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I've been trying to understand the relationship between the wave equation, eikonal equation, and reaction-diffusion equations. Is there an easy way to see the relation between these and the classical wave PDE?

I noticed in many cases (e.g. applications to the heart) wave propagation is computed using an eikonal equation or a reaction-diffusion equation. However, they tend to still call these solutions waves propagating in the medium. Indeed, for instance, in this unanswered question there is apparently a clearer duality.

I note that wikipedia says that the eikonal equation is an approximation to the wave equation. But I'm not sure why they wouldn't just numerically solve the wave equation directly (except maybe computational speed).

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    In the limit where the eikonal equation is relevant, the wave equation is exceedingly difficult to directly solve due to the waves having very high frequencies. – 2017-01-18
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    @Ian thanks; do you have any thoughts on the reaction-diffusion & wave equation duality? – 2017-01-20

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