I am trying to find the dispersion relations or wave numbers as a function of frequency for a layered medium in one-dimension. I have found a very brief breakdown of the problem here: https://en.wikipedia.org/wiki/Transfer-matrix_method_(optics) -- My question is how to solve for the wave number using this transfer method through a N-layered material with each layer having different properties (density, thickness, wave speed, etc). I have found some publications on the subject but I need to factor in layer thicknesses in order to run a non-linear optimization of the model. Any help is greatly appreciated.
Miles,
I can only speak on the field that I have some knowledge in (guided ultrasonic wave propagation), so please take with a grain of salt. The Transfer Matrix Method works okay, but has issues with large frequency-thickness products.
Some folks use a Global Matrix Method... provided the problem is solved appropriately, addresses the large frequency-thickness instability issue. The other approach is a semi-analytical finite element method (SAFEM) approach, which solves the N-layered harmonic wave propagation problem in a straight-forward FE approach, with no instability issues. I have some literature and Matlab based code if interested, just let me know.
Best,
Brian
If the wave is light and the layers are parallel and infinite, the problem is equivalent to solving Schrodinger's equation if the Electric field is parallel to the interface. Basically you need to replace the Potential V in Schrodinger's equation with -1 x the square of the refractive index of the medium. Any book on Quantum Electronics (e.g. Yariv) will have the details.
Regards
Paul
For finding dispersion curves for modes in layered solid & fluid media, I especially like Kennett's book "Seismic Wave Propagation in Stratified Media". He uses a matrix method that is numerically stable for the large frequency-thickness condition noted by Brian above. The method works for an solid-fluid media, not just seismic waves.
I've used Kennett's matrix method plus a mode-finding algorithm based on singular-value decomposition (SVD) to find dispersion curves for every possible kind of mode in layered media: Rayleigh, Scholte, Love, duct, flexural floating plate modes, evanescent & propagating modes, proper & improper modes, lossy and lossless, homogeneous and anisotropic materials; as well as identifying and separating close-mode pairs (necessary for navigating dispersion curves through close "osculation" points evident in dispersion curves.
The method is exhaustive and analytically exact, but it is non-trivial to implement with full rigor. (Be sure to manage branch lines & Reimann sheets correctly in complex wave-number plane when the media includes an infinite half space above and/or below the layered media, so that you know exactly which Riemann sheet you are working on --- a failing in some layered mode attempts I have come across.)
Kennett's book may be out of print, but there appears be a pdf available online.
Ronald
My recomendation is Волны в слоистых средах - Скачать
knigi.tor2.net/index.php?id=3324064
Название: Волны в слоистых средах Автор: Бреховских Л.М. Издательство: Наука Год: 1973
Hi!
The easiest and most reliable method of calculation is based on the analogy of the layered medium and a two-wire line with variable characteristic impedance. (Feldstein, A. L., and L. R. Yavich, Synthesis of Microwave Two- and
Four-Port Networks, Moscow, Russia: Svyaz, 1971.) Schematically the procedure of calculation is represented as: [a] - [A] - [T] - [S], where [S] - the wave scattering matrix. Optimization of layered medium described in the article: В.Б. Гласко, А.Н. Тихонов, А.В. Тихонравов. О синтезе многослойных покрытий. Журнал вычислительной математики и математической физики, Т. 14, №1, с.135-144. I think both the book and article were translated into English.
With respect, Jacob Reyzenkind.
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