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Chapter 21: Superposition and InterferencePowerPoint Presentation

Chapter 21: Superposition and Interference

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### 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.

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

Wave – periodic oscillations in space and in time of something

It is moving as a whole with some velocity

- 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

- 1) The sources must be coherent

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)

Interference in one dimension

- The phase:
- The phase difference is
- Constructive interference: ΔΦ = m(2π)
- Perfect destructive interference ΔΦ = (2m + 1 )π

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.

Rays 2 travels an additional distance of 2t equal-amplitude sound waves

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 …..

n(file)<n(glass) phase change at both equal-amplitude sound waves

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.

Standing wave equal-amplitude sound waves

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

Standing wave on a String equal-amplitude sound waves

- 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

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

Standing Sound Waves 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

- Open-open or closed-closed tube

m =1,2,3……

Open-closed tube 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

Problem 21.52 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

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?

When the air column length 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 producedL 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 mL, in a table:

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