# Why do we use numerical models to solve PDE's?

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
Check This : books.google.co.in/… – Inquest Feb 17 '12 at 14:57
If you find an explicit solution, let us know... – J. M. Feb 17 '12 at 14:59
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

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.