10.2.2 Necessary Conditions for Interference

Many people use the terms superposition and interference interchangeably. But strictly speaking, they are not exactly the same. Superposition occurs whenever two waves (of the same kind) meet or overlap. But there are 2 scenarios when two waves superpose but do not interfere with each other:

1) The waves are incoherent.

Two waves are said to be coherent if they maintain a constant phase difference between them. Conversely, if the phase relationship between them changes randomly and continuously, they are said to be incoherent.

(Take note. Coherence does not require the phase difference to be zero. It only requires that the phase difference to be unchanging.)

Let’s consider two waves with intensity I. If they superpose in-phase, the resultant intensity will be 4I due to C.I. If they superpose in antiphase, the resultant intensity would be 0 due to D.I.

Now, what if the phase difference between these two waves is changing randomly from time to time? This can happen if the waves transmit in short bursts. So the phase difference between the waves is only maintained for a short period of time before the next burst changes the phase difference again.

This will cause resultant intensity to continuously change between 0 and 4I, as the interference continuously change between D.I. and C.I. If this random phase changing occurs very frequently, then all we can detect is the time-averaged intensity of $\displaystyle \frac{{4\mathrm{I}+0}}{2}=2\mathrm{I}$ . Now think about it. Isn’t $\mathrm{I}+\mathrm{I}=2\mathrm{I}$ the same outcome as if there is no interference? For this reason, we say that incoherent waves do not interfere.

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2) The waves are polarised perpendicularly to each other.

Now let’s say wave X, with amplitude A_x and intensity I_x, is polarised horizontally. Wave Y, with amplitude A_y and intensity I_y, is polarised vertically. (Obviously, this section is applicable only to transverse waves since only transverse waves can be polarised.)

When they superpose, they neither add or subtract each other’s oscillation, since they have zero component in each other’s direction of oscillation. In fact, it can be shown mathematically (beyond H2 syllabus) that the intensity of the resultant wave is always ${{\mathrm{I}}_{x}}+{{\mathrm{I}}_{y}}$ regardless of the phase difference between X and Y. Since the resultant intensity is always ${{\mathrm{I}}_{x}}+{{\mathrm{I}}_{y}}$ , two perpendicularly polarised waves are as good as not interfering with each other.