I'm interrested in making computer simulation but I've run into rather physics oriented problem. I have to choose how to propagate my wave. Though I've found technique called FDTD (finite-difference time-domain), I could find only explanations how to work with this method while solving electromagnetic waves but I'm more interested in mechanical waves - water, earth,...
So there's first problem. EM waves are nicely discribed with Maxwells equations, but I couldn't find anything like that for mechanical ones. Only equation I could find is $y = A \sin(k x - \omega t + O)$. This is fine, one can choose a distance and a time and he gets height of that point (considering that we are working on a plane). But doesn't wave gets also damped - in time as well as distance? And what formula describes this?
And the second part of a question - in school I was teached a propagation theory called Huygens principle, and it goes like: at every point where wave envelope propagate it creates elementary waves, and frontface of those again create envelope. I don't fully understand how it should be imagined, because creating waves all around a circle like on this image: http://kr.cs.ait.ac.th/~radok/physics/fig300.jpg (sorry, but I got a little problem with tags..), there is nice envelope but inside of that is mess of curves that would gave us interference. So it's just a model and we don't care what's inside as long as we have envelope, or am I missing something?
And as I said, I'm creating computer simulation and propagating wave like that would be probably slow, so is there any other theory on how wave propagates or this is the best and most used one?