Talk:Capillary wave

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[edit] About derivation + thermal cw's

I have noticed my latest contribution has been reversed. First: the derivation of the dispersion relation is not "original research", I have adapted from a standard book, whose reference is included (I'm sure there are plenty more, this is just one I am familiar with). I think the derivation of something so common as waves on a puddle should be interesting to many. About the thermal part... OK, I guess that could be a separate entry, more related to statistical mechanics. Best. --Daniel (talk) 07:34, 9 April 2008 (UTC)

Your efforts and attempts are very much appreciated. Personally, I would very much welcome a (sub)section with the derivation of the dispersion relation. But then, since this is an encyclopedia, it has to be more than the sketch on the route how that might be done, as given yesterday. Which had several errors in the potential energy by surface tension, lacked expressions for the kinetic energy of (gravity-capillary) waves, and information on how to derive the dispersion relationship from the Lagrangian. It has to be as simple as possible, in order to be understandable for a large audience. In that respect, a derivation using variational principles (although more elegant) may be more difficult to grasp than one derived directly from the flow equations. Crowsnest (talk) 18:06, 9 April 2008 (UTC)
I am now not sure if it is best to expand the sketch or to shrink it to make it less technical. About the errors: I am not sure where those are for the potential energies and the kinetik one. About the Lagrangian, I have fortunately found out a way to derive the relation with no variational methods involved... but Fourier analysis must be used then (not much, though). Will do in the near future. Thanks for replacing the thermal part, but this is now in a different article (link added at the end). I will likely write down the whole derivation in sklogwiki, a more technical wiki I collaborate with. Best. --Daniel (talk) 20:01, 9 April 2008 (UTC)
The slogwiki is also erroneous, and not something I would call a reliable source. Also the used notations are very uncommon for gravity-capillary surface waves. The potential energy due to surface tension is in error: it is the surface tension times the deviation of the area from the smallest area (i.e. flat).In your notation:
E_{st}=\sigma\iint \text{d}x\;\text{d}y\; \left( \sqrt{1+(\partial h/\partial x)^2 + (\partial h/\partial y)^2} - 1 \right).
But also the remainder contains several erroneous assumptions, like for instance the separation of variables in a time function and a space function, which does not hold for progressive waves.
In my opinion, the present "sketch" of a possible derivation is not fit for Wikipedia. Please replace with something that is not original research, comes from a reliable source and is as simple as possible, and does not contain large unfinished steps. Otherwise I will remove it, see WP:V. Crowsnest (talk) 21:18, 11 April 2008 (UTC)
You are probably right about sklogwiki possibly not being reliable, as it is just another wiki. I just included a reference to a place where the proof is presented in more detail.
As for the rest: do you not know that an additional term in an energy is negigible? How can this be an "error" (other than from a pedagogical point of view?) Include the -1 if you want, it will not change a iota. Separation of variables does not always work, but when it does, it does (from unicity), and it certainly does in this case. Again, this is not original research (I wish!). Just get a copy of Dr. Safran's book, and read his proof. For crying out loud, this stuff is so well known he does not give references, this must go back to Laplace or so! I have tried to make it simple. The choice is either to simplify it to its bare bones ("this has to do with gravity, surface tension, hydrodynamics, but we won't go into the details"), or to fill the missing steps. Personally, I am now leaning toward the first option. Perhaps this is too technical a derivation for wikipedia, after all.
A final detail: I resent being told this is not verifiable, and that this is OR when before my input there were no references whatsover. Now there's one at least.--Daniel (talk) 20:19, 13 April 2008 (UTC)
An apology: sorry, I was misled by the trivial -1. There was indeed an important +1 missing inside the square bracket.--Daniel (talk) 12:10, 14 April 2008 (UTC)

[edit] New "derivation"

OK, whichever the criticism it seems clear that the derivation provided was too technical. I have followed the style of Feyman's book, and just pointed out the main assumptions made. No equations. This is probably for the better. Additional references to standard books on fluid dynamics and ocean science would be great. Best. --Daniel (talk) 09:12, 14 April 2008 (UTC)

[edit] Equipartition?

The fact that, on average, U=T, is relevant and interesting. But I doubt this can be called "equipartition". Afak, the later refers to the fact that every degree of freedom that enter quadratically in the Hamiltonian ends up contributing (1/2) kT, in classical statistical physics. See equipartition. (This is very important for thermal capillary waves, by the way.) But I may be wrong...--Daniel (talk) 07:21, 17 April 2008 (UTC)

Yes, it can, see Equipartition#Potential energy and harmonic oscillators. For water waves, this goes back to Rayleigh in the 19th century. Only in nonlinear water waves, this is not valid anymore. Crowsnest (talk) 16:14, 17 April 2008 (UTC)
Still, I think this would only apply when temperature is taken into account, in the common sense of the word. Equipartition#Potential energy and harmonic oscillators says just that: a harmonic oscillator contributes a quadratic term just like a kinetic term, and therefore contributes (1/2)kT. Anyway, this is perhaps not so important since the article explicitly says that the important result is that < U > = < T > .--Daniel (talk) 07:38, 18 April 2008 (UTC)
I removed the link to the thermal equipartition theorem, and added a reference to Rayleigh. Crowsnest (talk) 09:03, 18 April 2008 (UTC)
I have come across this equipartition term in an article on waves (F Behroozi, Eur. J. Phys. 25 115-122 (2004) ), so its use seems to be established. I have reintroduced the term back, but still do not dare to link to equipartition, where only the sense from statistical mechanics is discussed. I guess it is equipartition which would need some rewriting --Daniel (talk) 13:06, 23 April 2008 (UTC)
It is well established: linear wave systems in continuum mechanics have that the Lagrangian is zero at the critical point, so equipartition of average kinetic and potential energy (of the oscillatory motion, subtracting the "static" mean value of the potential energy). But it is different from the equipartition article for statistical mechanics. Rather than changing the equipartition article, a new article on equipartition in linear wave systems would be better, in my view. Crowsnest (talk) 18:29, 23 April 2008 (UTC)
Indeed. Then, a disambiguation should be added at the beginning of the existing entry on equipartition.--Daniel (talk) 10:14, 24 April 2008 (UTC)