# Similarities / diffs between diffusion & wave propagation

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• philmolz
In summary, the heat equation and the wave equation, both covered in a differential equations course, have wave-like solutions. However, they represent different phenomena and have different equations. The solution to the diffusion equation is not a propagating wave, while the solution to the wave equation is. This is due to the difference in time-reversal symmetry between the two equations. The wave equation allows for both forward and backward propagating waves, while the diffusion equation only allows for irreversible processes. The attenuating "wave" in the diffusion equation is a dissipative process and not a true propagating wave.
philmolz
Hi,

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

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

and the wave equation,

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

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

u= Ce^{-\sqrt{w/2D} x } \sin{(\sqrt{w/2D} x - wt)}

\Psi = \Psi_{0} e^{i(kx-wt)}

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.

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?

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.

## 1. What is the main difference between diffusion and wave propagation?

The main difference between diffusion and wave propagation is the mechanism of movement. Diffusion is the process of particles moving from an area of higher concentration to an area of lower concentration, while wave propagation is the transfer of energy through a medium without any physical movement of particles.

## 2. How does diffusion occur in a system?

Diffusion occurs due to the random motion of particles in a system. The particles move from areas of high concentration to areas of low concentration until the concentration becomes equal throughout the system.

## 3. What is wave propagation and how does it work?

Wave propagation is the transfer of energy through a medium. It occurs when a disturbance or vibration is introduced into the medium, causing the particles of the medium to vibrate and transfer the energy to neighboring particles.

## 4. Can diffusion and wave propagation occur simultaneously?

Yes, diffusion and wave propagation can occur simultaneously in a system. For example, in a gas, diffusion of individual particles occurs while the gas as a whole can propagate as a wave.

## 5. What are some similarities between diffusion and wave propagation?

Both diffusion and wave propagation involve the transfer of energy through a medium. They also both occur due to the random movement of particles in a system. Additionally, both processes can be described mathematically using similar equations.

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