Coherent and incoherent addition of waves

Two sources in a ripple tank

Narrated video. Transcript:

Take two needles, S₁ and S₂, and move them up and down together in a shallow trough of water. Each needle sends out circular ripples of the same wavelength.

Where the ripples overlap, a pattern appears. There are lines where the water heaves strongly, and between them, lines where the water is almost still.

The needles move in step, so the phase difference between them never changes. Two sources whose phase difference stays constant in time are called coherent sources.

Take a point P on the perpendicular bisector of S₁S₂, so S₁P = S₂P. The two waves travel equal distances, so they arrive in phase: crest meets crest.

If the displacement due to S₁ at P is y₁ = a cos ωt, the displacement due to S₂ is y₂ = a cos ωt too. The resultant is y = 2a cos ωt: twice the amplitude.

Intensity is proportional to the square of the amplitude. So the intensity at P is four times the intensity I₀ produced by one source alone: I = 4I₀. This is constructive interference.

The same happens at every point where the path difference S₂P − S₁P is a whole number of wavelengths: nλ, with n = 0, 1, 2, 3 and so on.

Now take a point Q where S₂Q − S₁Q = 2.5λ. The waves arrive exactly out of phase: crest meets trough. y₁ = a cos ωt and y₂ = −a cos ωt, so they cancel. The intensity is zero. This is destructive interference.

Destructive interference happens wherever the path difference is (n + ½)λ: half a wavelength, one and a half, two and a half, and so on. These points form the still lines in the water.

Example: the wavelength is 2 cm. At a point 30 cm from S₁ and 34 cm from S₂, the path difference is 4 cm, or 2λ: the water heaves strongly. At 30 cm and 35 cm, it is 5 cm, or 2.5λ: the water is still.

Coherent and incoherent addition

Narrated video. Transcript:

In general, the two waves reach a point with some phase difference φ. Then y₁ = a cos ωt and y₂ = a cos(ωt + φ).

Add them with the cos A + cos B formula. The resultant is y = 2a cos(φ/2) cos(ωt + φ/2). It is still a simple harmonic oscillation, with amplitude 2a cos(φ/2).

So the intensity is I = 4I₀ cos²(φ/2). For φ = 0, ±2π, ±4π, I = 4I₀. For φ = ±π, ±3π, I = 0. In between, I lies between 0 and 4I₀.

A path difference of one wavelength means a phase difference of 2π. So the phase difference is φ = (2π/λ) × path difference.

Example: where φ = π/2, I = 4I₀ cos² 45° = 4I₀ × ½ = 2I₀. Where φ = 2π/3, I = 4I₀ cos² 60° = 4I₀ × ¼ = I₀.

Now suppose the phase difference does not stay fixed. Let S₂ jump to a new random phase again and again, very rapidly. The bright and dark lines jump about all the time.

Our eyes and instruments see only the time average. The average value of cos²(φ/2), when φ changes at random, is ½. So the average intensity is 4I₀ × ½ = 2I₀, the same everywhere.

Such sources are called incoherent. For incoherent sources the intensities simply add: I₀ + I₀ = 2I₀. No interference pattern is seen.

This is why two separate sodium lamps lighting a wall never show fringes. Light from each lamp changes its phase randomly in less than a billionth of a second, and the two lamps do this independently.

To see interference with light, we need coherent sources. The trick, first used by Thomas Young, is to make two sources out of one. That is the next lesson.

Try it: a ripple tank

Interactive simulation. Things to try:

• Drag P around the tank. On a calm line the path difference S₂P − S₁P is (n + ½)λ; on a rough line it is nλ.
• Watch the small graph: on a bright line y₁ and y₂ rise and fall together; on a calm line they are opposite and cancel.
• Switch to “Intensity” and tick “Mark lines…” to see the lines of maxima (green) and minima (dashed).
• Change the extra phase of S₂: the pattern shifts but stays. Then untick “Coherent”: the pattern washes out to 2I₀.

Intensity at a point

Two coherent sources each produce intensity I₀ on their own. At a point P the waves from them have a phase difference of 2π/3. What is the intensity at P?

1. 0
2. I₀
3. 2I₀
4. 3I₀

Two sodium lamps

Two identical sodium lamps light the same white wall, and each alone gives an intensity I₀. What do you see on the wall?

1. Bright and dark fringes, from 0 to 4I₀
2. Uniform illumination of 4I₀
3. Uniform illumination of 2I₀
4. Uniform illumination of I₀

Summary

• Resultant of two equal waves: y = 2a cos(φ/2) cos(ωt + φ/2)
• Coherent sources: I = 4I₀ cos²(φ/2)
• Phase difference from path difference: φ = (2π/λ) × path difference
• Incoherent sources: I = I₀ + I₀ = 2I₀

• Coherent sources keep a constant phase difference. They produce a steady interference pattern.
• Constructive interference (I = 4I₀) where the path difference is nλ; destructive (I = 0) where it is (n + ½)λ.
• For incoherent sources the phase difference changes randomly; the average of cos²(φ/2) is ½, so the intensities just add.
• Two independent light sources are always incoherent, so they never show interference fringes.