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

Question

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.

Ref :

• https://www.rose-hulman.edu/~bryan/lottamath/wave.pdf
• http://farside.ph.utexas.edu/teaching/329/lectures/node90.html

Answer 1

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!