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**Net vibration at point P:**

Where:

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Coherent waves Two coherent waves must have parallel vibration directions identical frequency constant phase difference

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phase difference Interference constructively = Interference destructively

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**Path length difference**

Interference constructively = Interference destructively

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**§2-6 standing waves Traveling waves The energy of traveling waves**

What are standing waves? Standing waves are special cases of wave interference. The superposition of two coherent waves with identical amplitude, frequency and common vibration axis but opposite propagating direction is standing wave.

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y1 and y2 are identical x y y1 The propagating directions are the same. y is Not standing wave!

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x y y1 y2 y is Not standing wave!

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u x y u standing wave

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**Characteristics of standing wave**

antinode x y Every element in the medium vibrates steadily. There is no traveling of waveform. node antinode 2) The amplitude of standing wave is A(x). 3) The positions of maximum amplitude are called antinodes. 4) The positions of zero amplitude are called nodes.

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**Mathematical expression of standing wave**

Suppose there are two coherent waves with identical amplitude, frequency and common vibration axis but opposite propagating directions. u x y y1 y2 u Set the initial phase is 0.

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**The standing wave is the superposition of these two coherent waves.**

using interference of wave, we get: Where:

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**The initial phase of the superposed wave:**

take

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**The mathematical expression of standing wave is:**

standing wave equation

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**Amplitude of standing wave, A(x)**

Simple harmonic vibration Standing wave equation does not satisfy: Therefore, standing wave is different from traveling wave.

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**Every element of the medium in standing wave is**

doing simple harmonic vibration at identical frequency ω. 2) But the amplitude of every element might be different at various position. Since every element of the medium vibrates steadily, the disturbance does not propagate in standing wave.

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Amplitude when antinode Node: when node

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x y antinode node Adjacent antinodes are separated by a distance of λ/2. Adjacent nodes are also separated by λ/2. The distance between adjacent antinode and node is λ/4. We can measure the distance between two adjacent nodes to determine the wavelength of λ.

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Phase of standing wave If time is varying, for all the elements in the medium their phases are identical, that is ωt.

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The displacement, velocity and phase of the elements locating in the two sides of one node are opposite. But the displacement, velocity and phase of the elements locating between two nodes are with the same sign.

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**§2-7 half-wavelength loss**

If the incident wave reflects on a certain interface, the phase of the reflected wave is opposite to that of the incident wave. Such phenomena is called half-wavelength loss. Half-wavelength loss An electromagnetic wave undergoes a phase change of 180o on reflection from a medium of higher reflection index than the one in which it is traveling.

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**How can we get the reflected wave with half-wavelength loss?**

Wave with opposite vibration But the reflected wave with half-wavelength loss is not this opposite wave, but this wave with a phase difference of π.

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**The conditions which will cause half-wavelength loss:**

1) The reflection point is the fixed end of the medium. 2) When the wave propagates from a wave thinner medium to a wave denser medium, the reflected wave has half-wavelength loss. Wave denser medium: the medium with a larger reflection index n. Wave thinner medium: the medium with a smaller reflection index n.

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denser thinner glass, n=1.52 Air, n=1 Water, n=1.33 Water, n=1.33 thinner denser The reflected wave has half-wavelength loss The reflected wave has no half-wavelength loss Only reflection may have half-wavelength loss possibly. Refraction never has such phenomena.

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Example 2-3-1 The wave equation 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: 1) the wave equation of the reflected wave. 2) the wave equation of the superposition of these two waves. 3)the position of the nodes and antinodes.

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Solution: 1) Because the reflection point is one node, the reflected wave has half-wavelength loss. The wave equation of the reflected wave can be expressed as: The incident wave equation:

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