# Circular interference from the equation of a hyperbola

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1. Jan 24, 2015

### dimestorelaser

Hi all,

I had a question that I cant seem to find an answer too.
I was hoping people could point me in the right direction, or let me know if there is an "easy" method.

It has to do with the classic example of two stones in water producing constructive and destructive interference patterns, which create a hyperbola.

If we are given the closest hyperbola created by the interference patterns. Is there an "easy" way...to generate the equations for the other hyperbola that are also created?

Generally I have just been plotting some circles and "guessing" at the other ones. However it seems like there should be a pretty easy method...since they all have the same focus right?

For example, I am trying to create the other hyperbolic arcs created at the same time is this mildly complex hyperbola:

$y^2 = 4x^2 + 5x +632378$

Here is the wolframalpha of the equation:
http://www.wolframalpha.com/input/?i=y^2+=+4*x^2++5*x++632378
I would appreciate any feedback. Thank you!

2. Jan 24, 2015

### Staff: Mentor

There are formulas to get the focal points.

The manual way to find them: substitute $z=x+\frac{5}{4}$ or equivalently $x=z-\frac{5}{4}$, then $4x^2 + 5x = 4z^2 -\frac{25}{16}$.
That allows to convert it to standard form easily and then you get your focal points. It is possible to pack everything together in formulas, too, even for rotated hyperbolas (where you get xy terms).

3. Jan 27, 2015

### dimestorelaser

Hi thanks for the response,

Im not sure I was clear in what I was asking.

I know how to get the focus of the equation, thats not the question. My question is about the set of hyperbolas created by those foci. Consider a two point source at the locations of the foci, how do I determine the equations of ALL the hyperbolas associated with those foci. (Assuming that we have wavelength of 1).

In the above picture....If I know the equation of N3...can I easily determine the equations for N2 and N1?

I appreciate the response. Thank you.

4. Jan 28, 2015

### Staff: Mentor

The differences in distance to those foci have to be ... a-2, a-1, a, a+1, a+2, ... where a is some real number between 0 and 1, describing the relative phases of the emitters.

Calculate the distance difference for N3 and the focal points, reduce the distance difference by 1, 2, ... wavelengths.

5. Feb 6, 2015

### dimestorelaser

HI thanks for the help, I appreciate it.

Would you mind working out the example I gave? I would help me to understand what you are saying. Thank you,

6. Feb 7, 2015

### bubsir

Hi, I think there is an error in your drawing - the loci of "maximum constructive interference" should be straight lines.
This diagram is from "Young's Experiment" on double slit interference.

7. Feb 7, 2015

### sophiecentaur

The loci are hyperbolae, hence the Hyperbolic Radio Navigation Systems. The Young's slits pattern that you show is an approximation, and only holds when the distances are greater than the separation. Note that the pattern in the original diagram also gives straight lines, well outside the region of the two sources.

8. Feb 7, 2015

### dimestorelaser

Nevermind...I realized its actually quite simple.