Wave equations & propagation theories

Question

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?

Thanks

Answer 1

Your mechanical vibration expression can be rewritten (k subscript added by me) as the real part of:

$Ae^{j(k_1x-\omega t+\phi)}$

This describes a standing wave, or one component of a plane wave, created by a source which is constantly on, in a lossless medium. Notice that the entire exponent is multiplied by the imaginary factor 'j'. (EDIT: Scratch the 'standing wave' part: That would take two wave components, one with a -jkx factor in the exponent and another with a +jkx factor (traveling in the opposite direction).)

If you have a wave set up in a lossy medium by a constantly ON source, and you want to describe how it attenuates with distance, you need to add in a non-imaginary loss factor, $e^{-k_2x}$.

So then, assuming your source is at the origin, and the plane wave travels only in the positive 'x' direction, your expression for finding the instantaneous wave height at any point in the path of the plane wave would be the real part of:

$Ae^{-k_2x}e^{j(k_1x-\omega t+\phi)} = Ae^{(-k_2x+j(k_1x-\omega t+\phi))}$

This should be easy enough to code up. The tough part will be defining $k_2$.

And, on the Huygens thing: Don't worry about it too much if you are just doing plane waves or trying to analyze sound traveling down a lossy channel. Where Huygens really shines is explaining diffraction patterns or why light bends around a sharp corner.

Answer 2

The simplest wave-like phenomena are governed by a differential equation aptly named the "wave equation". In one dimension this is:

$\frac{\partial^2 \psi(x,t)}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 \psi(x,t)}{\partial t^2}$

where $\psi(x,t)$ is the height of wave at position $x$ and time $t$. Here $c$ is the speed of your waves. If you want your wave to be damped you can alter the equation to be:

$\frac{\partial^2 \psi(x,t)}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 \psi(x,t)}{\partial t^2}-r\frac{\partial \psi(x,t)}{\partial t}$ where $r$ is a parameter that controls how fast your wave is damped

See Wikipedia on the Wave Equation for more info. There's also a section on how to implement this as a computer program if my symbols mean nothing to you.