Physical meaning of KdV equation

[fian]

Here is one of the KdV form

u_t + u_x + uu_x + u_{xxx} = 0

Where u is elevation, x is spatial variable, and t is time variable. The first two terms describe the linear water wave, the third term represent the nonlinear effect, and the last term is the dispersion.

From what i understand, the nonlinear term explain the energy focusing that keeps the shape of the wave packet. But, how is u multiplied by u_x represents the energy focusing? For example, like in predator-prey model, the nonlinear term xy explain the interaction between the two species, where x and y are the number of predators and prey respectively.

Also, how does the last term, the third derivative of u with respect to x, explain the dispersion which is the deformation of the waves?

 

[Simon Bridge]

It can be useful to view this equation in context of others:
http://www.scholarpedia.org/article/Linear_and_nonlinear_waves#Nonlinear_wave_equation_solutions
... you'll find a discussion of dispersion relations if you scroll down.

You should also get a good idea from the derivation.
http://www.wikiwaves.org/KdV_Equation_Derivation

The thing that keeps, say, a water wave in it's shape is surface tension ... this depends on the height of the wave as well as how that height changes over the shape of the wave. The uu_x only produces that relationship in dimensionless units - it's not a case of "energy gets focussed so we need to multiply height with gradient".
Similarly with the dispersal term - if you think of dispersion in terms of the way u is distributed and see how that could work.
Once more the term is not the same as dispersion. Have you tried working out the dispersion relation with and without the term?

 

[fian]

Thank you for replying, Simon.

So, $uu_x$ explain how the tension works to waves as they propagate, doesn't it?
does tension relate to wave steepening? I read that the nonlinear term in KdV represent the nonlinear wave steepening (I forgot where I found it, but I wrote it in my note book and forgot to put the reference). Doesn't steepening also relate to energy focusing?
In the link about the derivation of KdV equation, it only shows the derivation KdV from Laplace eq., i didn't find the explanation about the terms' meaning.

The thing that keeps, say, a water wave in it's shape is surface tension ... this depends on the height of the wave as well as how that height changes over the shape of the wave. The uu_x only produces that relationship in dimensionless units - it's not a case of "energy gets focussed so we need to multiply height with gradient".
Similarly with the dispersal term - if you think of dispersion in terms of the way u is distributed and see how that could work.
Once more the term is not the same as dispersion. Have you tried working out the dispersion relation with and without the term?

Yes. I've found the dispersion relation with and without the dispersion term $u_{xxx}$. Without the term, the dispersion relation, $\omega = k$, is linear meaning that the phase velocity $C_p = \frac{\omega}{k}$ and group velocity $C_g = \frac{d\omega}{dk}$ are constant (these velocities formula also explained in the link you gave). In this case, the waves do not disperse. Meanwhile, with the term included, the dispersion relation is $\omega = k - k^3$, so there is change in both velocities, and the waves experience dispersion.

I still do not fully understand the physical meaning. Can we directly explain how the dispersion effect is from the term $u_{xxx}$ without deriving the dispersion relation first?I've just learned about water waves since several months ago, so the information in my head is still messy.

 

[Simon Bridge]

Strictly the term falls out from the initial assumptions in the derivation ... the maths emerges from there. That's probably why there's not much discussion on it: it's maths.
The initial assumptions give you the physics. You can sometimes get an idea how a term gives rise to an effect by watching the maths carefully. When you calculate the dispersion, how does the term u_xxx play a part? However, a mathematical term does not need to have any physical significance to give a physically significant outcome.

It may help to look at it like this though:
The first derivative is the gradient, the second is the curvature, the third is called "aberrancy", which means the tendency to deviate and misbehave ... it's the property of a function that, in a wave, drives dispersion ... so what does "dispersion" mean that it is affected by the way a wave changes it's curvature with distance?

 

[fian]

thank you, your hints help improve my understanding and insight. However, I do need to learn more.

Strictly the term falls out from the initial assumptions in the derivation ... the maths emerges from there. That's probably why there's not much discussion on it: it's maths.
The initial assumptions give you the physics. You can sometimes get an idea how a term gives rise to an effect by watching the maths carefully. When you calculate the dispersion, how does the term u_xxx play a part? However, a mathematical term does not need to have any physical significance to give a physically significant outcome.

what i can conclude from the contribution of term u_xxx to the dispersion relation is that the dispersion is nonlinear, resulting that there is change in velocities.

It may help to look at it like this though:
The first derivative is the gradient, the second is the curvature, the third is called "aberrancy", which means the tendency to deviate and misbehave ... it's the property of a function that, in a wave, drives dispersion ... so what does "dispersion" mean that it is affected by the way a wave changes it's curvature with distance?

I think, it means that the shape of waves can change to be steeper or more slope. Am I correct?

However, a mathematical term does not need to have any physical significance to give a physically significant outcome.

So, I can say that not every mathematical model can directly explain the physical condition of its system? some can't, because there are models which are obtained from derivation from other models and are not directly developed from physical phenomena.

 

[Simon Bridge]

Dispersion is the effect of the phase velicity depending on the frequency. In a pulse, the effect is to spread the pulse out as each component travels at a different speed. There are a lot of other effects but the word referrs to that spreading.

What we mean when we say that a particular model explains some physical phenomenon is philosophy... which is a banned topic here: look up "epistomology". What I am saying is that not every bit of an equation needs to have a specific physical meaning for the entire model to be useful. Even when you do have a physical analog of a mathematical operation, that does not mean the physical analog occurs in nature the way it appears in the equation or algorithm or whatever. I am concerned, in other words, that you may be over-thinking things.

 

[fian]

Alright, I see.

Thank you for your explanation. I got something new through this discussion beside what I asked.

 

[hunt_mat]

There are two things that balance each other, the steepening effect given by the nonlinear term uu_x, and the dispersive part u_xxx. There are many things which contribute to the uu_x term, they all come from velocity terms though