10 Interference of Waves, Standing Waves

1. 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.

2. 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）

4. Interference of Waves The combination of separate waves in the same region of space 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） (Example 10.1 Interference of two cosine waves)

5. And we have: For the case 1, waves are said to be everywherein phase (φ= 0) and to interfere constructively. That is, the crests of the 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）

6. And for the case 2, the waves are said to interfere destructively. That is, the crest of one wave coincides with the trough of the second and their displacements cancel at every 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） In general, when the phase constant has a value between 0 and π, the resultant wave has an amplitude whose value is somewhere between 0 and 2A. （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）

10. interfere constructively interfere destructively

11. If two waves are in phase everywhere, we get: interfere constructively interfere destructively path length path length difference

12. Interference of Waves Two waves with parallel vibration directions, identical frequency and constant phase difference superpose in the space. The result of the superposition is that the vibrations of certain points in medium are constructive while the vibrations of other points are destructive. （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） coherent waves （相干波）

13. (Example 10.2) Two speakers placed 3.00 m apart are driven by the same oscillator. A listener is originally at point O, that is located 8.00 m from the centre of the line connecting the two speakers. The listener then moves to point P, which is a perpendicular distance 0.350 m from O before reaching the first cancellation of waves, resulting in a minimum in sound intensity. What is the frequency of the oscillator?

14. Standing Waves （驻波） Standing waves are formed from the superposition of two waves that have the same frequency, amplitude, and wave length 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）

15. wave function of a standing wave The simple harmonic motion of every particle has an angular frequency ωand a position-dependentamplitude. （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）

16. The amplitude of the simple harmonic motion at x is: Maximum amplitude occurs where: antinodes （波腹） Zero amplitude occurs where: nodes （波节） Adjacent nodes and antinodes are both separated by a half wavelength.

17. Standing Waves in String P3388 Fig.13-19 The reflection of a traveling wave pulse at the fixed end of a stretched string. The reflected pulse is inverted, but its shape remains the same.

18. The inversion of a wave at a rigid end are named as a πphase 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, πphase 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）

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

20. 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 the fundamental frequency. The higher normal frequencies are 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）

21. (Example 10.3) Two waves traveling in opposite directions produces a standing wave. The individual wave functions are where x and y are measured in centimeters. (a) Find the amplitude of the simple harmonic motion of the element of the medium located atx= 2.3 cm. (b) Find the position of the nodes and antinodes if one end of the string is at x = 0. (c) What is the maximum value of the position in the simple harmonic motion of a point located at an anti-node?

22. (Example 10.4) The wave function of one wave propagating along x axis can be written as: 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) the position of the nodes and antinodes.

24. Beats: Interference in Time At given location x0, two waves disturbs as:

25. 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.

26. The frequency measured bythe observerreads: (observer moving toward source) (source moving toward observer) (Both the source and the observer are in motion)

27. (Example 10.5) A submarine (sub A) travels through water at a speed of 8.00 m/s, emitting a sonar wave at a frequency of 1400 Hz. The speed of sound in the water is 1533 m/s. A 2nd submarine (sub B) is located such that both submarines are traveling directly toward one another. The second submarine B is moving at 9.00 m/s. (a) What frequency is detected by an observer riding on sub B as the subs approach each other? (b) The subs barely miss each other and pass. What frequency is detected by an observer riding on sub B as the subs recede from each other?

28. Homework 7 P347 35, 46, 49 P368 35, 39, 60 28

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