Interference of Waves

In these revision notes for Interference of Waves, we cover the following key points:

• What is interference of waves?
• Which are the conditions to produce interference?
• How many types of interference are there in waves?
• What does the principle of waves' superposition say?
• What happens to the amplitude and frequency when interference occurs?
• How does the path difference affect the interference of waves?

Interference of Waves Revision Notes

By definition, interference is the combination of two or more waveforms to form a resultant wave in which the displacement is either reinforced or cancelled.

In other words, interference is the process of the overlapping of two or more coherent waves.

Interference is produced as a result of the principle of superposition, a key principle in waves theory, which says:

The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.

The principle of superposition may be applied to waves whenever two (or more) waves are travelling through the same medium at the same time.

There are two conditions the waves must meet to produce interference patterns.

1. Waves must be of the same kind. Obviously, a light wave cannot interfere with a sound wave as they are of different nature.
2. Waves must have the same frequency in order to produce a standing interference.

It is wavelength (and therefore, frequency) the quantity that determines the interference pattern. This means only like waves with the same frequency (wavelength) can produce interference in certain conditions.

Basically, there are two types of interference. They are:

a) Constructive Interference. In this kind of interference, waves enforce each other as they overlap at the same phase. As a result, we obtain a resultant wave whose amplitude is the arithmetic sum of the amplitudes of each single wave, i.e.

b) Destructive Interference. When two like waves have a phase shift of half a cycle, a destructive interference is produced. As a result, we obtain a resultant wave whose amplitude is the arithmetic difference of the amplitudes of each single wave, i.e.

The resultant wave will be in phase with the wave which has the greater amplitude.

In the special case when the amplitudes of the constituent values are equal, the resultant amplitude is zero and therefore, the waves cancel each other. As a result, no resultant wave will exist anymore.

When two coherent waves pass through one or more narrows gaps comparable to the amplitude, they (albeit initially in phase) may behave differently (may deviate) after leaving the gap because they often deviate their original path. Due to this deviation, waves will travel different paths after leaving the gap (d₁ ≠ d₂). If we place a screen at the position these waves meet, we will observe on the screen one of the three options mentioned above (constructive, destructive or no interference). The type of behavior depends on the path difference of the two waves.

a)If the path difference d₂ - d₁ is a whole multiple of wavelength, i.e.

where N = 0, 1, 2, ., the waves will be in phase when they fall on the screen. As a result, a constructive interference is produced on that point as the waves enforce each other.

b)On the other hand, when the path difference of the two waves is half a multiple of wavelength, i.e. when

where N = 0, 1, 2, ., there will be a destructive interference produced at that point of the screen, as waves try to cancel out each other.

If the waves 1 and 2 were light rays, a pattern with dark and bright regions is formed on the screen, where the maximum brightness is obtained on the central maximum and fringes around it, that become dimmer when moving away from the central maximum.