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10 Interference of Waves, Standing Waves

When two or more waves pass through the same region of space at the same time, the actual displacement is the vector sum of the separate displacement.

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The Principle of Superposition

Traveling Wave: a wave move through space without interacting with any other waves or objects.

The principle of superposition: When two or more waves

pass through the same region of space at the same time, it

is found that for many waves the actual displacement is

the vector sum of the separate of the separate

displacement.

（波的叠加原理：如果两列或更多的波同时传播到在空间的同一区域中，则介质中实际位移等于每列波引起的位移之和）

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Interference of Waves

The combination of separate waves in the same region ofspace to produce a resultant wave is called interference.

（由分立的波在传播的共同区域叠加产生新的波动行为的现象称为干涉）

1 2cos( ); cos( )y A kx t y A kx t

1 2 2 cos cos( )2 2

A kxy y y t

(Example 10.1 Interference of two cosine waves)

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1 2 2 cos sin( )2 2

A kxy y y t

And we have:

2 , for 2 ,2 cos

0, for (2 1) .2

A kA

k

For the case 1, waves are said to be everywhere in phase (φ = 0) and to interfere constructively. That is, the crests ofthe individual waves occur at the same positions.（对于第一种情况，我们称两列波处处同相位并且干涉增强 (相长干涉 )。也就是说，每列波的波峰出现在相同的位置）

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In general, when the phase constant has a value between0 and π, the resultant wave has an amplitude whose valueis somewhere between 0 and 2A.

And for the case 2, the waves are said to interfere destructively. That is, the crest of one wave coincides withthe trough of the second and their displacements cancel atevery point. (out of phase)

（第二种情况称为相消干涉。也就是说一列波的波峰和另一列波的波谷位置重叠，在空间中每一点位移相互抵消）

（一般情况下，相位介于 0 和 π 之间，叠加波的振幅介于 0 和 2A 之间）

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8

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1 1 1 1, cos( )y S t A t

2 2 2 2, cos( )y S t A t

1 1 1 1 1

2, cos( )y r t A t r

2 2 2 2 2

2, cos( )y r t A t r

1 2

cos( )

y y y

A t

2 21 2 1 2

2 1 2 1

2 cos

2( )

A A A A A

r r

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2 21 2 1 2 2 1 2 1

1 21 1 2 2

1 21 1 2 2

cos( )

22 cos ; ( )

sin( 2 ) sin( 2 )tan

cos( 2 ) cos( 2 )

y A t

A A A A A r r

r rA A

r rA A

1 2

1 2

2 ,

(2 1) ,

k A A A

k A A A

interfere constructively

interfere destructively

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If two waves are in phase everywhere, we get:

2 1 1 2

2 1 1 2

,

(2 1) , 2

r r k A A A

r r k A A A

interfere constructively

interfere destructivelypath length

2

;2 4 2

path length difference

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Two waves with parallel vibration directions, identical

frequency and constant phase difference superpose in th

e space. The result of the superposition is that the vibrati

ons of certain points in medium are constructive while th

e vibrations of other points are destructive.

Interference of Waves

coherent waves （相干波）

（两列振动方向和频率相同并且具有恒定相位差的波在空间中重叠，波的叠加的结果是重叠区域的某些质点的振动增强了，而另一些质点的振动减弱了）

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(Example 10.2) Two speakers placed 3.00 m apart are drivenby the same oscillator. A listener is originally at point O, thatis located 8.00 m from the centre of the line connecting thetwo speakers. The listener then moves to point P, which is aperpendicular distance 0.350 m from O before reaching thefirst cancellation of waves, resulting in a minimum in soundintensity. What is the frequency of the oscillator?

0.13 0.26 1.3 m; m; kHzr f

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Standing Waves（驻波）

Standing waves are formed from the superposition of twowaves that have the same frequency, amplitude, and wavelength but are traveling in opposite directions.

（两列具有相同频率、振幅和波长的波相向传播，相遇后叠加形成驻波）

v

v

1 cos( )y A t kx

2 cos( )y A t kx

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1 2

2 cos cos

y y y

y A kx t

wave function of a standing wave

The simple harmonic motion of every particle has anangular frequency ω and a position-dependent amplitude. （每一个质点作频率为ω 简谐振动，振幅则依赖于质点的位置）

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The amplitude of the simple harmonic motion at x is:

2 cosA kx

, or 1

2xkx m m

Maximum amplitude occurs where:

antinodes（波腹）

Zero amplitude occurs where:

1( ) , or

2

1( )

2 x mkx m nodes

（波节）

Adjacent nodes and antinodes are both separated bya half wavelength.

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Standing Waves in String

The reflection of a traveling wave pulse at the fixed end of astretched string. The reflected pulse is inverted, but its shaperemains the same.

P3388 Fig.13-19

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The inversion of a wave at a rigid end are named as a

π phase shift. A traveling wave is reflected at the free

end of a stretched string will not be inverted

In general, when a traveling wave in a medium strikes

the boundary of a more dense medium, the reflected wave

is inverted, otherwise, the wave will not be inverted. That

is, π phase shift occurs while the wave enters a dense

medium from a less dense medium.

（一般而言，当波在介质中传播时，从波疏介质进入波密介质时，波形被反转；反之，当波由波密介质进入波疏介质时，波形不发生反转。也就是说，当波由波疏介质进入波密介质中时，反射波出现半波损失）

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One can thus establish standing waves on a string bycombining incoming and reflected waves from a rigid end.And It is easily to find the wavelength of standing wavesin a string have to satisfy the following condition.

:

1

2n L

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Each possible wavelength specify a normal mode.（每一个可能的波长确定一种振动模式，称为简正模式）

The lowest frequency, corresponding to n = 1, is called thefundamental frequency. The higher normal frequenciesare called harmonics.

（ n = 1 对应于振动模式频率最低的情况，该频率称为基频。其他较大的简正频率称为谐频）

2, 1,2,

Ln

n

, 2

v nv Tf v

L

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(Example 10.3) Two waves traveling in opposite directionsproduces a standing wave. The individual wave functions are

1 24.0 sin 3.0 2.0 , 4.0 sin 3.0 2.0 cm cmy x t y x t

where x and y are measured in centimeters. (a) Find theamplitude of the simple harmonic motion of the elementof the medium located at x = 2.3 cm. (b) Find the positionof the nodes and antinodes if one end of the string is atx = 0. (c) What is the maximum value of the position in thesimple harmonic motion of a point located at an anti-node?

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The wave function of one wave propagating along x axis can be written as:

1 cos .y A t kx

The reflection occurs at x = 0 and the reflection point is one

node. Find a) the wave function of the reflected wave, b) the

wave function of the superposition of these two waves, c) th

e position of the nodes and antinodes.

(Example 10.4)

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2

Solution: (1) From the problem, phase shift occurs at 0.

That is the wave is inverted after reflection, and the reflected

wave reads: cos( ) .

(2) From the principle of superposition, the re

x

y A t kx

1 2

sultant wave is

2 sin( )sin( )

(3) From the wave equation, it is easily to find the antinodes

1locate at ( ) , and the nodes locate at

2 2

, 0,12

a

n

y y y A kx t

x k

kx k

, 2 .

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Beats: Interference in Time

1 1 2 2cos 2 , cos2y A f t y A f t

1 2 1 22 cos 2 cos 22 2

f f f fy A t t

At given location x0, two waves disturbs as:

1 2 1 2beat frequency: f f f f

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The Doppler Effect（多普勒效应）

When the wave source and the observer move relatively, the measured frequency by the observer is different from the true frequency (the frequency of the wave source). This phenomena is called Doppler effect.

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The frequency measured by the observer reads:

'' O Ov v v v v

f fv

(observer moving toward source)

'' S

v vf f

v v

(source moving toward observer)

''

'O

S

v v vf f

v v

(Both the source and the observer are in motion)

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(Example 10.5) A submarine (sub A) travels through water ata speed of 8.00 m/s, emitting a sonar wave at a frequency of1400 Hz. The speed of sound in the water is 1533 m/s. A 2ndsubmarine (sub B) is located such that both submarines aretraveling directly toward one another. The second submarineB is moving at 9.00 m/s.

(a) What frequency is detected by an observer riding on subB as the subs approach each other?(b) The subs barely miss each other and pass. What frequencyis detected by an observer riding on sub B as the subs recedefrom each other?

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Homework 7

P34735, 46, 49

P36835, 39, 60

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The End