Similarities / diffs between diffusion & wave propagation

[philmolz]

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.

 

[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.

 

[philmolz]

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]

The wave equation (1) does not allow an attenuating solution. You can check that by substituting the attenuating function (3) into (1). Any solution of the wave equation is of the form f(x ± vt). The decaying solution is not of that form.
The difference between the two equations (and their solutions) is in time-reversal symmetry:
In equation (1), if you change t to - t, the equation is the same. That translates to the fact (easy to check) that both a forward propagating wave (unattenuated), and a backward propagating wave are solutions of the same equation, and are actually possible phenomena. You see both waves happening all the time.
The diffusion equation (2), on the other hand, is not invariant under time reversal. It represents a macroscopic irreversible process, for example, heat conduction from a high temperature region to a low temperature region, spreading of a drop of ink through a body of water, etc. these processes never happen in the reverse direction. These are inherently dissipative processes. The reverse processes (conduction from low to high temperature, the ink drop gathering back together) would violate the second law of thermodynamics. They are not solutions of the diffusion equation.
The attenuating "wave" is actually a dissipative process, in which energy is transferred from the "wave" into several other modes. That process is also irreversible, and is a solution of the diffusion equation.