On acoustic–gravity wave triad resonance over an elastic half-space

Nonlinear interactions between free-surface gravity waves and acoustic–gravity motions provide a mechanism for energy exchange in weakly compressible fluids and have long been discussed in connection with microseisms and low-frequency underwater sound. While elastic solid-Earth response has been incorporated in microseism modelling, existing formulations of acoustic–gravity wave resonant triads are almost exclusively developed for a rigid seabed, which introduces a shallow-water cutoff where the acoustic–gravity member becomes evanescent and resonance cannot occur. Here, we extend the resonant-triad framework to a compressible fluid over an elastic half-space. Seabed elasticity modifies the acoustic–gravity eigenstructure and dispersion relation, admitting propagating or interface-guided acoustic–gravity modes in regimes inaccessible under rigid-bottom assumptions and thereby removing the rigid cutoff in frequency (or depth). Using a multiple-scale expansion in the small compressibility parameter, we derive modulation equations for two counter-propagating gravity waves coupled to a single elastic acoustic–gravity mode. A key feature is an interfacial term in the solvability condition arising from the parameter dependence of the elastic boundary operator, which alters the modal normalisation and the resulting coupling and detuning coefficients. The enlarged admissible parameter space also makes laboratory-scale realisations of acoustic–gravity triad resonance substantially more feasible, enabling controlled investigation of the triad mechanism.