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**Waves at Media Boundaries**

Section 9.2

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**Key Terms Media Boundary Free-end Reflection Fixed-end Reflection**

Transmission Standing Wave Node Antinode Fundamental Frequency/First Harmonic Harmonics Overtone

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Media Boundaries Wave speed depends on the properties of the medium through which the wave is travelling. All media have boundaries. The location where two media meet.

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Free-End Reflections If a wave travels from a more dense medium to a less dense medium, it will travel more quickly in the more dense medium. Wave moving towards the boundary will be reflected with the same orientation and amplitude

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Fixed-End Reflection As a wave moves towards a fixed boundary, it will reflect. Reflected pulse has the same shape as the incoming pulse, but its orientation is inverted.

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Amplitude When a wave encounters a boundary that is not strictly free-end or fixed-end, the wave will split in two. One wave is reflected Energy “bounces back”. The other is transmitted. Energy passes into new medium. Amplitude of the two waves may not be equal, but the sum of the amplitudes will be equal to that of the original wave.

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Media Boundaries Not all difference in media boundaries are as dramatic as fixed-end or free-end. Water Air If a wave travels from a medium in which the speed is faster (more dense) to a medium in which the speed is slower (less dense), the wave particles can move more freely Energy is transferred into new medium Reflected wave has same orientation

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**Media Boundaries The opposite is also true.**

Air Water If a wave travels from a medium in which the speed is slower (less dense) to a medium in which the speed is faster (more dense), the wave particles cannot move as freely Energy is transferred into new medium Reflected wave has inverted orientation

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Standing Waves Suppose a series of waves is sent down a string that is fixed at both ends. At a certain frequency, reflected waves will superimpose on the stream of incoming waves to produce waves that appear stationary The locations in which the particles of the medium do not move are nodes. The locations in which the particles of the medium move with the greatest speed are antinodes.

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Standing Wave

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Standing Waves Waves interfere according to principle of superposition. Waves are moving continuously At the antinodes, the amplitudes of the troughs and crests are double that of the original wave. At the nodes, the amplitudes are the same but one is a crest and the other is a trough. Interference pattern appears to be stationary because it is produced by otherwise identical waves travelling in opposite directions.

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**Standing Waves Between Two Fixed Ends**

Standing waves can be predicted mathematically. Consider a string with two fixed ends Standing wave with nodes at both ends. The shortest length of the string, L, is equal to one half of the wavelength. The frequency of the wave that produces this simplest standing wave is called the fundamental frequency First harmonic All standing waves to follow require frequencies that are whole-number multiples of the fundamental frequency. Additional standing wave frequencies are known as the nth harmonic of the fundamental frequency

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**Number of Nodes Between Ends**

Symbol Number of Nodes Between Ends Diagram Harmonic (n) Overtone f0 First Fundamental f1 1 Second f2 2 Third f 3 Fourth

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**Harmonics and Overtones**

Harmonics consist of the fundamental frequency of a musical sound as well as the frequencies that are the whole-number multiples of the first harmonic. When a string vibrates with more than one frequency, the resulting sounds are called overtones. Similar to harmonics, however the first overtone is equal to the second harmonic.

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**Calculations with Standing Waves**

The length of the medium is equal to the number of the harmonic times half the standing wave’s length. For a media with a combination of fixed and free ends (node at one end and antinode at the other), the equation is:

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Summary

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Homework Page 426 Questions 1-5

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Example: pulse on a string speed of pulse = wave speed = v

Example: pulse on a string speed of pulse = wave speed = v

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