Talk:Airy wave theory

I don't understand this formula

Equation 4 under the section Mathematical formulation of the wave motion states that

$(4)\qquad {\frac {\partial \Phi }{\partial t}}\,+\,g\,\eta \,=\,0\quad {\text{ at }}z\,=\,\eta (x,t).$

However, I don't understand what this formula actually tells us. It says in the paragraph before that it's obtained by supposing that the pressure at the surface is zero; nevertheless the pressure is never used in any equation. Moreover, I get a very strange result when I'm trying to use the equation. Assume that for a specific x- and a specific t-value, the velocity in z-direction is 0 at surface level:

$u_{z}=0,\qquad z=\eta (x,t)$

Now, since the surface stands still, we have ${\frac {\partial }{\partial t}}\eta (x,t)=0$. Using this, in combined with the fact that $u_{z}={\frac {\partial \Phi }{\partial z}}$, we can write

${\frac {\partial u_{z}}{\partial t}}={\frac {\partial }{\partial t}}{\frac {\partial \Phi }{\partial z}}={\frac {\partial }{\partial z}}{\frac {\partial \Phi }{\partial t}}$

And since we have

${\frac {\partial \Phi }{\partial t}}=-g\eta (x,t),\qquad z=\eta (x,t)$

(which follows from equation 4) we get

${\frac {\partial u_{z}}{\partial t}}=-{\frac {\partial }{\partial z}}g\eta (x,t)=0$

which means that if u_z is zero at surface level, u_z will stay zero?

--Kri (talk) 17:06, 15 October 2010 (UTC)

You cannot differentiate a boundary condition (4) into a fluid-interior (z) direction. As stated in the article this boundary condition is obtained from linearizing Bernoulli's equation for unsteay flow, assuming the atmospheric pressure above the free surface is a constant (taken zero for simplicity). -- Crowsnest (talk) 07:32, 19 October 2010 (UTC)

Ah, I see my mistake. Thanks. —Kri (talk) 16:02, 2 June 2012 (UTC)

What is Ω?

In the Table of quantities, what is Ω? There is already ω, which is the absolute angular frequency, and σ, which is the intrinsic angular frequency. Ω on the other hand, is introduced first in the expression for the observed angular frequency in the table, but is never explained. Besides, can ω and σ be negative? The article doesn't mention anything about it, so I guess they can. Maybe this is what is different between Ω and σ? Besides (since I've already touched it), can k be negative? Anyway, this should be made clear in the article. —Kri (talk) 15:53, 2 June 2012 (UTC)

The angular frequency ω and wavenumber k are not independent, but related through the dispersion relationship ω²=Ω²(k), e.g. in deep water ω²=gk. So Ω(k) is the dispersion relation, as said in the table. In the article, k is the length of the wavenumber vector k, and also given as k=2π/λ with λ the wavelength. And in the article it says that ω=2π/T with T the period. So in the present context both are positive. There are other wave problems where the mathematical treatment becomes easier by also allowing for negative (or complex) frequencies and wavenumbers, but these are out of the scope here. -- Crowsnest (talk) 19:48, 10 June 2012 (UTC)

Most treatments of dispersion relations don't use a different symbol for the frequency that is a function of k, but stick with lowercase omega(k); I think we should, too, unless there's a good reason and a source that suggests otherwise. And yes these frequencies and wavenumbers can be positive or negative (and probably also complex, though the analysis in this article doesn't support that). Dicklyon (talk) 20:53, 10 June 2012 (UTC)

It is quite common to use a different symbol for the functional relationship (the dispersion relation) itself, see e.g. Eq. (1.17) on page 9 of Whitham's "Linear and nonlinear waves" [1]; p. 99 on Craik's "Wave interactions and fluid flows" [2]. The reason to do so is for treating slowly-varying (narrow spectral width) waveforms A(μx,μt) exp(iΘ(x,t)), μ≪1, where ω and k are both defined in terms of a 3rd quantity, the wave phase Θ(x,t): ω=−∂Θ/∂t and k=+∇Θ. Even more generally, ω and k satisfy D(ω,k)=0.

To my opinion, positive and negative wavenumbers and frequencies should primarily be addressed in articles where that is directly appropriate, here it seems quite off-topic to me. -- Crowsnest (talk) 22:37, 10 June 2012 (UTC)

The cited source seems to do it all with lower case; I changed it but then reverted myself, as the the "observed angular frequency" doesn't make sense that way. I think we need to understand more about where these formulae come from... Dicklyon (talk) 00:20, 11 June 2012 (UTC)

Crowsnest, they article doesn't state that ω² = Ω²(k) — it states that (ω − k · U)² = (Ω(k))², where k = |k|.

Dicklyon, I agree with you that if Ω(k) and ω are essentially the same thing (or if they would be) they should also have the same symbol. Having three different symbols for, as far as I can see, two different things is just confusing. —Kri (talk) 18:25, 11 June 2012 (UTC)

Since ω² = Ω²(k) (without Doppler shift due to an ambient current), the two branches of the dispersion relation are: ω = +Ω(k) and ω = −Ω(k), so ω and Ω(k) are not the same thing. Even more, with ambient current U the dispersion relation is (ω − k•U)² = Ω²(k), where k = |k|. With solutions ω = k•U + Ω(k) and ω = k•U − Ω(k), even the more showing that ω and Ω(k) are not essentially the same. -- Crowsnest (talk) 13:08, 6 September 2012 (UTC)