# Similarities / diffs between diffusion & wave propagation

• I
Hi,

I'm a second year undergrad and we've covered the heat equation,

\begin{equation}
∇^{2}\Psi = \frac{1}{c^{2}}\frac{\partial^2 \Psi}{\partial t^2}
\end{equation}

and the wave equation,

\begin{equation}
D∇^{2}u= \frac{\partial u}{\partial t}
\end{equation}

in our differential equations course. Both Diffusion and wave propagation have wave like solutions, for example,

\begin{equation}
u= Ce^{-\sqrt{w/2D} x } \sin{(\sqrt{w/2D} x - wt)}
\end{equation}
\begin{equation}
\Psi = \Psi_{0} e^{i(kx-wt)}
\end{equation}

but are quite different phenomena. Could someone briefly explain the similarities/ differences in the phenomena and the solutions and how this relates to the differential equations please? Thanks.

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Chandra Prayaga
First, the difference in the equations themselves: The diffusion equation is first order in time, whereas the wave equation is second order. Now the solutions. The solution (3) to the diffusion equation is not a propagating wave, and is not a solution to the wave equation. Similarly, the solution (4) to the wave equation is not a solution of the diffusion equation.

That's great, thanks, but I thought (3) is a propagating wave that is attenuated with distance (propagating wave enveloped by a decaying exponential). Also, I was wondering what the physical difference between the two phenomena is that leads to the difference in the equations, ie why we don't just have a simple propagating wave solution for heat diffusion?

Chandra Prayaga