# Wave equations & propagation theories

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

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BTW: This site uses MathJax (a LaTeX alike math rendering egine) so you can write $A \sin(kx - \omega t + \phi)$ and similar expressions (that is A \sin(kx - \omega t + \phi) enclosed in dollar signs). You find a bare minimum of help in the FAQ. – dmckee Jun 28 '11 at 17:41
@dmckee: Thank you so much for that explanation. I had tried to make Mathjax work, to no avail. I will try again, while also attempting to answer Raven's question. – Vintage Jun 28 '11 at 19:43
If you want something appoximated by the linear wave equation, what you are looking for is the "fundamental solution" of the wave equation. If you google that you'll find lots of information. In the case of 3 (and other odd numbers) of spatial dimensions, for solution of the wave equation Huygen's principle is strong, in the sense that if you give a point perturbation there should in fact be nothing inside the envelope: the wave only lives on the wavefront. Compared against physical waves, this may suggest that the linear approximation is not the right one. – Willie Wong Jun 28 '11 at 19:47
Also: If you say more about what you are trying to simulate, maybe it would be easier (for readers) to suggest an appropriate model? – Willie Wong Jun 28 '11 at 19:49
If you actually solve the wave equation in 2D ($\partial_t^2 u = c(\partial_x^2 u + \partial_y^2 u)$) numerically, you will be able to incorporate diffraction automatically. Refraction can be handled by changing the wave-speed $c$ between different media (but you need to implement boundary matching rules to properly capture the effect of the wave being partially transmitted and partially reflected). – Willie Wong Jun 28 '11 at 23:39

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.

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Well equations are simple enough, I'm more concerned what you meant by defining k2. I'd await that it should be a constant I can search somewhere for every material (at least for that tested ones)... Well and I'm going to make every possible phenomenon, that means diffraction and refraction too, so Huygens looks best so far.. – Raven Jun 28 '11 at 21:19
@Raven: Sounds ambitious. Yes, k2 should be a real constant, assuming that the material and the cross section of the path remain constant. If you have changes in the path (geometry or materials) you run into equations for reflections at the boundaries. Sorry, I do not know where to look up what the numbers actually are. – Vintage Jun 30 '11 at 17:05

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.

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You guessed correctly I'm not much into those symbols, but I understand these equations. In fact I was looking for something like that because that FDTD methode I mentioned is starting also from differential equations (just for EM wave). I'll see how or if I could make a model from these, but for now it's a great start. – Raven Jun 28 '11 at 21:32