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If a wave is reflected off a surface, the angle of reflection is equal to the angle of incidence. But, how can we use this to find the actual path of the incident and reflected waves if we only know the positions of the wave origin and observer?

It seems clear that the reflection will occur in the plane normal to the reflective surface and including the origin and observer. We can then look at this problem in two dimensions:

Wave Reflection

(I've drawn the reflection off a horizontal line because my actual application involves a blast wave reflected off the ground in order to simulate the mach stem effect in a game. The blast wave also reaches the observer directly, but the mach stem effect is concerned with the constructive interference between this and the reflected wave.)

We know point O (the wave origin) and point P (the observer.) We know that R (the reflection point) lies on the ground line somewhere between where O and P project to the ground, but we don't know where. We also don't know theta although we do know it's the same on both sides.

I'm sure this is a relatively simple trigonometry problem but I'm terrible at seeing these things. Any ideas?

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2 Answers 2

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Reflect $O$ in the ground to $O'$, then $R$ is where $PO'$ meets the ground.

enter image description here

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  • $\begingroup$ Pretty much an answer. $\endgroup$
    – Kaster
    Jul 3, 2013 at 0:40
  • $\begingroup$ I knew I would smack myself. :) Many thanks! $\endgroup$
    – Deep Thought
    Jul 3, 2013 at 3:45
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The problem, as it is, is underdetermined. Given only the coordinates of $O$ and $P$, there are infinitely many $R$s (and $\theta$s) such that the ray originating from point $O$ will (upon reflection) pass through point $P$. To see why that is the case, imagine moving the "horizontal" line passing through $R$ up and down.

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  • $\begingroup$ It is determined if the altitudes of O and P are known, which I believe OP intended. $\endgroup$
    – Ross Millikan
    Jul 3, 2013 at 0:41
  • $\begingroup$ Is it undetermined even if the ground line (passing through R) is constrained? It can't move with respect to O and P as, in the application, those are actually specified relative to the ground line. I probably should have stated that in the question! Will edit it when I get back. $\endgroup$
    – Deep Thought
    Jul 3, 2013 at 0:42
  • $\begingroup$ @Kevin If we know the $x$-$y$ coordinates of $P$ and $O$, and if we know the "$y$-altitude" the horizontal line that passes through $R$, then the problem has a unique solution that is given by Zander. $\endgroup$
    – Lord Soth
    Jul 3, 2013 at 0:45

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