Talk:Lamb waves

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[edit] Giving the actual equations

I would like to see the actual equations for Lamb waves. I think it would be good to include them here. If they're really too complicated to include, then the article should tell the reader exactly where to look to find them -- preferably both a book and a website, since websites can disappear but books aren't as easy to find. It would also be useful, I think, to provide a link to where the reader can find computer code that calculates the equations, if possible.

Here's a link from a Google search. It has some equations -- are they the complete Lamb wave equations? --Coppertwig (talk) 16:08, 30 December 2007 (UTC)

Yes, these are the complete equations and they are written in a customary notation. It's a good article and should be linked and/or external-referenced. Also, the graphics (though not very beautiful) show just the things that should be added to the Wikipedia article in its next stage of development. I did not put such graphics into the article myself because of copyright / IP concerns.
Regarding computer code for calculating dispersion curves, it would certainly be good to provide a link. Code has been written in many universities and business, the only question is whether anyone has put such a program into the public domain. Adrian Pollock (talk) 19:03, 30 December 2007 (UTC)
I added the article to the External Links section. --Coppertwig (talk) 14:20, 1 January 2008 (UTC)
I don't see any equations in that externally-linked article that look too complicated to include here. But I'm not sure which equations you're identifying as "the" Lamb wave equations. There are "dispersion" equations and some other equations about group velociy. I still think it would be preferable to present the Lamb wave equations (solution to "the" wave equation given the boundary conditions) in this article. I can typeset the equations in LaTeX, I think, except that I don't know what the equations are. --Coppertwig (talk) 13:33, 15 January 2008 (UTC)
Equations 1(a) and 1(b) are the ones to include, along with the definitions of terms in the three sentences following. This particular notation for writing them may not be Lamb's original one (I'd have to look up his article to be sure of this), but it is a convenient one that has been quoted by many authors including our friend from China. Adrian Pollock (talk) 03:50, 16 January 2008 (UTC)
I don't think those equations are the solution to the wave equation. Those equations involve alpha, beta, d and k, where alpha is defined in terms of omega and cl and k, beta is defined in terms of omega and cl and k is defined in terms of omega and Cp. As I understand it, the various velocities C are constants (at least for the given material and Lamb wave mode). The solution to the wave equation should involve displacement as a function of position, or at least velocity or acceleration as a function of position. --Coppertwig (talk) 19:17, 19 January 2008 (UTC)
You are correct, those equations are not the solutions to the wave equation. What we have here is a boundary value problem. Those equations are the characteristic equations obtained by taking the wave equation (assuming sinusoidal solutions) and plugging into it the boundary condition of stress-free surfaces. Solution of these characteristic equations gives the relationship between omega and k that is necessary for the solution to exist (like, a determinant has to be zero for a set of simultaneous equations to be soluble). Having found (by numerical methods, there is no explicity solution) the relationship between omega and wavenumber k one has the phase velocity (equal to omega/k), and that is found to be NOT a constant, but a function of kd where d is the plate thickness. This is the phenomenon of velocity dispersion, wave velocity being a function not only of material but also of frequency; velocity dispersion is the general situation for acoustic waves in bounded structures. The characteristic equations are the heart of the matter, from which the wave velocities are determined; that's why they are the ones to go up first. We could also give the equations for particle displacement, they would add to the article; I would have to dig around a bit to find them.Adrian Pollock (talk) 06:11, 22 January 2008 (UTC)
Coppertwig, I have distributed the introductory material across the subsections per your suggestion. If you could put in the characteristics equations, it will help to pull it together. The equations can be a reference point for the text following, which is currently rather ragged after the restructuring. Thanks, Adrian Pollock (talk) 04:14, 26 January 2008 (UTC)
OK, I might do that within the next couple of days. --Coppertwig (talk) 16:38, 26 January 2008 (UTC)

[edit] Number of types of linear elastic waves in plates

It says "Lamb waves are one of several kinds of linear elastic wave that propagate in solid plates". This seems to me to imply that there are at least two other types of such waves that propagate in plates. I'm only aware at the moment of one other kind, i.e. SH or shear horizontal waves. I'd like to see either this wording changed to reflect only one other kind of wave, or else links to articles describing the various other kinds of waves (depending on what the actual correct number of types of such waves is), or at least mention of the names of the other types of waves. --Coppertwig (talk) 17:23, 30 December 2007 (UTC)

The SH modes are the only other straight-infinite-wavefront modes, but note also the cylindrical Lamb-like modes described further down in the article. It goes further. Lamb assumed isotropy; in anisotropic material, the thing is much more complicated. Also, there has been much study of wave propagation in multi-layered media and if you accept that the category "plates" includes laminates, there you go again. IMO these things would be best addressed in another article called "Plate waves" - plate waves to include "pure" Lamb waves as a subset. Adrian Pollock (talk) 18:59, 30 December 2007 (UTC)
I'm not sure that "plate waves" would be a topic with a sufficient amount of material, sufficiently connected into a single topic to justify an article. (Might or might not be.) I think a sentence or two in this (i.e. Lamb waves) article would suffice to classify the straight-infinite-wavefront isotropic wave types, and that articles on more general cases might perhaps be better off with names like "cylindrical wavefronts" or "anisotropic plate waves" or "waves in multi-layer plates" etc. These are just suggestions.
I thought of one more type of wave in a plate: a longitudinal wave travelling parallel to the plate boundaries. Perhaps that counts as a special case of Lamb waves? Which mode would it be?
Presumably Lamb also assumed homogeneous properties throughout the plate. I'm going to add that statement to the article -- please revert if I have it wrong. --Coppertwig (talk) 14:08, 1 January 2008 (UTC)
I inserted "The plate is assumed to be a homogenous, isotropic solid bounded by two parallel planes beyond which no wave energy can propagate." This brings in one more assumption: that energy doesn't propagate through the plate boundaries. Please revert if I have that wrong.
Are Lamb waves normally assumed to have a straight, infinite wavefront? (or to be present at the same amplitude everywhere in the plate?) --Coppertwig (talk) 14:16, 1 January 2008 (UTC)
Your "longitudinal wave travelling parallel to the plate boundaries" could be launched, but would be unstable and would quickly disperse. Look at it this way: at the plate boundaries, the physicals constraints would not be satisfied by the form of the longitudinal wave, with the outcome that energy would be progressively tapped off into shear wave energy travelling diagonally across the plate. That would mode-convert again when it reached the other surface, and so on. Mathematically this is not a very trackable/tractable way of looking at it, though it may be intuitively helpful. Formally, it's better to consider that initial "longitudinal-wave" as a starting condition which can expressed as a linear superposition of (symmetrical) Lamb wave modes. Following the propagation of these modes is the tractable way of determining the motion at substantial distances from your source. If you kept pushing and pulling your end of the plate in the x-direction at a given frequency indefinitely (while constraining it in thte z-direction - that's what you imply), you will end up with a distance-dependent interference pattern of however many symmetrical modes existed at that frequency. I expect there would also be evanescent waves in the near field - now that's a whole other subject !
I like your statement about the plate being homogeneous etc. Very nice polish !
"Are Lamb waves normally assumed to have a straight, infinite wavefront?..." - That depends on whether the speaker is a purist or a pragmatist. Pragmatists aren't too fussy and it's not a problem, everyone knows what they mean. Being a bit of a purist myself, I have tried to cover this issue in the section on "Guided Lamb Waves", and to offer a purist explanation of that phrase, which is increasingly used nowadays. People mostly toss in the word "guided" in a copycat manner without knowing precisely what they mean by it, so I am sticking my neck out to offer some guidance here. I don't know who actually originated the phrase. If someone feels strongly about this small detail they will no doubt edit or talk about it. Adrian Pollock (talk) 20:19, 1 January 2008 (UTC)


[edit] What does "out-of-plane" mean?

It says "in-plane and out-of-plane components". I don't understand this. What plane is meant? Since the motion is all in the x and z directions, never in the y direction, then the motion of each particle must be within a plane. Possibly in-plane means in the x direction only (not z), because it's within a plane parallel to the sides of the plate. If so, this needs to be explained to the reader. --Coppertwig (talk) 17:30, 30 December 2007 (UTC)

It means the plane of the plate. I'll clean up the text to clarify this. Adrian Pollock (talk) 19:03, 30 December 2007 (UTC)

[edit] Zero-order modes extend to all frequencies?

"At low frequencies, only the lowest or zero-order members of these families exist; these members extend to arbitrarily low frequencies." How about "...these members extend to all frequencies"? or maybe "...these members extend to all frequencies, including arbitrarily low frequencies"? (Unless there is some upper limit.) --Coppertwig (talk) 01:00, 12 January 2008 (UTC)

Yes, this discussion can probably be improved though I can't put my finger on exactly what is niggling you. It's true the "At low frequencies..." clause is not very good. Possibly let the mind start at a high frequency and come to a low. Like: as the frequency is lowered, fewer and fewer modes will be found;only the two zero order modes extend down to arbitrarily low frequencies. It's true that there is no upper frequency limit, not in the theory anyway. Let's keep chewing on it.
I think I see the problem now. The discussion of higher-order modes goes like: they exist only ABOVE their nascent frequencies. But the discussion of the zero-order modes goes like: they extend DOWN to (...). This is an incongruity and it makes for unsatisfying reading. The solution is to frame the discussion thus: all modes exist from their nascent frequency upwards, and the zero order modes are special in being the only ones that have a nascent frequency of zero. Please join me in an editing process along these lines and we'll surely be able to achieve a more pleasing read. Thanks for being persistent on this point. Adrian Pollock (talk) 01:44, 17 January 2008 (UTC)
Meanwhile I am starting to look for a formula for the ratio of z-displacements to x-displacements for the flexural mode in the low frequency limit. I did find a formula for its velocity just this evening. I'd like to show the velocity formula and the z/x ratio formula for the low-frequency limit of both zero order modes. It will add to the value of the article for people to be able to find all these four items conveniently in one place.Adrian Pollock (talk) 05:35, 12 January 2008 (UTC)

[edit] Lead section

I hope you don't mind. I deleted the Overview section heading, and I'm thinking of editing to somehow end up with a lead section of 2 to 4 paragraphs. It's my understanding of the Wikipedia Manual of Style that there isn't supposed to be a section with a title like "overview"; the introduction or lead is supposed to have no heading, and that the other sections are supposed to have more specific headings. --Coppertwig (talk) 17:15, 12 January 2008 (UTC)

Go for it ! I wasn't aware of these points of Wikipedia style when I started this article. The intent of the other named sections, once I had started working on them, was to expand on the content of the overview while maintaining a decently balanced coverage overall. You may be able to get the article into the desired style by making a short introduction and then checking the left-over material from the previous overview, to see whether any of it would be useful for introducing/starting the named sections. As first written, the named sections were intended to be based on information conveyed in the overview, so after deletion of the original overview they will need to be reviewed / edited for approachability. Thanks, Adrian Pollock (talk) 01:09, 15 January 2008 (UTC)
Right. Thanks. I might do this though not right now. --Coppertwig (talk) 03:24, 15 January 2008 (UTC)
Ah, I was misreading you. You only deleted the heading, in my paranoia I thought you had deleted the whole section. I only just now went to actually look. OK, what we are left with is simply an introductory section that is too long. So in the first approximation, what we need is simply a cut-paste-and-polish kind of operation one subtopic at a time - shouldn't be hard. Adrian Pollock (talk) 05:52, 15 January 2008 (UTC)

[edit] Equations

\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {4 \alpha \beta k^2} {(k^2 - \beta^2)^2}

OK, there's equation 1a from the page by Zhenqing. Are you sure these equations make sense?? The left-hand-sides of the two equations are identical to each other. If it makes sense to you, you can put this into the article and also I'm sure you can figure out how to express the other equation in LaTeX too. --Coppertwig (talk) 21:03, 26 January 2008 (UTC)

Yes, these equations do make sense and I will look forward to handling the second one. The reason there are two of them is that the first describes the symmetrical modes and the second describes the antisymmetric modes. The definitions of alpha and beta (etc.?) in Zhenqing's article have to be written into our text also; these bring in the angular frequency omega, which is how the characteristics equations permit computation of frequency as a function of wave number, hence of the phase and group velocities. Question: as a matter of Wikipedia style as well as feasibility, do you think the expressions for alpha, beta (?etc.) should be in full LaTeX with its big script, or is it acceptable to write them in some smaller format - after all, they are only definitions supporting the main equations ? Thanks, Adrian Pollock (talk) 22:23, 26 January 2008 (UTC)

I note that in the original paper, the "where's" don't typeset very well -- they're way above the text they're supposed to fit into. I expect something similar would happen here. So, I suggest:


where

\alpha^2 = \frac{\omega^2}{c_l^2} - k^2, \quad \quad \beta^2 = \frac{\omega^2}{c_t^2} - k^2,

and k = ω/Cp, where k is the ...

I inserted an extra word "where" before the last k because otherwise the k looks like part of the previous equation. (alternatively, delete "and" and insert another "and" where I put "where" :-) I found out how to spread equations out on a line. "\quad" adds whitespace within math mode -- just like in TeX. Reading the instruction pages can do wonders. I also just added a backslash before "tan" in the above equations -- I suppose it looks a little better or is more correct that way.

Actually, if you want opinions about what is the correct way to typeset math, Michael Hardy may be a good person to ask. (But read Help:math and Wikipedia:Manual of Style (mathematics) before asking him -- you might find answers there.)

Please check over the equations and make sure they're the same as on the original web page. I found some of the characters hard to see. And OK, if it's for two different modes, I guess the equations can be different -- but shouldn't some of the variables be marked with different subscripts or something then? Oh, well, maybe that would complicate things too much. I guess it's OK. --Coppertwig (talk) 00:03, 27 January 2008 (UTC)

Re the equations for ξ and ζ: If my guess is correct as to the meanings of these variables, I suggest inserting "(ξ and ζ respectively)" after "having x- and z-displacements", and then after the equations, inserting "where Ax and Az are constants, and fx(z) and fz(z) are functions for which Lamb found formulas (not given here)." --Coppertwig (talk) 13:08, 27 January 2008 (UTC)

I'm hoping to find the proper equations in a few days to put up instead of these, so I'm not putting more time into these at this point. Adrian Pollock (talk) 19:39, 27 January 2008 (UTC)