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Interference of waves of the same frequency - mathematical description  - equation of the phenomenon resulting from the superposition of two waves which are equal in frequency and amplitude

Let us assume that in the same propagation medium, two (harmonic) waves of the same frequency are superposed:

Example: two waves coming from sources which are emitting rhythmically (see figure at the side).

For simplicity, let us assume that they are also equal in amplitude:

Let us call x1 the distance from the point considered in the propagation medium to source 1, and x2 the distance to source 2.

When we say that the sources are emitting rhythmically, it means, for example, that at the instant t when y1 is a maximum (A1) at source 1 (x1 = 0), y2 is also a maximum (A2) at source 2 (x2 = 0).

Then, taking t = 0 at this instant of time, the equations of both waves can be written as follows:

By applying the preceding Superposition Principle together with standard trigonometric formulae (sum of trigonometric ratios), we obtain the following equation for the interference of waves of equal frequency and amplitude:

The first term (between square brackets) does not depend on time, but only on the position of the point considered in the propagation medium. Therefore, this is the equation of a wave of variable amplitude A', which depends on the position and more specifically on the difference between the distances to the sources (or path difference):

Problems: P10607.
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Where are the resulting oscillations of maximum or minimum amplitude?