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  5. Interference of waves

16.5 Interference of waves  (Page 5/12)

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y 1 ( x , t ) = A sin ( k x − ω t + ϕ ) , y 2 ( x , t ) = A sin ( k x − ω t ) .

When these two waves exist in the same medium, the resultant wave resulting from the superposition of the two individual waves is the sum of the two individual waves:

y R ( x , t ) = y 1 ( x , t ) + y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) + A sin ( k x − ω t ) .

The resultant wave can be better understood by using the trigonometric identity:

sin u + sin v = 2 sin ( u + v 2 ) cos ( u − v 2 ) ,

where u = k x − ω t + ϕ and v = k x − ω t . The resulting wave becomes

y R ( x , t ) = y 1 ( x , t ) + y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) + A sin ( k x − ω t ) = 2 A sin ( ( k x − ω t + ϕ ) + ( k x − ω t ) 2 ) cos ( ( k x − ω t + ϕ ) − ( k x − ω t ) 2 ) = 2 A sin ( k x − ω t + ϕ 2 ) cos ( ϕ 2 ) .

This equation is usually written as

y R ( x , t ) = [ 2 A cos ( ϕ 2 ) ] sin ( k x − ω t + ϕ 2 ) .

The resultant wave has the same wave number and angular frequency, an amplitude of A R = [ 2 A cos ( ϕ 2 ) ] , and a phase shift equal to half the original phase shift. Examples of waves that differ only in a phase shift are shown in [link] . The red and blue waves each have the same amplitude, wave number, and angular frequency, and differ only in a phase shift. They therefore have the same period, wavelength, and frequency. The green wave is the result of the superposition of the two waves. When the two waves have a phase difference of zero, the waves are in phase, and the resultant wave has the same wave number and angular frequency, and an amplitude equal to twice the individual amplitudes (part (a)). This is constructive interference. If the phase difference is 180 ° , the waves interfere in destructive interference (part (c)). The resultant wave has an amplitude of zero. Any other phase difference results in a wave with the same wave number and angular frequency as the two incident waves but with a phase shift of ϕ / 2 and an amplitude equal to 2 A cos ( ϕ / 2 ) . Examples are shown in parts (b) and (d).

radians. The crests of the blue wave coincide with the troughs of the red wave and vice versa. The green wave is absent. Figure d is labeled delta phi equal to 3 pi by 2 radians. Here, the red and blue waves each have an amplitude of 10 m and the green wave has an amplitude of 15 m. It has the same wavelength as the other two waves. The crests of the green wave are formed where the crests of the red and blue waves intersect each other.
Superposition of two waves with identical amplitudes, wavelengths, and frequency, but that differ in a phase shift. The red wave is defined by the wave function y 1 ( x , t ) = A sin ( k x − ω t ) and the blue wave is defined by the wave function y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) . The black line shows the result of adding the two waves. The phase difference between the two waves are (a) 0.00 rad , (b) π / 2 rad, (c) π rad, and (d) 3 π / 2 rad .

Summary

  • Superposition is the combination of two waves at the same location.
  • Constructive interference occurs from the superposition of two identical waves that are in phase.
  • Destructive interference occurs from the superposition of two identical waves that are 180 ° ( π radians ) out of phase.
  • The wave that results from the superposition of two sine waves that differ only by a phase shift is a wave with an amplitude that depends on the value of the phase difference.

Conceptual questions

An incident sinusoidal wave is sent along a string that is fixed to the wall with a wave speed of v . The wave reflects off the end of the string. Describe the reflected wave.

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A string of a length of 2.00 m with a linear mass density of μ = 0.006 kg/m is attached to the end of a 2.00-m-long string with a linear mass density of μ = 0.012 kg/m . The free end of the higher-density string is fixed to the wall, and a student holds the free end of the low-density string, keeping the tension constant in both strings. The student sends a pulse down the string. Describe what happens at the interface between the two strings.

At the interface, the incident pulse produces a reflected pulse and a transmitted pulse. The reflected pulse would be out of phase with respect to the incident pulse, and would move at the same propagation speed as the incident pulse, but would move in the opposite direction. The transmitted pulse would travel in the same direction as the incident pulse, but at half the speed. The transmitted pulse would be in phase with the incident pulse. Both the reflected pulse and the transmitted pulse would have amplitudes less than the amplitude of the incident pulse.

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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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