letter telephone fax …….,Small paper
In general,your information can be transmitted
by the waves that we will study in this chapter,
New York
斗南村
Small bush:
萨达姆
Old Bush,war
First step,war of information!!!
Second step:
Q Q
The types of waves
(1)Mechanical waves,earthquake waves,sound wave,
water wave,...
(2)Electromagnetic waves,light,sun,communication,...
(3) Gravity waves:
(4) Matter waves,electron,atom,molecule…….
(5)Ogle(秋波 ),a look of great interest,practising by
yourself.
The applications of wave
(1)The transmission of energy,solar energy,laser
weapon,...
(2)The transmission of information,radio,radar
system,communications satellite,B- 超, x-
ray,…,..
In a wave,information and energy move from
one point to another but no material makes
that journey.
In this chapter,for specific examples we shall
refer(涉及 ) to Mechanical Waves.
§ 15-1 Formation & Propagation of a mechanical
Wave 机械波的产生和传播
1,Conditions of mechanical waves:
Two conditions:
(1) There must be a vibrating
center called source of wave:
波源;
(2) There must be medium
propagating(传递 ) wave,
媒质 。
air空气
信号源
Molecules in medium
are vibrating
2.Transverse(横 ) and Longitudinal(纵 ) Waves
The particles of matter
experience an oscillating
displacement from equilibrium.
It is convenient to classify(分类 ) waves in terms of
how the motion of the individual(单个 ) particles of
medium is related to the movement of the waves itself:
?Transverse waves 横波, wave along the string;
?Longitudinal waves纵波, sound wave.
Particle wave
Transverse waves 横波,
The individual particles move up and down at
right angles(直角 ) to the direction in which the
wave propagates:
string
振动 波动
Longitudinal Wave:
A wave motion in which the individual particles
vibrate back and forth along (parallel to,平行 ) the
direction of the wave’s travel is called a longitudinal
or compressible(压缩 ) wave.
For examples,Sound waves as shown in Figure:
condensation rarefaction稀
振动 波动piston
Sound wave
The general waves are treated as the mixed waves
as a combination of longitudinal and transverse
wave,For example:
(1)Water wave.
(2)Earthquake waves.
Other classification:
(1)One-dimensional waves,a wave in a string;
(2)Two-dimensional waves,water wave;
(3)Three-dimensional waves,the flash of light.
3.Velocity wavelength and frequency
Take a sinusoidal(正弦波 ) wave in a string as example.
Crest:波峰 (peak)
Trough:波谷 (valley)
波速,扰动一
秒钟传播的距离为
波速。
V
波长, the distance from crest to adjacent(毗连的 )
crest or trough to adjacent trough.?
x
y
o
The shape of wave
?
V
One second


Water wave
周期 T,the time in
which wave traverses
a distance of a
wavelength.
频率 ?:
T
1??
即一秒钟通过横截面的完整波形的个数 。
波的周期和频率与波源的振动周期及频率相同。
x
y
o
The shape of wave
?
V
One second


波速:由媒质的性质
决定如空气声速不同
于钢轨中的速度
?
GV ? (固体中横波)
?
KV ? (纵波)
G为固体的切变模量,K为介质的体积模量,?为介质的密
度 。
x
y
o
The shape of wave
?
V
One second


波长:描述波的空间
周期性,与波速和频
率满足:
???V

T
V
?
?
x
y
o
The shape of wave
?
V
One second


4,Some concepts for the waves
同相面,A surface marking the points that have same
phase is called the same phase surface.
平面波

线
波面
球面波
波前 ( 波阵面 ), a surface
consisting of the points that
the disturbance has reached
is called the wave-front.
波线,the direction of wave transmission or wave
propagating line.
球面波和平面波:波阵面为球面(平面)。
§ 15-3 The wave equation of plane harmonic
waves
1,The plane harmonic wave
If the motion of the wave
source and the particles in
the medium are harmonic
motion(vibration),this kind
of wave is known harmonic
wave:
)c o s ( ?? ?? tAy
Note,here y is the displacement of particles at wave
source and in the medium.
t(s)
y
o
wave source
平面波

线
波面
球面波
P x
A A A
If the wave front of
the harmonic wave is
plane,this wave is called
as the plane harmonic
wave.
The motion of any point
in this plane is as same as
that of point P.
Choose one wave
line to study the
wave motion such
as x-axis.
y
o
Point P
t
2,The wave equation of plane harmonic wave(介绍)
在波线 ox上任选一
点 P来研究, 已知 o点
的运动方程为
)co s ( ?? ?? tAy o
P点的运动状态是由 o点的运动状态经一段时间传过来
的。
y
o x
Px
Vxt P??
V
t时刻 P点的运动
状态
时刻 o点
的运动状态 V
xt P?
与相同
经 时间达VxP
y
o x
Px
Vxt P??
V
t时刻 P点的运动
状态
时刻 P点
的运动状态 V
xt P?
与相同
经 时间达VxP
])(c o s [)( ?? ???? VxtAVxty PPo =
t时刻 P点的运动
状态
所以 t时刻 P点的运动状态为:
])(c o s [)()( ?? ????? VxtAVxtyty PPoP
因为 P为任意一点,去掉下标 P,x轴上任一点 (坐标 x)
满足:
])(c o s [ ?? ??? VxtAy
])(c o s [ ?? ??? VxtAy
y
o x
V
which is called the wave
equation of plane harmonic
wave.
])(2c o s [
])(2c o s [
?
?
??
?
?
?
???
???
x
tA
x
T
t
Ay
对横波和纵波都适用。
It can be rewritten as
Its importance:
(1) The equation of the plane harmonic wave can be
used to describe many types of waves observed in
nature;
(2)More complicated wave can be synthesized(综合 )
through the superposition of plane harmonic wave.
y
o x
V
])(c o s [ ?? ??? VxtAy
)c o s (
)c o s (
])(c o s [
0
0
P
P
tA
V
x
tA
V
x
tAy
??
?
??
??
??
???
???
3.Explanation of the wave equation
(1)when x is given to be x0,y is the function of time t,
which shows the displacement of particle at point
x0,That is the equation of vibration of the particles
at point x0.
V
x
P
0??? ??
The velocity of particle P is
)s in ( PPP tAdtdyu ??? ????
y
o
Point P
t
(2) When t is given,y is the function of x,which indicates
the shape of wave at time t(摄像法 ),
y
o x
V
时刻0t
)
2
c o s (
)c o s (
])(c o s [)(
0
0
0
t
x
A
t
V
x
A
V
x
tAxy
??
?
?
??
?
??
????
????
???
t0时刻的波形方程
o
V
t
x
y
(3) In general,y is the function of x and t,which
describes the traveling wave:
t+?t
tVd ???
( 4)前面讨论的波沿 x轴正向传播,如波沿 x轴负向传
播,如何写出相应波动方程?将上面所有方程中
的速度作以下变换:
VV ??
])(c o s [ ?? ??? VxtAy
V
y
o x
(5)已知 xP的振动方程,写出波动方程
y
o x
Px
x
Pxx?
V
)c o s ( PP tAy ?? ??
])(c o s [ PPP V xxtAy ?? ????
0?V如,波动方程?
( 6)如何判断波线上一点某一时刻的运动方向(以
横波为例)?
o
V
t
x
y t+?t
a
b
bu
波沿 x轴正向传播
0?au
4,Examples
Example 15-1:如图为一平面简谐波 t=0时刻的波形图。
求:( 1)该波的波动方程;( 2) P处质点的振动方程。
-0.04
o x(m)
y(m)
0.20
P
smV /08.0?
解:由图可知:
5
22
5/08.0
4.004.0
??
?
?
?
??
???
??
T
s
V
TsmV
mmA
( 1)先求 o点的振动方程,)52c o s (04.0 ?? ?? ty o
t=0时刻
??
?
?
?
0)0(
0)0(
o
o
u
y
2
?? ??
)252c o s (04.0 ?? ?? ty o
所以波的波动方程为:
]2)08.0(52c o s [04.0 ?? ??? xty
( 2) P点( )的振动方程mx P 20.0?
]).xt(c o s [.y PP ???????????? ??
])..t(c o s [,????? ?????????? ??
)tc o s (,????????? ??
Example 15-2:一平面简谐横波 沿 x轴正向传播, 波速为
30ms-1,波长 5m,振幅 3cm。 取坐标原点处质点通过平衡
位置向 y轴负向运动为计时起点 。 求,( 1) 该波的波动方
程; ( 2) x=15m处质点在 t=1s时的位移和速度; ( 3) 以
x=2m处的 B点为坐标原点写出波的波动方程 。
解:( 1)原点的振动方程:
???? 12261 ???? TsVT
)12co s (03.0 ?? ?? ty o

??
?
?
?
0)0(
0)0(
o
o
u
y
2
???
]2)30(12c o s [03.0 ?? ??? xty
o x
y
PB
( 2) x=15m处质点振动方程为:
)212c o s (03.0]2)3015(12c o s [03.0 ???? ????? tty P
)/(36.01203.0)1(,0)1(,1 smuyst PP ?? ??????? 时
( 3) x=2m处质点 B振动方程为:
)10312c o s (03.0]2)302(12c o s [03.0 ???? ????? tty B
如果现以 B点为坐标原点,波的波动方程为
]
10
3)
30
(12c os [03.0 ?? ??? xty
Example 15-3:如图为一平面简谐波原点的振动曲线, 沿 x
轴正向以速度 5m/s传播 。 试,( 1) 画出 x=25m处的质点
的振动曲线; ( 2) 画出 t=3s时的波形曲线 。
o t(s)
y(cm)
2 4
2
解:( 1) o点的振动方程为:
)22c o s (02.0 ?? ?? ty o
T
?? 2?
则波动方程为:
]2)5(2c o s [02.0 ?? ??? xty
x=25m处的质点的振动方程
)2c o s (02.0]2)525(2c o s [02.025 ???? ????? tty
?3?
x=25m 处的质
点的振动曲线
(2) t=3s时的波形方程:
)
10
c o s (02.0
)
10
c o s (02.0
]
2
)
5
3(
2
c o s [02.0
?
?
?
?
??
??
??
???
x
x
x
y
y(cm)
o t(s)
-2
1 3 5
sT 4?
y(cm)
o x(m)
-2
5 15 25
m20??
Example 15-4:一平面简谐波 ( 横波 ) 的波动方程为:
求,( 1) 此波的振幅, 波速, 频率和波长; ( 2) 波线上
质点的最大振动速度和最大加速度; ( 3) x1=0.2m和
x2=0.7m处质点的振动的位相差 。
)21 0 0c o s (05.0 xty ?? ??
解:( 1)波动方程:
)]
x
t(c o s [.
)xtc o s (.y
??
????????
?????????
?
??
则:
m
smVHzmA
0.1
/505005.0
?
???
?
?
( 2) 波线上质点的最大振动速度和最大加速度
smAAu /7.152m a x ??? ???
23222
m a x /1093.44 smAAa ???? ???
( 3) x1=0.2m和 x2=0.7m处质点的振动的位相差
????? ?????? )(2)( 1212 xxV xx
Example 15-5:一平面简谐横波 沿 x轴正向传播, 振幅
A=10cm,园频率 ?=7? rad.s-1.当 t=1.0s时, x=10cm处的
a质点的振动状态为, 此时 x=20cm处
的质点的振动状态为 。 设该波的波
长大于 10cm。 求波的波动方程 。
0)(,0 ?? aa dtdyy
0)(,0.5 ?? ab dtdycmy
解:由题意可写出波的波动方程:
)77c o s (10 ??? ??? V xty
质点的振动速度等于:
)77s in (70 ???? ???? V xtdtdy
对于 a点,因为 t=1.0s时,所以有 0)(,0 ?? aa dtdyy
§ 15-4 Energy of Wave 波的能量 波的强度
An important feature of wave motion is the transfer
of energy,The ability of the transfer of energy by wave
is characterized by the concept of energy flow density.
1,Energy of wave
Take the harmonic longitudinal wave as an example
to discuss the energy of wave.
The equation of wave in this case is
)c o s (
V
xtAy ?? ??
?=0
for simplicity
Consider an element in the wave line as shown in Fig.*:
S
x xx ??
S
x? yx ???
no
wave
VmxSV ?????? ?
)s in ( V xtAtyu ??? ??????
The kinetic energy of this
element is
)(s in
2
1
2
1
222
2
V
x
tVA
muW K
???
???
???
Tensile train
)(s in VxtV Axy ????? ??
x
y
?
?
The elastic potential energy is given by
)
V
x
t(s i nVA
V)
x
y
(YW P
?
?
?
?
?
?
?
?
?
???
?
????
??
?? 2VYYV ???
The total energy is
)
V
x
t(s i nVA
WWW PK
??
??
??? ????
???
Remarks:
(1) The kinetic,potential and total energy of the chosen
element vary with the time in the same way:
)Vxt(s i nW,W,W PK ?? ? ????
(2) The total energy of the element is not conservative;
this implies that the energy is transmitted from
one element to another;
Senergy into out energy
o x
V
形变小
(3) The kinetic,potential and total energy have
maximum values when the element is at the
equilibrium position.
Why? 形变大
(4) The density of energy and average energy density(能
量密度和平均能量密度 ):
)Vxt(s i nAVWw ??? ??? ????? per unit volume.
called the of density of energy(energy density),The
average energy density for the time is,
22
2
1 ?? Aw ?
which is depend on the angular frequency ?.
2,Energy flow and intensity of wave(Energy flow
density) 能流和波的强度
S
energy
Experiment,measure the energy
through an area,
Energy transmitted in unit time
through an area is called the
energy flow of this area:
S
One second
3,The amplitudes of plane wave and sphere(球面 ) wave
Assuming that he energy is not
absorbed(吸收 ) by the medium.
平面波

线
波面
球面波
1A
A
2A
1P 2P
21 PP ?
AAA ?? 21
)c o s (
V
xtAy ?? ??
VSA 2221 ??
2A
1A
21 PP ?
CrArA ?? 2211
)c o s (
V
xt
r
Cy ?? ??
r
CA ?
平面波

线
波面
球面波
24 rS ??
4,Same elementary questions:
(1)the intensity of sunlight
reaching the upper levels of
Earth’s atmosphere(大气 ) is
23 /1038.1 mW??
40%of which is reflected back
into the space.
(2)The minimum intensity of sound waves( at a
frequency of 1000Hz) to which the human ear is
sensitive is approximately:
212
0 /10 mWI
?? )()(lo g10
0
10 dBI
I??
The intensity of ordinary speech is on the order of
10-6W/m2,which is 60 dB(Decibels).
(3) The intensity of laser can reach:
218 /10 cmW?
High power laser
Laser weapon in USA
1,Christian Huygens(惠更斯 ),1629~1695
§ 15-5 Huygen’s Principle and its applications
惠更斯原理及其应用
荷兰物理学家、数学家和天文学家;
1656年制成了第一座机械钟;
1678年, 光论)提出了光的波动学
说,建立了著名的惠更斯原理,成
功地解释了光的直线传播;
1665年发现了土星的光环和木星的卫星
(木卫六)。
2,Huygens’s principle
Huygens observed that in a typical wave motion
each particle is set into vibration by a
neighboring particle,This let him to
postulate that every point on a wavefront acts as a
new source sending out secondary wavelets(子波 ).
波源 次级波源
横波的念
球模型
wave wave
Huygens’s principle:
(1) 媒质中波动传到的各点
(波前 )都可以看作是新的次
级波源 (secondary source);
(2)这些次级波源发射与 原
波相同 (速度,波长 ….)
的次级子波 ;
(3)其后任意时刻这些次级
子波的前方 包迹 就是新的
波阵面,
平面波
·
u ? t
波传播方向
球面波
t + ? t
平面波
·
u ? t
波传播方向
?
Problem,Why isn’t there a backward wave if every
particle on the wave front sends out wavelets with
the same amplitude in all directions?
About 1815 Fresnel considered
that the wavelets do not have the
same amplitude in all directions
and the amplitude of Huygens
wavelets varies according to a
factor:
No backward wave
??
?
?
?
w hen
w hen 0
??
???
0
2c os1 ?
3.The diffraction of wave
The bending of waves into the shadow region of
obstacles and related phenomena are called diffraction.
wave
Yes or noobstacle
light
波会绕过障碍物,继续在媒质中传播,且到达沿
直线传播所不能到达的区域,这种波传播偏离直线
的现象叫波的衍射现象或绕射现象,
All types of waves exhibit(展现 ) the diffraction:
??d?,most prominent(显著 )
Here d is the size of obstacle or opening,
?<(<<)d?,less apparent(不明显 )
Kiss me
more!
Silent!
没有不透风的墙 ! 隔墙有耳 !!!
For example:
ms o u n d 1~?
easy to result in diffraction.(1)sound:
(2) water wave:
(3) light,wavelength is small to be difficult to
exhibit the diffraction which we will see in chapter
18.
Huygens’s principle,can provide a
good explanation for the diffraction of
waves.
3.波的反射与折射
反射与折射也是波的特征,当波传播到 两种介质
的分界面时,波的一部分在界面返回,形成反射
波,另一部分进入另一种介质形成折射波。
用作图法求出折射波的传播方向
BC=u1(t2-t1)
AE=u2 (t2 - t1)
折射波传播方向
· ·
媒质 1
媒质 2
·A C
i
1
i2
t1
t2
B
E波的折射定律
2
1
2
1s i ns i n uuii ?
i1--入射角,i2--折射角

(自学 )
§ 15-6 Principle of Superposition of Waves and
Interference(干涉 ) of Wave
1.Principle of Superposition of Waves 波的叠加
原理 (波的独立传播原理 )
There are many occasions
(机会,场合 ) in which two
more wave meet in the same
medium,What’s happen?
Many satellites send the
messages to Earth’s
surface.
总经理 总书记
有两个问题必须回答,
(1)两种或多种波在同一种介质中是怎样传递?
(2)有多钟波在一种介质中传递,介质粒子 怎样 反应?
Principle of Superposition of Waves:
当几列波在同一介质中传播时,
(1)各列波仍保持原有的特性不变,按照原来的方
向继续前进,就象没有遇到其他的波一样 ;
(2)在其相遇区域内,任一点的振动为各个波单独
存在时在该点引起的振动的矢量和,
21 yyy ??
before
meeting
after
2,Interference of waves
In general,the superposition of arbitrary(任意 )
waves may result in the complicated vibration(大街
上的噪声,真让人心烦 ).
相干波,
两个频率相同, 振动
方向相同, 初位相相同
或保持恒定的两个波源
发射的波称为相干波;
这两个波源称为相干波
源 。 播音室
9.11 event
9.11 event
两列相干波相遇,会发生什么?由于两列波引起
质点振动,因各质点的空间位置不同,有的点的合振
动的振幅最大(加强),有的点合振动的振幅最小
(减弱),这样形成一幅稳定的‘图样’,这种现象
叫波的干涉。
播音室
9.11 event
9.11 event


Where?
)tc o s (Ay o ??? ?? ??
)tc o s (Ay o ??? ?? ??
设两相干波源的振动,
Their wave equations are
])(c o s [ 1111 ?? ???
V
rtAy
])(c o s [ 2222 ?? ???
V
rtAy
P1S
2S
1r
2r
P1S
2S
1r
2r
For point P:
]c o s [ 1111 ??? ???
V
rtAy
P
]c o s [ 2222 ??? ???
V
rtAy
P
the resultant vibration is,
)c o s (11 ?? ???? tAyyy PPP
where ??c o sAAAAA
??
?
?
?
? ????
??????
??
??
????? rr
Discusses:
(1) The amplitude of point P is determined by
12 rr ?
?????
12
12 2
rr ?????
Hence,different points have the different amplitudes.
(2) when for point P
.....2,1,022 1212 ???????? kkrr ??????
the amplitude of resultant vibration has maximum value:
21 AAA ??
Constructive 干涉极大
(3) when for point P
.....2,1,0)12(2 1212 ????????? kkrr ??????
the amplitude of resultant vibration has minimum value:
21 AAA ??
destructive 干涉极 小
( 4)特例,21 AA ?
]2,0[ 1AA ?
有些点不动( A=0),有些点振动最强( A=2A1),形
成一定的稳定分布,即 干涉图样 。
( 5)特例,?? ???
?
?
?
??
?
?
???
??
???
...2,1,0
2
)12(
...2,1,0
2
2
12
kk
kk
rr
?
?
?
极大
极小
where ? is the difference of distances( path difference:路
程差) traveled by two wave to point P,That is:
?
?
?
?
?
极小数倍路程差等于半波长的奇
极大数倍路程差等于半波长的偶
点P
Experiments,sound
receiver
1S 2
S
shift(1)
(2)共鸣管实验,
Example 15-6,设 P点到波源 S1和 S2的距离相等,若 P点的
振幅为零,则由 S1和 S2分别发出的两列简谐波在 P点引起
的两个简谐振动应满足什么条件?
解,因为 r1=r2 和 AP=0,S1和
S2分别发出的两列简谐波
在 P点引起的两个简谐振
动应满足,
1S
2S
P
r?
r?
(1)振幅相等 (A1=A2); (2) 振动方向相同 ;
(3)频率相等 ; (4) 初位相差, ??? )12(12 ???? k
§ 15-7 Standing Wave (stationary waves) 驻波
1,Standing waves
(1)A particular type of interference of great
importance in connection with musical instruments(乐
器 ) is interference between two waves traveling in
opposite direction.
V V
(2)物理实验, 共鸣管和绳波 ?the two
waves with the same frequency and
amplitude traveling in opposite directions
in a straight line result in standing waves.
A
B
条件, 适当的距离 AB(试推出 ).
波节,始终静止不动,
波腹,振幅最大,
看不见‘波传播’
驻波,standing wave
波源 反射
Node
波节
Antinode
波腹
A B
NO
2.Conditions of node and antinode,
Simply,consider the
two harmonic waves of
the same frequency and
amplitude along a
straight line:
)(2c o s2 ??? xtAy ??
)(2c o s1 ??? xtAy ??
txAyyy ??
?
? 2c o s2c o s221 ???
The combined wave is( the equation for all particles in
the line)
V<0
V>0
txAyyy ??
?
? 2c o s2c o s221 ???
This equation does not describe a traveling wave because
it is not of the form of wave equation.
For the given x such as x0,it is an equation of vibration
with the amplitude:
?
? xAxA 2c o s2)( ?
That is:
)2c o s ()( ??? ?? txAy
??
?
?
?
?
?
????
???
02c os2
002c os2
xA
xA
重要特点, 满足下式的点,因振幅为零,则始终不动
02c os2 ??? xA
没有信号和能量传递
驻波,standing wave
波源 反射
A B
4)12(02c o s2
?
?? ????? nx
xA 波节
212c o s2
?
?? nx
xA ???? 波腹
4)12(
???? nx
2
?nx ??
成立条件, 对上面用的方程形式 !!!
重要结论,
2
?
2
?
4
?
(1)两相邻波腹或波节之间的距离为 ;
2
?
(2) 一波腹与相邻波节之间的距离为 ;
4
?
(3)两相邻波节之间的点振动位相相同 ;
(4)两相邻波节之间为一段,则两相邻段振动位相相反 ;
普遍成立 !!!
波源
反射
A B
(5)B点为波节,表明反射波与入射波位相相
反,振动相位突变 ?,这等价于半个波长,
这现象叫半波损失,
node
3.波在两媒质界面的反射,
波速媒质的密度波阻 ??
媒质 1
媒质 2
波垂直入射到两界面,
(1)如果波从波阻小的媒质
反射回来,在反射出反射波
与入射波的位相相同 ;
(2)如果波从波阻大的媒质反射回来,在反射出反射
波与入射波的位相相反,有半波损失 ;
(3)如果波从波阻很大的媒质反射回来,反射波的振
幅与入射波的振幅近似相等 (即等能量没有损失 ).
Example 15-7,两列相干波 (振幅相等 ),相向在同一条直
线上传播 (x 轴 ),已知 x=0处为波节,写出波腹点和波节点
的坐标,
o
波节
解,
2
?nx ??
波节
4)12(24
??? ?????? nnx
波腹
Example 15-8:设沿弦线传播的一入射波的表达式为,
波在 x=L处发生反射,反射点为固定端如图,设波在传
播和反射过程中振幅不变,试写出反射波的表达式,
])(2c o s [ ??? ??? xTtAy 入
解,反射波来自反射点
B,B点相当于‘波源’,
波源 B的振动方程,
入射 反射
B
x
y
x
L
L-x
]
2
2c o s [
])(2c o s [
?
?
???
??
?
?
L
T
t
A
L
T
t
Ay B
????
????
半波损失
]
4
)(2c o s [
]
2
)(2c o s [
?
?
??
?
?
?
?
??
?
?
Lx
T
t
A
LxL
T
t
Ay
?????
???
?
??

入射 反射
B
x
y
x
L
L-x
反射波的波动方程为,
§ 15-8 The Doppler Effect 多普勒效应
1.Christian Doppler,1803~1853
奥地利物理学家,主要贡献:多普勒效应。
多普勒效应有十分广泛的应用:( 1)宇宙学
( big bang);( 2)原子光谱分析; (3)航空和航
天,飞行器的导航;( 4)速度测量 ……….,。
雷达
2,The Doppler Effect(特别对:声波和电磁波)
昆明站
61次
62次
昆明站
The relative motion between a source of wave and an
observer or with respect to the medium result in the
Doppler effect,a change in the observed frequency which
is different from that of the wave source.
(1)The velocity V of waves through the medium is
determined by the medium;
Note:
(2) The frequency observed by the observer is the
number of the complete wave segment measured by
the observer in unit time.
波源
观察者
x
sV oV
波源相对于媒质的速度。 观察者相对于媒质的速度。
两者大于零,表沿 x轴正向运动;
两者小于零,表沿 x轴负向运动;
以下分几种情况讨论机械波的多普勒效应:
( 1) 00 ??
os Va n dV
波源
观察者
x
频率 ?,周期 T
这完整的波形 T时间通过
V
?
? V
T
?? 1
等于波源的振动频率!
( 2) 00 ??
os Va n dV
设这完整的波形通过观察者的时间为 T?
TVVTTV o ???? T
VV
VT
o?
?? ?? V VVT 01 ?????
波源 x
波源
观察者
x
V
V
TVo ?
VT
TV ?
oV 感觉
( 3) 00 ??
os Va n dV
波源在 T时间发射一完整的波形:
TV s??? ?? ??? sVVV ??? ??
sVV
V
??
?
波源
观察者
x
V
sV
?
波源
观察者
x
V
??
TVs
波形变
( 4) 00 ??
os Va n dV
波源
观察者
x
V
??
TVs
波形变
sV
oV
运动观察者对因波源运动而 变形 的波的 感觉 的频率为:
??
s
o
VV
VV
?
???
Discusses:
??
s
o
VV
VV
?
???
(1)波源和观察者的速度在公式中,位置不对称; Why?
(2)如果,上式不成立,对应什么情况?VV ?0
(3)如果,,对应什么情况? This means
that the source is moving so fast that it keeps pace with
its own wavefronts;
VVs ? ????
波源
What happens when the speed of the source exceeds
( 超过 ) the speed sound such as Supersonic
speeds(超音速 )?
Shock waves(冲击波)
So
c
V
c
V
?
?
?
c o s
1
1
2
2
?
?
?
V,波源与观察者相对速度的绝对值 ( 电磁
波的传播不需要媒质, 不存在一个绝对坐
标系 ) 。
( 4) 光波的多普勒效应 观察者
source
?
V
c,光速