Waves in slowly varying band-gap media. (English) Zbl 1380.34090
This paper is concerned with waves in locally periodic media in the high-frequency limit where the wavelength is commensurate with the period. A key issue is that the Bloch-dispersion curves vary with the local microstructure, giving rise to hidden singularities associated with band-gap edges and branch crossings. An asymptotic approach is proposed to overcome this difficulty. This approach is demonstrated in the simplest case of time-harmonic waves in a one-dimensional locally periodic medium. The method entails matching adiabatically propagating Bloch waves, captured by a two-variable Wentzel-Kramers-Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localized about dispersion singularities. WKB analysis is characterized by the following aspects: the frequency domain is used, the complex-valued transport equation is decomposed into two simpler real equations, respectively, governing the amplitude and a slow phase. Additionally, by considering perturbations of the Bloch eigenvalue problem, exact identities that allow to simplify the asymptotic formulae are derived. Also a more general multiple-scale representation of the locally periodic medium is considered.
The latter solutions, obtained following the method of high-frequency homogenization (HFH), hold over dynamic length scales intermediate between the periodicity (wavelength) and the macro-scale. In particular, close to a spatial band-gap edge the solution is an Airy function modulated on the short scale by a standing-wave Bloch eigenfunction.
The key message of this paper is that multiple-scale WKB approximations and HFH are complementary and have overlapping domains of validity. The two methods can therefore be systematically combined, through the method of matched asymptotic expansions, to furnish a more complete scheme for high-frequency waves in periodic, or locally periodic, media.
The latter solutions, obtained following the method of high-frequency homogenization (HFH), hold over dynamic length scales intermediate between the periodicity (wavelength) and the macro-scale. In particular, close to a spatial band-gap edge the solution is an Airy function modulated on the short scale by a standing-wave Bloch eigenfunction.
The key message of this paper is that multiple-scale WKB approximations and HFH are complementary and have overlapping domains of validity. The two methods can therefore be systematically combined, through the method of matched asymptotic expansions, to furnish a more complete scheme for high-frequency waves in periodic, or locally periodic, media.
Reviewer: Vasile Dragan (Bucureşti)
MSC:
| 34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 78M35 | Asymptotic analysis in optics and electromagnetic theory |
Keywords:
Bloch waves; periodic media; singular perturbations; WKB approximation; high-frequency homogenization (HFH)References:
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