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When waves move into another medium, such as from air to water, we can calculate how much they transmit, reflect, and diffract as we know the angle of incidence they make with the interface, and we know the refractive index of the materials. And this is the theory behind basic microscopy.

But what happens when we don't have a clear well defined interface such as waves entering water from air and instead there are innumerable extremely small interfaces? For example when acoustic waves are travelling through the human body? Maybe there are some interfaces that are pretty straightforward, such as a section of bone..but most of the time the body will be extremely inhomogeneous with a spatially dependent index of refraction $n(x)$.

How can we handle this situation mathematically/physically when it is not possible to do simple calculations anymore like we did for the air-water case?

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It depends on what is your objective. Sometimes you are interested in a net effect, while in others you may be curious about a specific region.

As an example, let's say you're studying how a certain type of electromagnetic radiation interacts with the human body. There are several possible questions regarding this subject:

  1. Will the radiation pass through the body?
  2. How does that specific kind of radiation interact with a certain type of complicated tissue or organ?
  3. Will it give you cancer?

and so on. I can give you some ideas about how to model some of those problems:

  1. To see if the radiation will be able to exit your body, you might want to model it as a thin wall of the most impenetrable tissue in the body with respect to the kind of radiation you're using (in general those are bones). The thickness of the wall can be used to approximate experimental data.
  2. If you're worried about the effects in a specific organ, you might want to model that organ first, which means obtaining its approximated geometrical structure and using interpolation techniques. You can then use numerical methods to solve some approximation of Maxwell's equations inside a cavity to give you some results about bouncing, or maybe you have a greater interest at modelling the wall of that organ as a uniform tissue with a mean density. Perhaps your interest lies exactly in the fact that your problem is inhomogeneous, and in that case you can take an infinitesimal section of inhomogeneous tissue, do some Monte Carlo or any other method to solve some numerical equations, and then numerically integrate.
  3. In this case you'll probably going to model things at the Chemical limit, which means a sort of semiclassical limit where sometimes you need Quantum Mechanics and sometimes Classical Mechanics is enough.

I don't even know if the things I told you make complete sense or if they answer your question, but I said them just to show you that modelling nature requires simplification and, almost always, numerical techniques. The satisfying part is that for a lot of experiments modelling the human body as a ball of water gives you extremely good results.

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