Mechanism for slow waves near cutoff frequencies in periodic waveguides

Slow waves are of considerable topical interest; we show that they can be created within a simple waveguiding structure, a planar waveguide with periodic corrugations, and we describe the physical mechanism responsible. We show that this is a general feature of waveguides in both scalar systems, we use the Helmholtz equation within a guide with either Dirichlet or Neumann wall conditions as our main pedagogic example, and in fully coupled systems such as in-plane elasticity. The presence of slow modes within elastic waveguides has remained unexplored and so potential applications have not been exploited. We demonstrate that the same physics is responsible for the elastic slow modes with a subtle nuance connected to the presence of negative group-velocity modes within the elastic system. In addition to describing the mechanism we derive the frequencies at which they arise using a high-frequency asymptotic algorithm. These asymptotics predict the existence of slow modes associated with a countable set of discrete eigenfrequencies around the cutoff frequencies of an unperturbed waveguide. The asymptotic scheme leads to a natural physical explanation for the presence of such slow modes. Finite element computations for both dispersion curves and associated eigenmodes provide further evidence of the existence of slow modes and allow for comparison with the asymptotics. Many new applications are possible, particularly in elasticity, to generate analogies of optical delay lines. Furthermore the guiding structure we describe is simple to construct.

Figure 1

The periodic waveguide and geometric parameters. In the scalar system we have either Neumann, $\partial \varphi /\partial n=0$, or Dirichlet, $\varphi =0$, conditions on the guide walls.

Figure 2

Dispersion curves for a homogeneous topographic waveguide with height variation given by Eq. (2) with $-1<\epsilon x<1$, $\epsilon =0.1$, and $\alpha =10$. Panel (a) shows typical dispersion curves for a range of frequencies. Dashed lines correspond to finite element computations, and solid lines correspond to predictions from asymptotic formula (7) for panels (b) $n=1$, (c) $n=2$, and (d) $n=3$.

Figure 3

(a) Dispersion curves for the same values as Fig. 2 but for a thinned guide, with $\alpha =-10$. Dashed lines are from finite element computations and solid lines are from Eq. (7) for (b) $n=1$, (c) $n=2$, and (d) $n=3$. In (d) these are completely indistinguishable from the lowest two flat curves predicted by finite elements.

Figure 4

Mode shapes along the centerline from the asymptotic method (solid lines) and finite elements (dots). These are for the lowest two flat modes, for $n=2$ and for a thinned guide from Fig. 3c.

Figure 5

Dispersion curves near the two lowest transverse resonance frequencies of an elastic waveguide (left: traction-free walls; right: clamped walls). The asymptotics for the slow modes are shown as crosses as, for $n=1$, otherwise visually indistinguishable from the finite element dispersion curves (shown as solid lines). The geometrical parameters are the same as in Fig. 2.

Figure 6

(Color online) Distribution of energy of slow elastodynamic modes corresponding to panels (c) and (d) of Fig. 5. There is notable similarity between (a) and (c), up to a shift of half a cell.