Phase difference

  •  The phase constant tells what the wave is doing at t = 0, x = 0
  •  The phase difference: difference between the phases of two waves

These two waves are completely out of phase. Where one has a crest, the other has a trough, so they exhibit completely destructive interference.

The path length difference is Dx For completely in phase sources, Df0 = 0 For completely out of phase sources, Df0 = p Completely constructive interference: The waves interfere completely constructively when they are in phase, when Df is an even multiple of pi.
For the special case when the sources are identical, the phase difference equation reduces to Dx = ml. The path length difference is an integer number of wavelengths. Completely destructive interference: The waves interfere completely constructively when they are completely out of phase, when Df is an odd multiple of pi.
Practice problems: 1. Two speakers emit waves with l = 2.0 m. Speaker 2 is 1.0 m in front of Speaker 1. What can be done to cause completely constructive interference between the two waves?

A. Move speaker 1 forward 1.0 m

 

B. Move speaker 1 forward 0.5 m

 

C. Move speaker 1 backward 0.5 m

 

D. Move speaker 1 backward 1.0 m

 

E. Don't move anything, they already have completely constructive interference

 

2. Assume the blue circles in the diagram represent wave crests. Which points are positions of complete constructive interference?

Use this interactive PhET simulation to explore superposition, and find locations of completely constructive interference and completely destructive interference for waves emanating from two sources.

Practice problems

 

1. Two completely out-of-phase radio antennas at x = +300 m and -300 m are emitting 3.0 MHz waves. Is the point (x, y) = (300 m, 800 m) a point of completely constructive interference, completely destructive interference, or neither?

2. Two speakers are in phase, emitting a note of frequency 686 Hz (l = 0.50 m). Speaker A is at the origin, while Speaker B is at (0.0 m, -2.2 m). Where is the first point along the x-axis, to the right of the origin, that you would be able to hear a maximum sound intensity?

Amplitude function

The amplitude function is useful for the special case where the two sources have the same amplitude, and their displacement from each other is known, as well as their initial phase difference.

The black dots in the images above represent two sources of sound waves

  • in phase (left)
  • out of phase (right)

The colors indicate the amplitude of the superposed sound.

The pale green lines show where destructive interference occurs in space.

Thin-film optical coatings

When light strikes a surface from a lower index of refraction to a higher index of refraction, the light wave undergoes a phase shift of pi radians.

 

When light comes from a higher index of refraction to a lower index of refraction, there is no phase shift.

 

You can think of this in terms of the wave on a string encountering a hard or soft boundary. Reflecting from the hard boundary causes a phase shift, where reflecting from the soft boundary does not.

 

In thin-film coatings, the phase-shifted, light reflected from the outer boundary interferes with the light reflected from the inner boundary.

 

If the interfering waves are in phase, they produce a "strong reflection" because they undergo constructive interference. However, the thickness of the film can be adjusted such that the waves undergo destructive interference, by ensuring that the waves are out of phase when they interfere. Thin-film coatings can be used to create anti-reflective coatings for optical lenses.

The path length difference is Dx For completely in phase sources, Df0 = 0 For completely out of phase sources, Df0 = p Completely constructive interference: The waves interfere completely constructively when they are in phase, when Df is an even multiple of pi.
Practice problems: 1. Two speakers emit waves with l = 2.0 m. Speaker 2 is 1.0 m in front of Speaker 1. What can be done to cause completely constructive interference between the two waves? Completely destructive interference: The waves interfere completely constructively when they are completely out of phase, when Df is an odd multiple of pi. For the special case when the sources are identical, the phase difference equation reduces to Dx = ml. The path length difference is an integer number of wavelengths.
2. Two speakers are in phase, emitting a note of frequency 686 Hz (l = 0.50 m). Speaker A is at the origin, while Speaker B is at (0.0 m, -2.2 m). Where is the first point along the x-axis, to the right of the origin, that you would be able to hear a maximum sound intensity?
The path length difference is Dx For completely in phase sources, Df0 = 0 For completely out of phase sources, Df0 = p Completely constructive interference: The waves interfere completely constructively when they are in phase, when Df is an even multiple of pi.
Practice problems: 1. Two speakers emit waves with l = 2.0 m. Speaker 2 is 1.0 m in front of Speaker 1. What can be done to cause completely constructive interference between the two waves? Completely destructive interference: The waves interfere completely constructively when they are completely out of phase, when Df is an odd multiple of pi. For the special case when the sources are identical, the phase difference equation reduces to Dx = ml. The path length difference is an integer number of wavelengths.
2. Two speakers are in phase, emitting a note of frequency 686 Hz (l = 0.50 m). Speaker A is at the origin, while Speaker B is at (0.0 m, -2.2 m). Where is the first point along the x-axis, to the right of the origin, that you would be able to hear a maximum sound intensity?
The path length difference is Dx For completely in phase sources, Df0 = 0 For completely out of phase sources, Df0 = p Completely constructive interference: The waves interfere completely constructively when they are in phase, when Df is an even multiple of pi.
Practice problems: 1. Two speakers emit waves with l = 2.0 m. Speaker 2 is 1.0 m in front of Speaker 1. What can be done to cause completely constructive interference between the two waves? Completely destructive interference: The waves interfere completely constructively when they are completely out of phase, when Df is an odd multiple of pi. For the special case when the sources are identical, the phase difference equation reduces to Dx = ml. The path length difference is an integer number of wavelengths.
2. Two speakers are in phase, emitting a note of frequency 686 Hz (l = 0.50 m). Speaker A is at the origin, while Speaker B is at (0.0 m, -2.2 m). Where is the first point along the x-axis, to the right of the origin, that you would be able to hear a maximum sound intensity?