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Chapter 21: Superposition and standing wave. 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 standing wave

Chapter 21: Superposition and standing wave

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 1 page 635 stop to think 21 2 page 641 stop to think 21 3 page 645
Stop to think 21.1 page 635Stop to think 21.2 page 641Stop to think 21.3 page 645
  • Example 21.1 page 638
  • Example 21.2 page 641
  • Example 21.5 page 643
  • Example 21.6 page 645
standing wave
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

standing wave on a string
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

slide5

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
Standing Sound Waves
  • Open-open or closed-closed tube

m =1,2,3……

problem 21 52
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?

slide9

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