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Consider two point sources which generate circular wave disturbances (e.g. speakers, radio towers, etc.) which propagate uniformly in all directions.

Provided that each source emits its signal in-phase, and that the signals share a common wavelength, the interference maxima can be described easily by a set of hyperbolic equations:

$$\frac{4 x^{2}}{n^{2} \lambda^{2}} - \frac{4 y^{2}}{4 d^{2} - n^{2} \lambda^{2}} = 1, \qquad n \in \Big\{ \mathbb{Z}\, \Big\vert\, n < \tfrac{2d}{\lambda} \Big\}$$

where $d$ is the distance between the point sources, the x-axis is given as the line connecting them ($x=0$ being the midpoint between them) and $\lambda$ is the wavelength of the signals they emit.

A similar equation can be derived for interference minima.

However, things become much more difficult if the waves do not share a common wavelength or are not emitted in-phase. Yet the points of interference maxima/minima should still be well-described by a set of hyperbolae.

Can we generalize this expression to accurately describe cases where the signals may not have the same wavelength and may not be emitted in-phase?

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  • $\begingroup$ A hyperbola is an extremely well behaved geometric figure. What makes you think that it's goint go be hyperbolae in the general case? $\endgroup$ – Gabriel Golfetti Sep 18 '18 at 1:04
  • $\begingroup$ We require that the phase difference between the two waves at any given point equal some constant in order for the degree of interference to be the same. With no initial phase offset & a common wavelength, this implies the path difference is e.g. $n \lambda$ for completely constructive interference. I may be wrong but it seems not much really changes when we generalize it — the phase shift is just a constant-valued adjustment to the needed path difference; and with unequal wavelengths we now also require that path difference be equal to both $n_{1} \lambda_{1}$ as well as $n_{2} \lambda_{2}$ $\endgroup$ – corvus Sep 18 '18 at 2:35
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If the sources have a common wavelength and not emitted in phase then all that will happen is that the position of the interference pattern will move.
Changing from the sources being in phase to being exactly out of phase will produce a minimum where a maximum was and vica versa.

If there is a constant difference between the wavelengths then you can think of the result of the superposition of the waves from the two sources as a moving interference pattern or beats.

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