In the late 1500s, Vincenzo Galilei, father of Galileo, first noticed the relation between the frequency of a vibrating string, its density, and tension on the string (made precise by Marin Mersenne in 1637). In 1747, Jean le Rond d’Alembert found the general solution with which we are all familiar today. The one-dimensional wave equation with spatially periodically-varying moduli—studied by Gaston Floquet in 1883—provides the canonical example of a band-gap, a range of frequencies where no waves propagate. In 1958, Philip Anderson found that localization would occur if the moduli were disordered. Thus, one would think that no more big surprises remain in the study of the linear, one-dimensional wave equation with varying moduli. However, this is not the case if the moduli vary in both space and time, a problem studied by electrical engineers in the 1950s and extensively explored by Konstantin Lurie and his collaborators . Examples of materials with spatially and temporally-varying moduli abound: think of a liquid crystal display, or a small amplitude wave traveling on top of a large amplitude (pump) wave in a nonlinear medium. More recently, time crystals have generated some excitement.
To understand the complexity involved, one need only look at a space-time checkerboard, i.e., the wave equation in a one-dimensional, two-phase medium. Here, the moduli alternate both in space and time, having not only the familiar spatial interfaces, but also temporal interfaces, across which the material properties switch from one phase to the other. At each temporal interface, a wave splits into an outgoing wave and an incoming wave;  offers a brilliant experimental demonstration in water waves where one jolts the water tray upwards at a specific time to create a temporal interface. After the jolt, the waves generated by a model of the Eiffel Tower reconverge to the original shape. The Green’s function generated by an instantaneous disturbance in the space-time checkerboard is thus concentrated on a complicated cascade of characteristic lines, as shown in Figure 1a. Note that the horizontal axis denotes the spatial variable and the vertical axis denotes the time variable.
Figure 1. The characteristic lines of Green’s functions in space-time checkerboards. 1a. An example of a generic space-time checkerboard. The initial disturbance propagates giving rise to a cascade of disturbances. 1b. A two-phase space-time checkerboard in which the diagonals of the rectangles have the same slope as the characteristic lines in both phases (the two phases have the same wave speed c1=c2=c). 1c. A three-phase space-time checkerboard (phase 1 is colored in white, phase 2 in light gray, and phase 3 in gray) in which the wave speeds of the phases satisfy the following relation:c2/c1=c1/c3=3. Image credit: Ornella Mattei.
One way to avoid this complexity is to assume that there is no impedance mismatch, and therefore no reflected wave occurring at spatial boundaries and no incoming wave created at temporal interfaces. Even though the characteristics no longer split, a surprise still exists: neighbouring characteristics can converge, creating the analog of a shock in a linear medium . Our approach to avoiding the complexity is different; if the material properties are suitably chosen—as in Figure 1b and 1c—the disturbance propagates along an orderly pattern of characteristic lines that we call a field pattern. In particular, Figure 1b represents a two-phase space-time checkerboard in which the dimensions of the rectangles are related to the wave speed of the two phases, \(c_1 = c_2 = c\). In contrast, Figure 1c shows a three-phase space-time checkerboard in which the wave speeds of the three phases are such that \(c_2\!/\!c_1 = c_1\!/\!c_3 = 3\).
We found other examples—called field pattern materials—of space-time microstructures giving rise to field patterns [6, 7]. To discover field pattern materials, one need only draw the appropriate lines on a candidate space-time geometry. All of the microstructures in [6, 7] have the common property of P- and T-symmetry (see Figure 1b and 1c), where P and T stand for parity and time respectively. Field patterns are PT-symmetric but not separately P- and T-symmetric. Systems with PT-symmetry have been studied extensively—particularly in the context of quantum mechanics—, and our field pattern materials share some features with them. In particular, if the PT-symmetry is “broken,” then field patterns support both propagating waves and those that can blow up or decay exponentially with time. But if the symmetry is “unbroken,” all modes are propagating modes with no blow-up [6, 7]. Whether the condition of PT-symmetry is broken or unbroken depends on the type of space-time microstructure and the material parameters. For instance, the condition of PT-symmetry is always unbroken for any range of properties of the two phases for the two-phase checkerboard (see Figure 1b), whereas both propagating modes and growing modes (coupled with decaying modes) could exist for the three-phase checkerboard (see Figure 1c), depending on the choice 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 remarkably does not zero away from the wave front . Figure 2 depicts this new type of linear wave, representing the amplitude of a wave propagating through the space-time checkerboards of Figure 1a and 1b; Figure 2a corresponds to 1b, and 2b corresponds to 1c (the latter clearly corresponds to a choice of material parameters such that the condition of PT-symmetry is unbroken, thus avoiding a blow-up). As each field pattern lives on its own discrete network of characteristic lines, we determine the solution only on such a network—at specific points—denoted by the parameter \(p\) at specific times \(n\), both chosen in accordance to the field pattern’s periodicity. In Figure 2a, the wave propagates with the same intensity at the edge of a triangular shape with its vertex at the point of injection of the disturbance, and the solution is periodic inside the triangular shape. In Figure 2b, on the other hand, the solution inside the triangular shape is no longer periodic.
Figure 2. Novel types of linear waves. Left. The propagation of a disturbance in the two-phase space-time checkerboard of Figure 1b. Right. The propagation of a disturbance in the three-phase space-time checkerboard of Figure 1c. Here n represents time in discrete snapshots and p represents space, indexing the characteristic lines at those times. Image credit: Ornella Mattei.
Another very interesting feature of field pattern materials is the infinite degeneracy of their dispersion diagrams, determined by applying Bloch-Floquet theory . To understand this concept, it is important to realize the existence of a family of field patterns rather than a single field pattern. A single field pattern can be excited in many different ways; choosing a different point for the initial disturbance generally (but not always) yields a field pattern with a different geometry. The families of field patterns (notice that the number of families is infinite) all have the same dispersion law, relating the Bloch wavenumber \(k\) to the crystal frequency \(\omega\). An infinite space of Bloch functions—a basis for which are generalized functions, each concentrated on a field pattern—is therefore associated with every point on the dispersion diagram. The dynamics separate into independent dynamics on the different field patterns, each with the same dispersion relation.
For the two-phase checkerboard geometry of Figure 1b, the dispersion relation is simply \(\omega = \pm k\), independent of the choice of material parameters. In particular, the dispersion relation is exactly the same as the trivial case of a homogeneous material. However, the wave propagation in the homogeneous case is significantly different: an instantaneous disturbance would have fronts that travel at the group velocity \(\partial \omega\!/\!\partial k = \pm1\), in contrast to the inhomogeneous case where the fronts are trailed by an oscillating wave with an amplitude that depends on the material parameters (see Figure 2a). Besides proving that field patterns are a new type of wave, this example also shows that the band structure alone is insufficient in determining the longtime response to a localized disturbance.
Figure 3 depicts the dispersion diagrams of Figure 1c’s three-phase checkerboard for different choices of the material parameters. For some combinations of parameters, the condition of PT-symmetry is unbroken and \(\omega\) is real. Yet for other combinations, the PT-symmetry is broken so that growing modes (coupled with decaying modes) exist in addition to propagating modes, and \(\omega\) is complex.
Figure 3. Dispersion diagrams for the three-phase space-time checkerboard shown in Figure 1c for different values of the material parameters. 3a corresponds to a condition of unbroken PT-symmetry, whereas 3b corresponds to a condition of broken PT-symmetry, in which takes nonzero values indicative of exponentially growing and decaying modes. Image credit: Ornella Mattei.
There is a wealth of problems still to be studied. What happens when one perturbs the field pattern material geometry? Is a chaotic description using the concepts of statistical physics then appropriate? How does this generalize to higher dimensions and other wave equations? What happens when there is a small amount of dispersion, loss, or nonlinearity? Like the discoveries of anomalous resonance, cloaking, nonreciprocity, and topological insulators, this research shows that the study of linear systems is far from a dead subject. On the contrary, many exciting discoveries lie ahead.
Acknowledgments: Our work was supported by the National Science Foundation through grant DMS-1211359, and largely completed at the Institute for Mathematics and its Applications at the University of Minnesota during the special program on mathematics and optics.
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