10.1.3 Path Difference

Imagine you make a speaker play a monotone. You walk around the speaker and hear the same boring monotone everywhere. Now you find another speaker and make it play the exact same monotone. Is the sound twice as loud now as before? You are not quite sure, but as you walk around, you distinctly hear the tone alternates between louds and softs. There are locations where having two speakers actually produce a softer sound, or even silence.

What on earth is happening? Let’s think through the situation. Let’s assume that the sound waves from speaker A and speaker B are in phase with each other at the time when they leave the speakers. However, these two sound waves take two different paths, and therefore travel different distances, before they arrive at your position P. The so-called path difference, $\displaystyle \delta =|\text{AP}-\text{BP}|$, results in a phase difference between the two waves when they arrive at P. This means that we can have constructive interference at some locations but destructive interference at other locations.

Let’s use a few examples.

Example 1

Suppose A, B and P are positioned such that AP and BP correspond to 1λ and 3λ respectively. This means that the wave from B travels a longer distance of $|3\lambda -\lambda |=2\lambda$ to arrive at P. So two waves are arriving at P, but the one from B is lagging the one from A by two complete cycles, which makes the two waves exactly aligned (upon arrival at P). So as far as P is concerned, there is a superposition of two waves which are in-phase with each other. Constructive interference will occur at P, resulting in a loud sound being heard at P.

Example 2

How about the above scenario? Wave A travels $|2\lambda -0.5\lambda |=1.5\lambda$ longer than wave B before arriving at P. So wave A will arrive 1.5 cycles behind wave B, which makes them line up exactly opposite to each other at P. As far as P is concerned, there is a superposition of two waves which are in antiphase with each other. Destructive interference between the two waves results in silence at P.

Example 3

One more example. Wave A travels $|2.25\lambda -1.75\lambda |=0.5\lambda$ longer than wave B before arriving at P. The path difference of 0.5l introduces a phase difference of π radians between the waves arriving at P. Destructive interference between the two waves results in silence at P.

Conclusion

In a nutshell, the path difference δ determines the phase difference of the waves arriving at the destination. It is thus the quantity to evaluate if we want to know whether the interference at a location is constructive interference (C.I.) or destructive interference (D.I.). For two wave that started out in-phase at the sources,

C.I. occurs at locations where $\displaystyle \delta =n\lambda$, $n=0,1,2...$

D.I. occurs at locations where $\displaystyle \delta =(n+\frac{1}{2})\lambda$  , $n=0,1,2...$

Video Explanation

How Path Difference Determines the Outcome of Superposition

Concept Test

2011