Existence of interior trapped-mode resonances

Determine whether the compactly supported interior hydroelastic operator associated with the submerged-plate problem possesses interior resonance frequencies, which are the only frequencies at which loss of uniqueness can occur under the stated radiation and unique-continuation arguments.

Background

The paper reduces possible nonuniqueness of the continuous hydroelastic problem to the existence of compactly supported interior eigenfunctions. Any nontrivial homogeneous solution must have zero reflection and transmission coefficients for all propagating modes and therefore carry no radiated energy.

Unique continuation then implies that the fluid potential vanishes outside the plate region, leaving an interior hydroelastic eigenvalue problem. The authors note that related geometries exhibit trapped modes at isolated frequencies, but do not establish whether such resonances occur for the present submerged-plate configurations. The well-posedness theorem consequently assumes a continuous non-resonance condition.

References

This does not rule out the absence of such resonances. Trapped modes are known to occur for related floating-elastic-plate geometries at isolated frequencies. Nonetheless, Proposition~\ref{prop:reduction} narrows \autoref{ass:continuous_nonresonance} from an open-ended statement about the full exterior problem to a finite-dimensional interior eigenvalue condition, which can be verified numerically.

— A discontinuous finite element method for the hydroelastic analysis of submerged structures  (2609.08313 - Wegert et al., 8 Sep 2026) in Remark following Proposition [Reduction of injectivity], Section 4.5

Near such a common zero, the function $E$ is still holomorphic for $Im c\neq0$, but we do not study its behavior at real phase speeds and leave the resonant case open.

— A Compressible Miles Instability for Wind-Wave Generation in a Neutral Atmosphere  (2610.02628 - Li, 2 Oct 2026) in Remark \ref{rem:determinant}, Section 3.4 (subsection on the holomorphic form of the dispersion relation)