Order of Glancing Undamped Points in Damped Waves

• The Order of a Glancing Undamped Point is a geometric microlocal parameter that quantifies the tangential interaction between undamped trajectories and damping boundaries.
• It encodes the rate at which the damping region recedes from a trajectory, refining decay rate predictions in various settings such as polygonal and smooth curves.
• It also appears in spectral theory and networked systems, influencing the persistence of oscillatory modes and sensitivity of non-Hermitian resonant systems.

The order of a glancing undamped point precisely quantifies, in microlocal and geometric terms, the tangential interaction between undamped trajectories (e.g., geodesics) and the boundary of a damping region in the context of energy decay for damped wave equations. It encodes the rate at which the damping set recedes from such a trajectory in local affine coordinates and provides a critical refinement in the prediction of decay exponents for energy as a function of time. This order also arises naturally in related contexts involving undamped or marginally stable oscillatory modes at the boundary of stability, as in networked dynamical systems and nanophotonic chains, as well as at points of non-trivial algebraic or geometric multiplicity in spectral theory.

1. Geometric Definition of the Order

Let $\theta$ be a point where an undamped geodesic (or more generally, a Hamiltonian flow line) is tangent to the boundary $\partial\omega$ of a damping set $\omega$ on a compact manifold (such as a torus). The order $\eta$ of the glancing undamped point at $z\in\partial\omega$ is defined as follows (Datchev et al., 5 Sep 2025):

There exists an affine chart $(\psi,U)$ around $z$ such that the image of the glancing line $L$ is $\{x=0\}$ in coordinates $(x,y)$, and two model sets:

• $$\Omega_{(\eta)} = \{(x,y): |y|^\eta \leq C_\text{out} |x|\}$$
• $$\mathcal{F}_{(\eta)} = \{(x,y): C_\text{in}^{-1} |y|^\eta \leq |x| \leq C_\text{in} |y|^\eta\}$$

for suitable constants $1<C_\text{in}<C_\text{out}$, such that:

• in the one-sided case: $$\mathcal{F}_{(\eta)} \cap \{x\geq 0, y\geq 0\} \subseteq \psi(\omega\cap U) \subseteq \Omega_{(\eta)}\cap\{x\geq 0\}$$,
• in the two-sided case: $$\mathcal{F}_{(\eta)} \cap \{y\geq 0\} \subseteq \psi(\omega\cap U)\subseteq \Omega_{(\eta)}$$.

The exponent $\eta$ is the order of the glancing point. For polygonal boundaries, $\eta=1$ at a vertex; for smooth curves with nonvanishing curvature, $\eta=2$.

2. Role in Wave Damping and Energy Decay Rates

The order $\eta$ directly determines the energy decay rate for solutions to the damped wave equation where the damping coefficient $W$ behaves like a power $\beta$ near the glancing point ($W(z) \sim d(z)^\beta$, $d(z)$ distance to the undamped region).

An averaging argument yields an improved decay exponent $\alpha$, with

$$E(u, t) \leq C t^{-2\alpha}, \quad \text{where } \alpha = 1 - \frac{1}{\beta'+3}$$

and

$\beta' = \frac{\beta}{\min\{\eta,1\} + 1/\eta}$

For example, if $\eta = 2$ (smooth, curved boundary), $\beta' = 2\beta/3$; for $\eta=1$ (vertex or corner), $\beta' = \beta/(1+1) = \beta/2$. This mechanism demonstrates that higher order (more “rounded”) glancing points yield improved decay rates.

Averaging along periodic directions, via

$$A_v(W)(s) = \frac{1}{T_v} \int_0^{T_v} W(s v^\perp + t v) dt$$

translates local polynomial vanishing into global decay bounds, with the improvement in exponent determined by $\eta$ (Datchev et al., 5 Sep 2025).

3. Genericity and Typical Examples

The analysis of the order of glancing undamped points reveals that for generic damping sets among polygons and smooth curves, energetically favorable values of $\eta$ predominate:

• For non-self-intersecting polygons on the 2-torus, generic position ensures all glancing points are at vertices, so $\eta=1$ for almost all cases.
• For a generic smooth curve, one can perturb so that all glancing points have strictly positive curvature, leading to $\eta = 2$ at all such points.

Thus, the improved energy decay bounds derived via the order $\eta$ are generically attained by both polygonal and smooth damping sets (Datchev et al., 5 Sep 2025).

4. Relation to Microlocal Analysis and Glancing Phenomena

The geometric concept of order parallels similar ideas in microlocal spectral analysis. For eigenfunctions of the Laplacian restricted to hypersurfaces, the degeneracy at glancing points (where the conormal derivative of the symbol vanishes) is associated with a sharp loss of regularity and an optimal one-fourth ($s_0 = 1/4$) smoothing exponent in $L^2$ bounds for the restriction operator—reflecting the underlying order of tangency (Galkowski, 2016).

Thus, the notion of order encodes the minimal smoothing or decay one must impose microlocally to compensate for concentration phenomena at glancing (tangency) points.

5. Analogous Notions in Dynamical and Networked Systems

In diverse dynamical settings, the concept of a glancing undamped point generalizes to situations where system trajectories, eigenmodes, or spectral points reside "on the rim of stability." In network theory, marginally stable oscillatory modes associated with undamped subspaces correspond to glancing points in the spectral domain. The geometric or algebraic multiplicity ("order") of such undamped modes dictates the persistence of oscillatory behavior and, in network consensus, the failure or success of output convergence (Koerts et al., 2016).

Similarly, in plasmonic nano-chains, undamped (glancing) propagation modes occur at the boundary between stability and instability. Their fixed amplitude is determined only after introducing nonlinear stabilization, and their existence or multiplicity reflects the underlying system parameters—a functional analog of the glancing order concept (Jacak, 2012).

6. Implications for Non-Hermitian and Resonant Systems

In spectral problems with non-Hermitian degeneracies (exceptional points), the order of the exceptional point quantifies how many eigenvalues and eigenvectors coalesce, dictating eigenvalue splitting and sensitivity to perturbations. As the order increases, the corresponding "spectral response strength" diverges—a feature intimately connected with the enhanced response of the system at glancing undamped points (Wiersig, 2023). This reflects a generalized order parameter that controls both the qualitative and quantitative behavior of the system near criticality.

7. Summary Table: Order and Its Consequences

Geometric Setting	Typical Order ($\eta$)	Consequence for Decay/Sensitivity
Smooth curved boundary	2	Improved decay rate ($\beta' = 2\beta/3$)
Polygonal vertex	1	Standard decay rate ($\beta' = \beta/2$)
Higher-order tangency	$\eta>2$	Further improved (fractional) decay rates, possible in degenerate cases
Spectral/Networked system	Algebraic multiplicity of undamped mode	Number of persistent oscillatory directions

The order of a glancing undamped point provides a unifying geometric and analytic parameter governing the interaction between localized undamped trajectories and dissipative regions, refining both energy decay predictions and the qualitative stability of solutions in a broad variety of wave, network, and open quantum systems.