Talk:Helmholtz equation

Paraxial Helmholtz Equation

This section is confusing, to be honest. Saleh for instance gives that the equation indeed should be

$\nabla _{\perp }^{2}A+2ik{\frac {\partial A}{\partial z}}=0,$

Rather than u. I am unclear why the final result has the 2k^2A = 0. This conflicts with easily available and canonical sources. Both u(r) and A(r) are called complex amplitudes - to my understanding, u(r) should be the complex wave function. I think there is some kind of error here. Can someone explain what's going on here?

AnyyVen (talk) 23:16, 24 May 2016 (UTC)

I was confused about this section too. ${\frac {\partial u}{\partial z}}$ would always be zero by definition, since u is a function of r only. 74.140.199.156 (talk) 18:17, 11 April 2021 (UTC)

Hi @AnyyVen, things were probably modified in the article since 2016, so maybe your questions were about things that already changed and I don't even see; I will answer as if the question is about the current version (with the extra 2 k^2 A term in the last equation — which I restored today).

A is the complex amplitude of the wave, and u is the (also complex) amplitude of the slowly-varying envelope of the wave, after the "fast" carrier wave is taken out of it.

Usually the paraxial Helmholtz equation is an equation for this envelope, and not for the full wave (to obtain the full wave from the solution for the envelope, one needs only to multiply by the carrier wave).

I don't have Saleh in front of me, but could it be that in his notation, A is actually the envelope and not the full wave? That would explain the discrepancy.

As I mentioned above, I reverted the equation to the form it had until May 2020 (with the 2 k^2 A term). Nevertheless, I'm not sure how useful the paraxial equation for A is. I just fixed what I understand to be an error in it, but maybe the article would be better off without having it at all.

Regarding the question asked by the anonymous user: both A and u are functions of the vector r, not the coordinate r — meaning that these quantities are functions of all three coordinates (x,y,z), so du/dz doesn't trivially vanish. E L Yekutiel (talk) 21:08, 3 January 2024 (UTC)

Hi again @AnyyVen (or anyone else reading this), I now have Saleh in front of me, and as I guessed above, it's a matter of notation.

Saleh calls the slowly-varying envelope A, while in the wikipedia article it is denoted as u (you can see the assumption about u a little further down, with the derivative comparison).

So the equation obtained in Saleh for A is valid here for u.

It's worth mentioning that the wikipedia article and Saleh use opposite phases (in Saleh it's exp(-jkz), in the wikipedia article it's exp(ikz)), but it's a matter of convension and makes no difference; the only thing that really matters is which function represents the full wave and which is the slowly varying envelope. E L Yekutiel (talk) 09:27, 7 January 2024 (UTC)

Helmholtz equation for guitar sound-holes

Sound Hole Diameter in inches = F^2 (f square) x W x L x D x 0.00000023895

where: W = box inside width L = box inside length D = box inside depth F = frequency of 2nd lowest string in Hz

Simple Explanation ?

http://www.phys.unsw.edu.au/~jw/Helmholtz.htm

http://physics.hallym.ac.kr/education/wave/music/textwaves.htm

http://members.aol.com/briarbass/resocalc.xls (soundbox volume calculator)

— Preceding unsigned comment added by 117.248.61.172 (talk) 17:42, 16 February 2017 (UTC)

Standing Wave

When k is real then the solution is always a standing wave trapped inside a finite domain, with infinite phase speed. Standing wave solutions, by def, do not change with time. Thus, the separation of vars is valid. For complex k, separation of vars is not a valid assumption since the wave energy is not trapped inside a domain, and ends up being a travelling wave. A good example is the Schrodinger eqn. When the particle is trapped in a potential well (E-V is negative) then k is real and the static solutions can be found. These are are quantized in energy/momentum/spin. When a particle escapes the well the solutions are no longer quantized.203.221.203.224 (talk) 23:24, 3 December 2017 (UTC)

Etymology

Why is the Helmholtz equation called the Helmholtz equation? Who is/was Helmholtz? Other people are mentioned in this article, but there is no mention of Helmholtz himself. —Kri (talk) 16:11, 14 December 2023 (UTC)

Hi @Kri, I added a sentence linking to him. E L Yekutiel (talk) 21:22, 3 January 2024 (UTC)