Coupled physical systems — two waveguides, a wing and its airflow, a plate with bending and shear modes — have dispersion relations that combine the individual dispersion relations of each subsystem. The question is how.
The paper on factorized dispersion relations (arXiv: 2603.23664) proves a general form: G₁ · G₂ = γ · G_c, where G₁ and G₂ are the uncoupled dispersion functions, G_c captures the coupling, and γ measures the hybridization.
The proof uses determinant expansion theorems for block matrices: the coupled system's dispersion relation (a determinant of a block matrix) factors into the product of the subsystem determinants times a coupling correction. The factorization is exact, not approximate.
Three examples illustrate: traveling wave tubes (beam-wave interaction), airplane wing vibrations (structure-aerodynamics coupling), and Mindlin-Reissner plate theory (bending-shear coupling). In the plate case, the factorized form gives a quantitative measure of mode hybridization: at any nonzero coupling, branches show characteristics of both subsystems, but at large frequencies and wavenumbers, they recover the pure uncoupled modes.
The through-claim: coupling between systems factorizes the dispersion relation into recognizable pieces. The coupled system's wave behavior is the product of the uncoupled behaviors, corrected by a coupling term. This factorization is algebraic (it follows from block-matrix structure) and universal (it applies to any two coupled linear wave systems). The hybridization is measurable as the departure from the factored form.
2603.23664. Mathematical physics / wave propagation / dispersion relations / coupled systems / mode hybridization.