第13章 机械波

§13.1 机械波的产生和传播

§13.2 平面简谐波

§13.3 波的能量和能流

§13.4 波的干涉

§13.5 驻波

§13.6 多普勒效应

Interference of wave§13.4 波的干涉

两列波在空间相遇，某些点的振动始

终加强，另一些点的振动始终减弱。

干涉现象

产生干涉现象的波 相干波

CAI

Coherent condition相干条件

⑴频率相同 ⑵振动方向相互平行 ⑶相位差恒定

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Pr2

r1

S1

S2

)cos( 11010 ϕω += tAy

)cos( 22020 ϕω += tAy

)cos( 1111 krtAy −+= ϕω

)cos( 2222 krtAy −+= ϕω

21 yyy += )cos( ϕω += tA

ϕ∆cos2 2122

21

2 AAAAA ++=

( )2121 rrk −−−= ϕϕϕ∆退出返回

Pr2

r1

S1

S2

ϕ∆cos2 2122

21

2 AAAAA ++=

ϕ∆cos2 2121 IIIII ++=

( )2121 rrk −−−= ϕϕϕ∆

πϕ∆ n2=

21 AAA +=…±±= ，，， 210n

干涉加强 (干涉相长 ) 2121 2 IIIII ++=

πϕ∆ )12( += n

21 AAA −=干涉减弱 (干涉相消)

2121 2 IIIII −+=

…±±= ，，， 210n

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( )2121 rrk −−−= ϕϕϕ∆

12 ϕϕ =

( ) ( )12122 rrrrk −=−=λπϕ∆

Pr2

r1

S1

S2

波程差

…±±==− ，，， 2 1 0 12 nnrr λ波程差

212121 2 IIIIIAAA ++=+= 干涉加强

( ) …±±=+=− ，，， 210 2

1212 nnrr λ波程差

212121 2 IIIIIAAA −+=−= 干涉减弱

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Geometrically, how can the minima and maxima be located?

For maxima:

L 2 1 0 ,,,nnL ±±== λ∆

θ∆ sin12 dLLL =−=

Here, S1 and S2 are the sources, and P is either a maximum or a minimum.

The difference in path lengths can be written in terms of the angle , assuming that R>>d .

θ

dn λθ =sin

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dn λθ =sin L 2 1 0 ,,,n ±±=

0 0 =→= θn 中央极大

For minima:

( ) 2

12 λ∆ += nL

L 2 1 0 ,,,n ±±=θ∆ sin12 dLLL =−=

dn λθ )

21(sin +=

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Two coherent sources are 3.0 cm apart and make harmonic ripples of the same frequency. Consider the ripples along a straight line parallel to the line that connects the sources and 40.0 cm away from line . If the distance between the central maximum at and the next maximum on is 8.0 cm， what is the wavelength of the ripples?

1L

2L

2L

1L 1L

The angle of the next maximum is given by

LD

=θtancm040

cm08..

=

= 0.20o311.=θ

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dnλθ =sin

For the next maximum：

= d sinθλ

= (3.0 cm) sin 11.3°

= 0.59 cm

在均匀介质中沿直线传播的波在遇到另外一种介质时，

会发生反射和折射现象。

vZ ρ= 介质的特性阻抗

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undulatory thinner mediummaller svρ 波疏介质

larger vρ undulatory denser medium波密介质

Incident wave

Reflected wave

π反射时相位突变

phase jump

half-wave loss 波疏介质 波密介质半波损失

CAI退出返回

Standing waves§13.5 驻波

具有这种波形特征的波

驻波 standing waves

两列振幅相同的相干波在同一直线

上沿相反方向传播时叠加

一种特殊的干涉现象 CAI

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没有波形的推进，参与波动的各

个质元处于稳定的振动状态。

各振动质元的振幅各不相同，但

却保持不变，有些点振幅始终最

大，有些点振幅始终为零。

波腹 antinode 波节 node

CAI

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Two harmonic waves of the same amplitude and wavelengthtravel in opposite directions along a stretched string

Their interference with each other produces a standing wave

CAI退出返回

2At = 0

y

0 x

0t = T/ 8 x

x0t = T/2

0 xt = T/4

波节波腹 λ /4-λ /4

x0

2A

-2A

λ/2

振动范围

xt = 3T/8 0

CAI

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