Revision Notes for Interference of Waves

When light waves from two coherent sources meet at a point, they interact with each other to yield interference.

Coherent sources are those two or more sources of light that produce light waves of the same color (monochromatic), the same wavelength, and a constant phase difference.

When light waves from two coherent sources meet at a point in phase, they produce constructive interference. If they have the same amplitudes and each equal to a, then the amplitude of the resultant wave is 2a. The condition is that the path difference between the two waves must be an integral multiple of its wavelength.

When light waves from two coherent sources meet at a point out of phase, they produce destructive interference. If they have the same amplitudes and each equal to a, then the amplitude of the results waves is 0. The condition is that the path difference between the two waves must be an integral and a half multiple of its wavelength.

Let pd denote the path difference then,

  • Condition of constructive interference, pd = m$\lambda$
  • Condition of destructive interference, pd = (m + 1/2) $\lambda$

where m is an integer and $\lambda$ is the wavelength.

In Young's double-slit experiment, we observe that two coherent light sources produce waves that meet at the screen to produce an interference pattern. To memorize the variety of formulas, we only need to understand two concepts:

1) Concept discussed above for the condition of constructive interference and condition of destructive interference.

2) Geometrical relationship present in the experimental setup.

Let us proceed to a geometrical relationship of this experimental setup by looking at the given figure cited from the University Physics textbook.


It is time to understand the various variables we will be utilizing throughout the experiment,

  • d: slit width. It is the distance between the two slits, measured from their centers.
  • $\theta$: angle made by the rays at the point of contact on the screen with the line passing perpendicular through the center of d.
  • y: distance of constructive interference or destructive interference from the center.
  • R: distance between sources (slits) and the screen.

From geometry, we can observe that the path difference, pd, is pd = d $\rm \sin \theta$.

Recalling the above formulae, we get,

  • For constructive interference, d $\rm \sin \theta$ = m $\rm \lambda$ -- (1)
  • For destructive interference, d $\rm \sin \theta$ = (m + 1/2) $\rm \lambda$ -- (2)

Similarly, from geometry, we get, y$\rm _m$ = R $\rm \tan \theta$. For very small angle $\rm \theta$, we have,

$\rm \tan \theta \approx \sin \theta$. Therefore, y$\rm _m$ = R $\rm \sin \theta$ -- (3)

By combining equations (1) and (3), and equations (2) and (3), we can get the expressions for y.

Then, finding the difference between two consecutive y gives us the value of fringe width.

Hence, we cover the entire Young's Double Slit Experiment and Interference of Waves in one blog post.

The sad part, you will have to memorize the derivations. However, they are just theoretical measures to validate a theory and people spend years generating the derivation, firsthand.