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This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.

An asymptotic one-dimensional theory, with minimal essential parameters, is constructed to help elucidate (two-dimensional) low-frequency dynamic motion in a pre-stressed incompressible elastic plate. In contrast with the classical theory, the long-wave limit of the fundamental mode of antisymmetric motion is non-zero. The occurrence of an associated quasi-front therefore o¬ers considerable deviation from the classical case. Moreover, the presence of pre-stress makes the plate sti¬er and thus may preclude bending, in the classical sense. Discontinuities on the associated leading-order wavefronts are smoothed by deriving higher-order theories. Both quasi-fronts are shown to be either receding or advancing, but of di¬ering type. The problems of surface and edge loading are considered and in the latter case a speci c problem is formulated and solved to illustrate the theory. In the case of antisymmetric motion, and an appropriate form of pre-stress, it is shown that the leading-order governing equation for the mid-surface de®ection is essentially that of waves propagating along an in nite string, a higher-order equation for which is derived.

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The counterintuitive properties of photonic crystals, such as all-angle negative refraction (AANR) [J. Mod. Opt.34, 1589 (1987)] and high-directivity via ultrarefraction [Phys. Rev. Lett.89, 213902 (2002)], as well as localized defect modes, are known to be associated with anomalous dispersion near the edge of stop bands. We explore the implications of an asymptotic approach to uncover the underlying structure behind these phenomena. Conventional homogenization is widely assumed to be ineffective for modeling photonic crystals as it is limited to low frequencies when the wavelength is long relative to the microstructural length scales. Here a recently developed high-frequency homogenization (HFH) theory [Proc. R. Soc. Lond. A466, 2341 (2010)] is used to generate effective partial differential equations on a macroscale, which have the microscale embedded within them through averaged quantities, for checkerboard media. For physical applications, ultrarefraction is well described by an equivalent homogeneous medium with an effective refractive index given by the HFH procedure, the decay behavior of localized defect modes is characterized completely, and frequencies at which AANR occurs are all determined analytically. We illustrate our findings numerically with a finite-size checkerboard using finite elements, and we emphasize that conventional effective medium theory cannot handle such high frequencies. Finally, we look at light confinement effects in finite-size checkerboards behaving as open resonators when the condition for AANR is met [J. Phys. Condens. Matter 15, 6345 (2003)].

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