Field patterns without blow up

Abstract

Field patterns, first proposed by the authors in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819), are a new type of wave propagating along orderly patterns of characteristic lines which arise in specific space-time microstructures whose geometry in one spatial dimension plus time is somehow commensurate with the slope of the characteristic lines. In particular, in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819) the authors propose two examples of space-time geometries in which field patterns occur: they are two-phase microstructures in which rectangular space-time inclusions of one material are embedded in another material. After a sufficiently long interval of time, field patterns have local periodicity both in time and space.This allows one to focus only on solving the problem on the discrete network on which a field pattern lives and to define a suitable transfer matrix that, given the solution at a certain time, provides the solution after one time period. For the aforementioned microstructures, many of the eigenvalues of this  -symmetric transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. However, there are also modes that blow up exponentially with time coupled with modes that decrease exponentially with time. The question arises as to whether there are space-time microstructures such that the transfer matrix only has eigenvalues on the unit circle, so that there are no growing modes (modes that blow-up)? The answer is found here, where we see that certain space-time checkerboards have the property that all the modes are propagating modes, within a certain range of the material parameters. Interestingly, when there is no blow-up, the waves generated by an instantaneous disturbance at a point look like shocks with a wake of oscillatory waves, whose amplitude, very remarkably, does not tend to zero away from the wave front.

Photonics of Time-Varying Media

“…These objects arise from the consecutive spatial and temporal scattering in a material that features a checkerboard-like structure of its response parameters in space and time, and may offer new angles to study space-time metamaterials in the future. 162,187,188 Finally, beside photonic systems, non-electromagnetic dynamical metamaterials have already acquired significant momentum, with several theoretical proposals [189][190][191] and experimental realizations, including asymmetric charge diffusion, 192 nonreciprocity, 140,193,194 Floquet topological insulators 183 and topological pumping 195 amongst many others. In particular, digitally controlled acoustic meta-atoms have recently attracted significant interest to probe time-modulation physics in acoustics.…”

A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

“…Sometimes there are no bandgaps in k. An example occurs within the special class of "spacetime" periodic microstructures that generate "field patterns" where the characteristic lines form a pattern commensurate with the material, but not neccessarily with the same unit cell of periodicity [28]. For these structures the waves propagate with no exponential increase or decay in amplitude [23]: they are an example of "P T -symmetry" [3]. An interesting question is whether such propagating waves always occur when there are no bandgaps in any "timelike" direction in (k, ω) space.…”

Field patterns: A new type of wave with infinitely degenerate band structure

“…We denote the Green function by G l,l (m − m ) to indicate that it provides the current at point l of cell m, given the current at point l of cell m . Then, j(l, m, n) = l ,m T (l,m),(l ,m ) j(l , m , n − 1), Figure 1: a) represents a two-phase space-time checkerboard in which the two phases have the same wave speed, so that the slope of the characteristic lines is the same in the two phases (see [1]). The field pattern is shown (in red), and within each unit cell (in magenta) we calculate the current distribution only at four points, denoted by blue dots in the picture.…”

“…However we find that there are some spectacularly novel features associated with waves in certain one-dimensional linear media with moduli that vary both in space and in time. One feature, remarked on in [1], is the appearance of a new type of wave generated from an instantaneous source whose amplitude, unlike a conventional wake, does not go to zero away from the wave-front. Here we show that the associated band structure in the dispersion diagram is infinitely degenerate.…”