Phase-resolved settling velocity change of inertial particles in surface gravity waves

We investigate how negatively buoyant inertial spheres settle in progressive surface gravity waves, focusing on the phase-dependent modulation of the particle–fluid relative motion. Starting from a Maxey–Riley framework with a quasi-steady nonlinear-drag closure, we formulate a phase-resolved equation of motion in terms of the slip velocity and obtain a perturbation solution in the limit of small wave amplitude. The leading state corresponds to the quiescent-water terminal slip, the first-order response yields a dominant harmonic modulation induced by the wave kinematics and second-order terms introduce higher harmonics arising from higher-order interactions. A key element is that, under nonlinear drag with finite particle Reynolds number, the particle relaxation time and hence the Stokes number depend on the instantaneous slip. This dependence is retained consistently in the perturbation expansion, allowing the vertical-slip phase at finite Stokes number to be predicted without the additional ordering adopted in previous work. We test predictions using wave flume experiments with simultaneous particle tracking velocimetry/particle image velocimetry, which provide slip estimates along particle trajectories and phase-binned ensemble statistics. The analysis first assesses whether particles attain a terminal settling regime within the observation window, then quantifies the phase-dependent oscillation of the horizontal and vertical slip. For cases that attain the terminal regime, the perturbation solution reproduces the amplitude and the phase of the dominant slip oscillation. However, cases that remain in a wave-acceleration-dominated relaxation regime show settling hindrance near the water surface; the wave-averaged vertical slip approaches the quiescent terminal velocity with depth, and no clear settling enhancement is observed.