Wave interference due to a path difference

When two harmonic waves travelling along different directions meet at a certain point, the resultant interference pattern will depend on the difference in distance traveled by the two waves, known as the path difference.

Let us consider two wave sources S1 and S2 such as the ones shown in the figure below. It will be assumed that they are producing harmonic waves with identical amplitudes and frequencies (and therefore identical wavelengths). We will also assume that the phase difference between them is zero.

Both waves meet at point P and the interference between them will be constructive, destructive or something in between depending on the distances traveled by both, which we will call respectively x1 and x2 .

In order to find the wave function of the wave resulting from the interference of the two waves 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.

The wave functions of the two waves are given by:

where A is the amplitude, k the wavenumber and ω the angular frequency.

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When both waves arrive at point P, their wave functions are respectively given by:

And the resultant wave function is their sum:

In order to simplify the expression above we can use the following trigonometric identity:

where:

And substituting in the resultant wave function we get:

The resultant wave is a harmonic wave whose amplitude is given by:

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The interference will be constructive when the amplitude A’ has its maximum value (2A), and that will occur when the cosine reaches its maximum value:

The interference will be destructive (A’ = 0) when the cosine is zero:

For any other value of the path difference, the interference of the two waves will not be completely constructive or completely destructive, but somewhere in between.

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