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wave propagation in Hamilton Systems

Teng zhang's picture

I am a junior graduate student now, and very interesting in wave motion. My advisor Prof. Zhong wanxie and his PHD student qiang Gao have developed a precise numerical technique to solve the Rayleigh wave frequency equation, which can avoid the missing root. They did a systematic work involving surface wave propagation in a transversely isotropic stratified solid resting on an elastic semi-infinte space, wave propagation in the anisotropic layered media and the propagation of stationary and non-stationary random waves in a viscoelastic, transversely isotropic and stratified half space.

The essence of these work is the idea of introducing dual variables, then tranforming them into duality system. The surface wave frequency equation was solved via the precise integration method (PIM) and the extended Wittrick-Williams (W-W) algorithm. As for the third problem, they transformed the random wave problems into deterministic problems and transformed governing equations of viscoelastic materials into Hamilton equations in which the dual variables were specially chosen due to the viscoelastic. And pseudo-excitation method (PEM) was used for the random waves solution. 

 I am just following their work and maybe do some work based on that in the future. We have some puzzles now. Although we think our numerical technique has advantage, we find that it is a little difficult in extension--not many people used these methods. So I want to discuss the advantage and disadvantage of these symplectic methods and the traditional methods, and hope to hearing the discussion from others. 

 Advantage

1. precise and avoid the missing root for the surface wave frequency equation

2. the same simple formula and uniform steps for anisotropic layered media and the isotropic

3. high efficent for random wave problems owe to PEM

Disadvantage

1. strict restriction of the geometric shape--only for layered structure now

2. only for the linear elastic now, most time the traditional methods are efficient enough in these areas

3. people are not very used to these, so they maybe not want to change their familiar method to these new ones

I think the last two disadvantages may be the reason for the less use of these new methods. We are trying to find new areas to use these methods such as piezoelectric crystals, phononic/photonic crystals now, however, we have not found a clear idea by now. I want to know the main weakness of the traditional methods of the wave propagation, how can we improved our methods for more widely used and some potential use of these methods.   

That is just my own opinion. Thank you for your attention and I really look forward to listening to your opinions.

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