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R. Field 11/14/2013 University of Florida PHY 2053Page 1 Traveling Waves: Superposition Wave Superposition: Add the two waves together (superposition of wave 1 and wave 2) as follows:

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R. Field 11/14/2013 University of Florida PHY 2053Page 2 Traveling Waves: Superposition Wave Superposition: Wave 1 Wave 2 Superposition! The intensity of the new wave is proportional to A 12 squared!

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R. Field 11/14/2013 University of Florida PHY 2053Page 3 Traveling Waves: Superposition Wave Superposition: Consider two waves with the same amplitude, frequency, and wavelength but with an overall phase difference of = . New intensity! Superposition! New amplitude! sinA+sinB = 2sin[(A+B)/2]cos[(A-B)/2]

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R. Field 11/14/2013 University of Florida PHY 2053Page 4 Traveling Waves: Interference Maximal Constructive Interference: Consider two waves with the same amplitude, frequency, and wavelength but with an overall phase difference of = 2 n, where n = 0, ±1, ±2, … Max Constructive! Maximal Destructive Interference: Consider two waves with the same amplitude, frequency, and wavelength but with an overall phase difference of = +2 n, where n = 0, ±1, ±2, … Max Destructive!

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R. Field 11/14/2013 University of Florida PHY 2053Page 5 Example Problem: Superposition Two traveling pressure waves (wave A and wave B) have the same frequency and wavelength. The waves are superimposed upon each other. The amplitude of the resulting wave (wave C) is 13 kPa. If the amplitude of wave A is 12 kPa and the phase difference between wave B and wave A is B – A = 90 o, what is the amplitude of wave B and the magnitude of the phase difference between wave A and wave C, respectively? Answer: 5 kPa, o

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R. Field 11/14/2013 University of Florida PHY 2053Page 6 Traveling Waves: Superposition Wave 1 distance d 1 Lateral Phase Shift: Consider two waves with the same amplitude, frequency, and wavelength that are in phase at x = 0. Wave 2 distance d 2 Max Constructive Max Destructive

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R. Field 11/14/2013 University of Florida PHY 2053Page 7 Examples: Superposition d = max constructive d = max destructive d = /4

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R. Field 11/14/2013 University of Florida PHY 2053Page 8 Example Problem: Superposition The figure shows four isotropic point sources of sound that are uniformly spaced on the x-axis. The sources emit sound at the same wavelength and the same amplitude A, and they emit in phase. A point P is shown on the x-axis. Assume that as the sound waves travel to the point P, the decrease in their amplitude is negligible. What is the amplitude of the net wave at P if d = /4? Answer: Zero x Max Destructive

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R. Field 11/14/2013 University of Florida PHY 2053Page 9 Example Problem: Superposition Sound with a 40 cm wavelength travels rightward from a source and through a tube that consists of a straight portion and a half-circle as shown in the figure. Part of the sound wave travels through the half-circle and then rejoins the rest of the wave, which goes directly through the straight portion. This rejoining results in interference. What is the smallest radius r that results in an intensity minimum at the detector? Answer: 17.5 cm Point A Point B At point A the waves have the same amplitude, wavelength, and frequency and are in phase. Wave 1 travels a distance d 1 = 2r to reach the point B, while wave 2 travels a distance d 2 = r to reach the point B. Max Destructive

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