Signatures of space-time resonances in the spatiotemporal spectrum of water waves

In weakly nonlinear dispersive wave systems, long-time dynamics are typically governed by time resonances, where wave phases evolve coherently due to exact frequency matching. In this work, we identify and characterize an alternative mechanism - space resonances - which arise when wave packets share the same group velocity and remain co-located in space over time. We show that in systems where leading-order time resonances are absent, space resonances dominate the long-time behavior. Furthermore, their signature persists in the spatio-temporal spectrum even when subleading time resonances contribute. In the context of surface gravity waves, where triadic interactions occur without exact three-wave time resonances, the dominant interactions correspond primarily to self-interactions. These generate the nonlinear corrections responsible for the classical Stokes wave expansion. We also demonstrate that gauge-breaking terms in the Hamiltonian introduce space resonances supported on negative frequencies. Our analysis combines sea surface elevation data, numerical simulations, and theoretical developments, resulting in an analytical derivation of the leading-order spatio-temporal spectrum for weakly interacting water waves.