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Let’s say we have a 2D “maze” which represents walls in a space. If we place a sound source somewhere in it, what would be the amplitude at a different point in the maze. Or in a simpler case, given a sound source, a wall (possibly enclosing it from all sides but with a gap), and a sound receiver along the outside wall, how do I calculate the amplitude there.

I found sources online saying that amplitude does decrease after diffraction but none about how exactly I calculate it.

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  • $\begingroup$ Do you let your source run indefinitely? Is it a wave with a single frequency? $\endgroup$
    – nicoguaro
    May 15 at 18:49
  • $\begingroup$ Well, let's say that its a constant frequency wave and it emits only for a short while, so only the immediate results are considered from the diffraction $\endgroup$
    – Evgeny
    May 16 at 10:48
  • $\begingroup$ If it's a pulse it can't be made of a single frequency. $\endgroup$
    – nicoguaro
    May 16 at 16:59
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If you have a point source that produces a pulse then, in general, the amplitude will decrease as moving away. For wavelengths that are short in comparison with your maze you could estimate the decaying by measuring the length of the path of a ray. If you have larger wavelengths the diffraction is more relevant and it would be more difficult.

If you're interested in computing the amplitude you need to solve the wave equation

$$\nabla^2 P = \frac{1}{c^2} \frac{\partial^2 P}{\partial t^2} + F(x, y, t)\, ,$$

where the boundary conditions are, probably, Neumann in all the walls. That is, null normal derivatives.

For an arbitrary geometry you would need a numerical method to solve this problem. I would suggest the finite element method.

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