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  5. Interference of waves

16.5 Interference of waves  (Page 4/12)

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Figures a and b each show a wave with amplitude A and wavelength lambda. They are in phase with one another. Figure a is labeled y1 parentheses x, t parentheses equal to A sine parentheses kx minus omega t parentheses. Figure b is labeled y2 parentheses x, t parentheses equal to A sine parentheses kx minus omega t parentheses. Figure c shows a wave that is in phase with the other two. It has amplitude 2A and wavelength lambda. It is labeled y parentheses x, t parentheses equal to y1 plus y2 equal to 2A sine parentheses kx minus omega t parentheses.
Constructive interference of two identical waves produces a wave with twice the amplitude, but the same wavelength.
Figures a and b each show a wave with amplitude A and wavelength lambda. They are out of phase with one another by an angle pi. Figure a is labeled y1 parentheses x, t parentheses equal to A sine parentheses kx minus omega t plus pi parentheses. Figure b is labeled y2 parentheses x, t parentheses equal to A sine parentheses kx minus omega t parentheses. Figure c shows the absence of any wave. It is labeled y parentheses x, t parentheses equal to y1 plus y2 equal to 0.
Destructive interference of two identical waves, one with a phase shift of 180 ° ( π rad ) , produces zero amplitude, or complete cancellation.

When linear waves interfere, the resultant wave is just the algebraic sum of the individual waves as stated in the principle of superposition. [link] shows two waves (red and blue) and the resultant wave (black). The resultant wave is the algebraic sum of the two individual waves.

Figure shows three waves. Two of these, blue and red have y values varying from -10 to plus 10 and the same wavelength. They are slightly out of phase. The third, which is black, has the same wavelength but a larger amplitude. Another figure shows a blown up portion of this graph. At x approximately equal to 0.74, the y values of the red and blue waves are y1 = 8 and y2 = 10 respectively. The y value of the black wave is y1 + y2 = 18. At x equal to 1, the y values of the red and blue waves are both 9.5. The y value of the black wave is y1 + y2 = 19.
When two linear waves in the same medium interfere, the height of resulting wave is the sum of the heights of the individual waves, taken point by point. This plot shows two waves (red and blue) added together, along with the resulting wave (black). These graphs represent the height of the wave at each point. The waves may be any linear wave, including ripples on a pond, disturbances on a string, sound, or electromagnetic waves.

The superposition of most waves produces a combination of constructive and destructive interference, and can vary from place to place and time to time. Sound from a stereo, for example, can be loud in one spot and quiet in another. Varying loudness means the sound waves add partially constructively and partially destructively at different locations. A stereo has at least two speakers creating sound waves, and waves can reflect from walls. All these waves interfere, and the resulting wave is the superposition of the waves.

We have shown several examples of the superposition of waves that are similar. [link] illustrates an example of the superposition of two dissimilar waves. Here again, the disturbances add, producing a resultant wave.

Figure shows three waves. Wave 1 has larger wavelength and amplitude compared to wave 2. The third wave, labeled resultant wave is irregularly shaped.
Superposition of nonidentical waves exhibits both constructive and destructive interference.

At times, when two or more mechanical waves interfere, the pattern produced by the resulting wave can be rich in complexity, some without any readily discernable patterns. For example, plotting the sound wave of your favorite music can look quite complex and is the superposition of the individual sound waves from many instruments; it is the complexity that makes the music interesting and worth listening to. At other times, waves can interfere and produce interesting phenomena, which are complex in their appearance and yet beautiful in simplicity of the physical principle of superposition, which formed the resulting wave. One example is the phenomenon known as standing waves, produced by two identical waves moving in different directions. We will look more closely at this phenomenon in the next section.

Try this simulation to make waves with a dripping faucet, audio speaker, or laser! Add a second source or a pair of slits to create an interference pattern. You can observe one source or two sources. Using two sources, you can observe the interference patterns that result from varying the frequencies and the amplitudes of the sources.

Superposition of sinusoidal waves that differ by a phase shift

Many examples in physics consist of two sinusoidal waves that are identical in amplitude, wave number, and angular frequency, but differ by a phase shift    :
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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