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# Lect02 - Lecture 2 Interference S1 d S2 Interference of...

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Lecture 2, p.1 Lecture 2: Interference S 1 S 2 d λ Interference of sound waves Two-Slit Interference Phasors

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Lecture 2, p.2 Adding Sine Waves with Different Phases Suppose we have two sinusoidal waves with the same A 1 , ϖ , and k : y 1 = A 1 cos(kx - ϖ t) and y 2 = A 1 cos(kx - ϖ t + φ ) One starts at phase φ after the other: ( 29 1 1 cos cos 2 cos cos 2 2 A A β α β α α β - + + = 1 2 y y ( / 2 φ ( / 2 kx t ϖ φ - + 1 2 cos( / 2) cos( / 2) y A kx t φ ϖ φ = - + Use this trig identity: Spatial dependence of 2 waves at t = 0: Resultant wave: Amplitude Oscillation y = y 1 +y 2 φ
Lecture 2, p.3 ACT 1: Noise-cancelling Headphones 1. What must be the phase of the signal from the speaker relative to the external noise? a. 0 b. 90˚ c. π d. -180˚ e. 2 π 2. What must be the intensity I s of the signal from the speaker relative to the external noise I n ? a. I s = I n b. I s < I n c. I s > I n Noise-canceling headphones work using interference. A microphone on the earpiece monitors the instantaneous amplitude of the external sound wave, and a speaker on the inside of the earpiece produces a sound wave to cancel it.

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Lecture 2, p.4
Lecture 2, p.5 Interference Exercise The two waves at this point are “out of phase”. Their phase difference φ depends on the path difference δ ≡ r 2 - r 1 . The relative phase of two waves also depends on the relative distances to the sources: δ φ Ι 0 λ/4 λ/2 λ Path difference Phase difference A = 2A 1 cos( φ /2) r 2 r 1 2 360 δ δ φ π λ λ = = ° Each fraction of a wavelength of path difference gives that fraction of 360º (or 2 π ) of phase difference:

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Lecture 2, p.6 Amplitude vs. Intensity for 2 Interfering Waves Q: What is the spatial average intensity? A: I av = 4I 1 *0.5 = 2I 1 Plot 2A 1 cos( cos( φ /2) /2) and and 4A 1 2 cos cos 2 ( φ /2) as a function of φ . 0λ2λ 3λ4λ5λ 3λ4λ5λ φ φ 02π 4π6π8π10π 4π6π8π10π δ Constructive Interference Destructive Interference 2A 1 4A 1 2 Does this make sense?
Example: Path-Length Dependent Phase Each speaker alone produces intensity I 1 = 1W/m 2 at the listener, and f = 300 Hz.

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