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Interference of Sound Waves

Interference of Sound Waves. Principle of Superposition. 2 Waves In The Same Medium : The observed displacement y(x,t) is the algebraic sum of the individual displacements: y(x,t) = y 1 (x,t) + y 2 (x,t). (for a “linear medium”). What’s Special about Harmonic Waves?.

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Interference of Sound Waves

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  1. Interference of Sound Waves

  2. Principle of Superposition 2 Waves In The Same Medium: The observed displacement y(x,t) is the algebraic sum of the individual displacements: y(x,t) =y1(x,t) + y2(x,t) (for a “linear medium”)

  3. What’s Special about Harmonic Waves? 2 waves, of the same amplitude, same angular frequency and wave number (and therefore same wavelength) traveling in the same direction in a medium but are out of phase: Trig: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] Result: Resultant amplitude

  4. Assume  is positive: For what phase difference  is the total amplitude 2A? For what phase difference  is the total amplitude A? For what phase difference  is the total amplitude 0? For what phase difference  is the total amplitude A/2?

  5. wave 1 wave 2 Resultant: Sine wave, same f, different A, intermediate f

  6. Interference 2 waves, of the same frequency; out of phase. Eg. y1=A0sin (kx - wt) y2=A0sin (kx - wt +f) Then yR=ARsin(kx-wt+f /2), and the resultant amplitude is AR=2A0cos(½f ). Identical waves which travel different distances will arrive out of phase and will interfere, so that the resultant amplitude varies with location.

  7. Example: Two sources, in phase; waves arrive at P by paths of different lengths: P S1 x1 detector x2 At P: S2

  8. Phase difference : = Define x to be the path difference Then (using trig), at detector: kx terms don’t cancel!

  9. Example 0.35m 8.0m A pair of speakers is separated by 3.0m and driven by the same oscillator. The listener walks perpendicular from a point on the centerline 8m away to a distance of 0.35m before reaching the first minimum in sound intenstiy. What is the frequency of the oscillator? 3m

  10. Solution

  11. y Quiz You are located at position y, where you can hear a loud sound - the first maximum in intensity from two speakers.The speakers are then connected ‘out of phase’ (difference of π). What will you hear?A) no change – same loud soundB) no soundC) something between ‘no sound’ and ‘loud sound’

  12. Intensity I I= Power per unit area Units: W / m2 (the area is measured perpendicular to the wave velocity) Intensity is proportional to (resultant amplitude)2 , since I=P/Area and power is proportional to Amp2 Two sources, each with amplitude Ao, intensity Io , phase difference f:

  13. Notes: • Maximum IR is 4 x IO • Maxima when f = 0, 2π, 4π, 6π , … • Minima (zero intensity) when f = π, 3π, 5π , … x = 0, ± λ, ± 2λ,…  x = ± λ/2, ± 3λ/2, ± 5λ/2,… Note: The sources are in phase !!!

  14. Example 6 m The detector x I think I hear something! 2 speakers, same intensity, in phase; f = 170 Hz (so l= 2.0 m when speed of sound is 340 m/s) At the position x=9 m, find the intensity in terms of the intensity of a single speaker

  15. Solution

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