Interference of waves

When two or more waves meet while propagating in the same medium, they interfere. In this page we will analyze how is the resultant wave caused by the interference of two harmonic waves of equal amplitude, frequency and wavelength, but with a phase difference between them. In order to do so, we use the principle of superposition. It states that:

The wave resulting from the superposition of two or more waves is the sum of the individual waves.

When these waves propagate in a material medium the net displacement at any point of the medium is given by the sum of the displacements caused by each wave at the same point.

Mathematically:

That is, the wave function of the resultant wave is the sum of the individual waves.

Let us consider two harmonic waves of equal amplitude, frequency and wavelength, with a phase difference between them. Their wave functions are respectively given by:

where A is the amplitude, ω the frequency, k the wavenumber and Φ the phase constant. Graphically:

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The resultant wave is their sum:

To simplify the expression above we can use the following trigonometric identity:

where:

Now, substituting into the resultant wave function:

As it can be seen in the equation above, the resultant wave is also a harmonic wave which has the same frequency and wavelength. Its amplitude A’ (in red) is:


This amplitude A’ depends on the phase difference between the interfering waves. Its minimum and maximum values will respectively occur when:

When the amplitude A’ of the resultant wave is zero, the waves interfere in destructive interference. On the other hand, when the amplitude A’ has its maximum value (2A), the waves interfere in constructive interference.

Any other phase difference between the two waves will result in a wave of amplitude A’ between 0 and 2A.

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In the figure below both the constructive (upper figure) and destructive (lower figure) interference between the two waves (blue and green) are shown in red:

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