When two waves superimpose one over the other, their sum produces a different wave in terms of amplitude. This new wave has a different mathematical representation and results in a different shape.
Types of interference
The attributes of a wave are the amplitude ‘A’, the frequency ‘f’, and the phase ‘θ’. The amplitude indicates the excursion of the wave, the frequency is the number of oscillations per unit time (usually a second), and the phase is the position of the wave in space relative to a reference point.
To clarify the ideas and concepts behind amplitude, frequency, and phase, let’s examine an example where we have the sinusoidal functions \(\sin (x)\) and \(\sin{(x + \frac{\pi}{2})}\). Both have the same shape because the amplitude and frequency are the same. But let’s have a look at their graphs:
Figure 1. Sin (x) (red) and sin (x + π/2) (black).
The phase is different, which causes a backward shift of the second wave (in black). We notice the shift because the point where the function passes from zero isn’t the origin anymore. This is very important because it changes the way the waves overlap.
We study phase differences with the help of a reference point. In the graph above, the reference point is taken as zero. The phase angle of the red wave sin (x) is zero and can be shown as sin (x+0). The starting point of the wave, therefore, is the same as the reference point with no phase difference.
For the black wave, however, the phase difference is positive (+ \(\frac{\pi}{2}\)), which means that the origin of the wave (its starting point) is said to be ‘before’. On the graph, it can be seen to the left of the reference point. When the phase difference is negative, the origin of the wave is said to be ‘after’ the reference point. On the graph, it would, therefore, be to the right of the reference point.
Constructive and destructive interference
If we add up two identical waves, the resulting amplitude doubles:
\[\sin (x) + \sin (x) = 2 \sin (x)\]
While mathematically obvious, have a look at what it means visually, with two sinusoids one above the other, as in figure 2.
Figure 2. Two sin (x) functions translated above and below the x-axis.
The points that have the same x coordinate also have the same amplitude. Applying the sum causes the wave to stretch. Where the amplitude was 1, it is now 2, and where it was -1, it is now -2. This is called constructive interference. An example from everyday life is two speakers playing the same track. The volume of the music perceived is maximum when the waves produced by the speakers are in phase, interfering constructively.
When the waves have different phases, the result of the superposition changes, especially if they are in phase opposition or counter-phase when every point is added to one of the opposite value, e.g., 1 + (-1), as in the graph below.
Figure 3: Sin (x - π/2) (blue), sin (x + π/2) (red). Their sum is a straight line.
The sum of these waves is zero due to destructive interference. Notice how, in this case, the phase causes the same result as the negative sign before the function.
\(\sin (x - \frac{\pi}{2}) + sin (x + \frac{\pi}{2}) = 0\)
\(\sin (x - \frac{\pi}{2}) - \sin (x - \frac{\pi}{2}) = 0\)
We defined the two ends of the line. In between, there are all the combinations of the two waves. The phase of the resulting wave is shifted to somewhere between the phases of the interfering waves, depending on their amplitudes, and the value of its amplitude will be between zero and twice the amplitude of the interfering waves.
Figure 4: Sin (x - π/2) (blue), sin (x) (red), and their sum (green).
Interference patterns
We talked about the interference between one-dimensional waves. The same phenomenon occurs when the propagation happens along two or more dimensions. In this case, two waves interfere and create an interference pattern.
When two stones are thrown into a lake, with one being thrown from a slightly different spot close to the one from which the first stone was thrown, a bi-dimensional wave forms on the surface of the water. In this scenario, the surface of the water gets corrugated but still exhibits a regularity, hence the name of this type of interference.
Figure 5. In the case of a bi-dimensional wave, the position of the source (the point from which the stone is thrown) influences the pattern as well. Notice how the waves are much more corrugated near the sources and almost unaffected moving away from them.
In the image, two circular waves propagate towards each other at an angle of \(\frac{\pi}{2}\). The wave fronts interfere almost orthogonally, giving the water a grid-shaped surface. The lines of the grid are points of destructive interference, while between them, there are points of constructive interference.
Interference - key takeaways
- When we say that two waves are superimposed or interfere, we are talking about the same phenomenon.
- The interference can be constructive or destructive, but generally speaking, the interference will be a combination of the two.
- The phenomenon of interference occurs in any case, with the propagation on one or more dimensions. In the latter case, we speak of interference patterns.
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