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I'm currently learning about modelling the propagation of acoustic waves using numerical models. This is done by solving the wave equation (expressed as a partial differential equation) with something like a finite difference model or a finite element model or something similar.

My question is: Why can we not solve the partial differential equation directly? Why must we use a numerical model to solve the wave equation?

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If you have a propagation in a nonlinear medium, for example, it could be hard to find the solutions of the wave equation analitycally, so you could need to solve it numerically. –  Riccardo.Alestra Feb 17 '12 at 14:56
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Check This : books.google.co.in/… –  Inquest Feb 17 '12 at 14:57
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If you find an explicit solution, let us know... –  J. M. Feb 17 '12 at 14:59
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Even an explicit solution is a rather broad term: if it is expressed as series over some exotic function - are you sure that it is faster to compute (approximately) such series? –  Ilya Feb 17 '12 at 15:08

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up vote 4 down vote accepted

You do not need to, but in the context of your course it makes more sense to do it numerically from the start:

Take the standard wave equation $$\partial^2_t u(\vec x, t)=c(u)^2\cdot\nabla^2u(\vec x, t)$$ If $c(u)=\mathrm{const}.$, then there is no problem in using the standard analytical methods. But as Riccardo said, if it is not constant, for example in a dispersive or non-isotropic medium, you can solve it by only a simple extension to the numerical model, whereas there might not even exist an analytic solution.

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