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I don't have a background in numerical modeling so this question is rather broad.

What I am interested in is modeling the propagation of an ultrasonic acoustic wave in 3d space. The basic 3d wave equation is tractable to me but I don't understand how boundary interaction is computed or how to get accurate simulations of things like diffraction.

What is a good starting point for understanding this information?

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    $\begingroup$ The answer to this question encompasses an entire field of research, with thousands of published papers. You should specify whether you're concerned with solid (physical) boundary conditions, or artificial boundaries that exist only to truncate the computational domain to a finite size. It sounds like you mean physical boundaries, in which case it's important to specify whether they are fully reflecting or not. $\endgroup$
    – David Ketcheson
    Jan 3, 2021 at 14:03
  • $\begingroup$ Solid boundary conditions, partially reflected but we could start with full reflection. For the full reflection model how does energy leave the system? $\endgroup$
    – FourierFlux
    Jan 3, 2021 at 21:25
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    $\begingroup$ If all boundaries are fully reflecting then no energy leaves the system. I recommend editing your question so that people don't need to read the comments to know exactly what you're asking. $\endgroup$
    – David Ketcheson
    Jan 4, 2021 at 9:50
  • $\begingroup$ Do you have access to a software with this model in it or do you want to write a toy software to learn some of the details? $\endgroup$
    – nicoguaro ♦
    Jan 6, 2021 at 16:29
  • $\begingroup$ I don't have any software but I can find some, I have access to a lot. I also would be interested in learning the methods. $\endgroup$
    – FourierFlux
    Jan 6, 2021 at 21:58

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Let's consider the one-dimensional string first. Standard text-book physics considers the three usual boundary conditions here, namely Dirichlet (endpoints of the string are fixed), Neumann (endpoints are free) and Robin conditions (obtained e.g. when the endpoints are attached to a spring).

Now, for real sonic propagation, those boundary conditions won't apply exactly, but serve only as an approximation (as usual in physics). When a sound wave hits an object, it will likely be reflected only partly while a certain component is damped out. Moreover, this damping will likely depend on the frequency, the input direction, the geometry and material of the object and so on. That is a very tough proble in general.

For the first steps, however, it is usually sufficient to use ideal reflection and use zero-Dirichlet boundary conditions. The assumption there is, that when a sound wave hits a wall, it won't lead to any oscillation there, and thus will reflect the wave completely with a phase shift of 180°. Using Dirichlet conditions is also the easiest you can do: In a cartesian grid treatment is suffices to leave the boundary planes out of the propagation.

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