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Suppose a simple spherically-symmetric cosine wave of wavelength $\lambda$ and velocity $c$ is transmitted from an unknown point outwards in $2D$ (in circles). We'll denote this unknown point as $(0,0)$. The wave is received at a distance of $r=r_0$ from this point. How can you measure the distance $r_0$, knowing only its wavelength and speed (and therefore, frequency)?

What I'm thinking about is measuring the phase of the wave, but I'm not sure about the math. I was told that the phase measurement would yield an infinite amount of solutions for $r_0$, and in order to figure out what $r_0$ truly is, another measurement is required, from a different radius or rather angle. I'm not sure how to do it. The most important thing for me is to see the math behind it because at this point, I'm even confused as to what the equation of the wave is. I tried researching and it seems like phase arrays might do it but again, not sure about the math behind it.

Thank you very much!

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You need to measure the phase at 3 (or more) points.

At some point $A$ and time $t$, you measure the phase.

You pick another point $B$, say a distance of $\lambda/4$ from $A$. You measure the phase at time $t$.

Suppose the ray from $(0,0)$ through the midpoint of segment $AB$ happens to be perpendicular to $AB$. Then the two phases will be the same. Suppose the ray happens to be parallel to $AB$. Then the phases will be different by $2\pi/4$.

You should think about how to get the angle between the ray and $AB$ from the phase difference. Once you have that angle, you know the direction from the midpoint to $(0,0)$.

Now choose another point $C$, perhaps a distance $\lambda/4$ from $A$ in another direction. You can get the direction from another point to the origin. From that, you can find the distance to the origin.


Note that phase might not be what you directly measure. Suppose the $2$-D surface is the surface of the ocean, and you are measuring water waves. You would measure the height of the surface at time $t$. From that you might figure out the phase.

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