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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.2012-02-17
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    Check This : http://books.google.co.in/books?id=HftVSGIWVdYC&pg=PA21&lpg=PA21&dq=analytical+solution+PDE+impossible&source=bl&ots=O1HrH_hCfH&sig=NwTnX_L5tJgVE546tYCCYCrYZWc&hl=en&sa=X&ei=qmo-T_a4OZGyrAeorPi3Bw&ved=0CEgQ6AEwBQ#v=onepage&q=analytical%20solution%20PDE%20impossible&f=false2012-02-17
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    If you find an explicit solution, let us know...2012-02-17
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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?2012-02-17

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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.