PH 201-4A spring 2007

[Pages:26]PH 201-4A spring 2007

Interference Phenomena Lecture 28

Chapter 17 (Cutnell & Johnson, Physics 7th edition)

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The Superposition of Waves

A superposition principle: when two or more waves arrive at any given point simultaneously, the resultant instantaneous deformation is the sum of the individual instantaneous deformations. ? The waves do not interact, they have no effect on one another ? Each wave propagates as though the others were not present ? The net displacement of the medium is the vector sum of the individual displacements The superposition principle is very well satisfied for waves of low amplitude. For waves of very large amplitude the superposition principle fails, because the first wave alters the properties of the medium and therefore affects the behavior of a second wave propagating on the same string, air, etc. (medium)

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Consider two waves with the following properties:

? propagate in the same direction ? same frequency ? same amplitude ? in phase (wave crests and troughs coincide)

The superposition yields a wave of twice the amplitude of the individual waves.

Reinforcement of one wave by another is called constructive interference.

y1 = Asin(Kx ? t) y2 = Asin(Kx ? t) y = y1 + y2 = 2Asin(Kx ? t)

3

Consider two waves with the following properties:

? propagate in the same direction ? same frequency ? same amplitude ? out of phase (wave crests of one wave match the wave troughs of the other) The superposition yields a wave of zero amplitude. A cancellation of one wave by another is called destructive interference.

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Consider two waves with the following properties: ? propagate in the same direction ? same frequency ? amplitudes are not equal ? out of phase The superposition yields a wave, amplitude does not equal zero Destructive interference ? cancellation is not total

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Interference

Like waves of any kind, sound waves interfere when they meet. 2 loudspeakers emit sound waves in phase.

The phase difference at particular points in space is related to the path difference (PD) by a simple relationship:

y = AcosKx; = Kx; = kx=2/ * (PD)

Constructive interference ? when =2n; or 2n=2/ * (PD). or PD=n is a multiple of the wavelength

= 2/ (AC ? BC) = 2/(4 - 3) = 2 PD = n n = 0, 1, 2, 3

= 2/ (3 ? 2.5) = out of phase

Destructive interference ? when the path difference is an odd number of half-wavelengths

PD = m(/2) m = 1, 3, 5, ...

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At an open-air concert on a hot day (Tc = 25?C, Vs = 346.5 m/s), a person sits at a location that 7.0 m and 9.1 m respectively from speakers at each side of the stage. A musician, warming up, plays a single 494 Hz tone. What does the spectator hear?

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Constructive and Destructive Interference of Sound Waves

Assume that two loudspeakers in the figure are vibrating out of phase instead of in phase. (see example 2 from the text) The speed of sound is 343 m/s. What is the smallest frequency that will produce destructive interference at point C?

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