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Chapter 21: Superposition and Interference

Chapter 21: Superposition and Interference. The principle of Superposition: When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

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Chapter 21: Superposition and Interference

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  1. Chapter 21: Superposition and Interference The principle of Superposition: When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

  2. Stop to think 21.4 page 650Stop to think 21.5 page 655Stop to think 21.6 page 658 • Example 21.8 page 649 • Example 21.11 page 656 • Example 21.12 page 658

  3. Waves Wave – periodic oscillations in space and in time of something It is moving as a whole with some velocity

  4. Conditions for Interference • To observe interference the following two conditions must be met: • 1) The sources must be coherent • - They must maintain a constant phase with respect to each other • 2) The sources should be monochromatic • - Monochromatic means they have a single (the same) wavelength

  5. Anti-phase Perfect destructive interference Two identical sources produce perfect destructive interference when the path- Length difference is a half-integer number of wavelength ( or say odd number of Half-wavelength)

  6. Interference in one dimension • The phase: • The phase difference is • Constructive interference: ΔΦ = m(2π) • Perfect destructive interference ΔΦ = (2m + 1 )π

  7. These two loudspeakers are in phase. They emit equal-amplitude sound waves with a wavelength of 1.0m. At the point indicated, is the interference maximum Constructive, perfect destructive, or something in between? The path-length difference is Δr = λ. For identical sources, interference is constructive.

  8. Rays 2 travels an additional distance of 2t before the waves recombine. Suppose the Index of refraction of film is n (n>1) For constructive interference 2nt+λ/2 = mλ m = 1, 2,3 ….. For destructive interference 2nt+λ/2 =(m+1/2)λ m = 0,1, 2,3 …..

  9. n(file)<n(glass) phase change at both surface of the film For constructive interference 2nt+λ/2+ λ/2 = mλ m = 1, 2,3 ….. →2nt = mλ m = 0,1,…… For destructive interference 2nt =(m+1/2)λ m = 0,1, 2,3 ….. Antireflection coatings use the interference of light waves to nearly eliminate reflections from glass surfaces.

  10. Standing wave Ex: there are two waves: The resultant wave function is: Notice, in this function, does not contain a function of (kx±ωt). So it is not an expression for a traveling wave

  11. Standing wave on a String • A standing wave can exist on the string only if its wavelength is one of the values given by • F1=V/2L fundamental frequency. • The higher-frequency standing waves are called harmonics, ex. m = 2, second harmonics m=3 third harmonics Node Antinode

  12. Stop to think: A standing wave on a string vibrates as shown at the figure. Suppose the tension is quadrupled while the frequency and the length of the string are held constant. which standing-wave pattern is produced Answer: a

  13. Standing Sound Waves • Open-open or closed-closed tube m =1,2,3……

  14. Open-closed tube

  15. Problem 21.52 A 1.0m tall vertical tube is filled with 20 C water. A tuning fork vibrating at 580 Hz is held just over the top of the tube as the water is slowly drained from the bottom. At what water heights, measured from the bottom of the tube, will there be a standing wave in the tube above the water?

  16. When the air column length L is the proper length for a 580 Hz standing wave, a standing wave resonance will be created and the sound will be loud. From Equation 21.18, the standing wave frequencies of an open-closed tube are fm = m(v/4L), where v is the speed of sound in air and m is an odd integer: m = 1, 3, 5, … The frequency is fixed at 580 Hz, but as the length L changes, 580 Hz standing waves will occur for different values of m. The length that causes the mth standing wave mode to be at 580 Hz is • We can place the values of L, and corresponding values of h =1 mL, in a table:

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