# What is the difference between the wave equation and the advection equation?

I am trying to understand what is the difference between the wave equation ad the advection equation? Bot the equation seems to move a quantity. While in literature it is said that the advection equation is the simpler of the two I don't see how the wave equation is that much different than the advection equation except moving the quantity in both directions when talking in terms of 2d.

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• Which source are you referring to? – Solar Mike Feb 25 '19 at 15:26
• Edit your question to include that information properly - don't leave it dangling in comments. – Solar Mike Feb 25 '19 at 16:59
• The fact that "the quantity can move in both directions" is a HUGE difference. For example waves can be reflected from boundaries. But looking at the table of contents of these notes, it's not very clear what they are intended to teach you about math and/or numerical methods and/or physics (which are three different subjects, of course). – alephzero Feb 25 '19 at 17:19
• @alephzero If we consider the physics isn't wave just transfer of energy? Running the 1D equations, both advection and the wave equation seems to do the same thing ie transfer of a quantity. So what is the difference between them? – GRANZER Feb 26 '19 at 13:24

## 1 Answer

As alephzero mentioned the difference is huge.

I will explain mathematically and then physically.

Mathematically,

For any given partial differential equation of the form, $$A \frac {\partial^2\phi}{ \partial x^2} + B \frac {\partial^2\phi}{ \partial x \partial y} + C \frac {\partial^2\phi}{ \partial y^2} + \rm first\ derivative \ terms\ =0$$ We get,

$$\rm B^2 - 4AC = 0 , - Parabolic,$$

$$\rm B^2 - 4AC < 0 , - Elliptic$$

$$\rm B^2 - 4AC > 0 , - Hyperbolic.$$

Based on the above conditions the wave equation is a hyperbolic equation and the diffusion equation is a parabolic equation.

These conical conditions decides the zone of influence and zone of dependance in your domain of interest. i.e., To be get disturbed at any point of interest in the domain, you should know where and when you should pinch.

Physically,

1. Consider a 2D square plane, apply heat on the west side edge, then the heat will flow from west to east. if you are running from north to south you will feel the same temerature. $$->$$ This process is goverend by unsteady heat diffusion equation mathematically parabolic and zone of dependance is all east side to where apply heat.

2. Throw a stone in the water pool which is still. after some time you will find, the wave reach all the corners of the pool. So you disturb anywhere to get disturbed anywhere. These waves are goverend by Laplace equation which is elliptic.

3. Now consider a moving channel, throw a stone, then see the waves now. Do the disturbances reach upstream of the channel? Nope. The disturbances created by the stone are carried downstream by the moving stream. This is in the 2D sense. Imagine same thing for 1D also. The created disturbances are carried by the material elasticity (noted by the wave/sound speed). when and where (the disturbed region) is decided by the wave speed. These inrteresting problems are governed by hyperbolic equations. So disturb only certain location to get disturbed somewhere.

You can google more to get pictorial explanations for these.

Hope this answer helps!

• Thank you for the great explanation. Yes, even I have been trying to study up on the matter since I posted the question and have found that both advection and wave equation are hyperbolic in nature. So mathematically they are similar. Physically the advection eq is modelling the moment of a quantity along with the flow (and any similar application) and wave equation is modelling the wave motion which model reflection, interference and can support shocks etc(depending on boundary conditions). – GRANZER Feb 28 '19 at 13:31
• but I thing you have confused between the advection/convection equation and the heat equation. – GRANZER Feb 28 '19 at 13:33
• Hi @GRANZER, The characteristics equation are similar for wave and advection equation. So I took advantage to explain these phenomena. If you see carefully the wave equation could actually be derived from advection of the disturbances. In the 3rd point, the stream-stone is advection and 1D string is wave equation. – mustang Feb 28 '19 at 14:38
• Or I could say ( the wave equation could actually be derived from advection) the advection is one of the characterics of wave equation. – mustang Feb 28 '19 at 14:41