The near-resonance liquid sloshing under surge excitations: theory and experiment

Under surge excitations, the steady-state liquid sloshing in a rectangular container exhibits substantial differences on the sub- and super-sides of the secondary resonance, while the underlying physics remains unclear. To better understand these behaviours and quantify their influences on wave dynamics, we formulate a new closed-form analytical solution up to second-order accuracy based on the twin-wavemaker model by projecting onto wavenumber space. Different from the Stokes wave, this second-order solution exhibits a π / 2 phase difference compared with the first-order, due to the change in the parity of the wavemaker boundaries. This difference causes crest–trough superposition on the sub-side, but crest–crest superposition on the super-side, with the latter exhibiting a steeper surface slope and being more prone to breaking. The temporal evolution near the side wall transitions from a W-shaped to an M-shaped curve upon exceeding the resonance. We have demonstrated that the physical mechanism of this transition is attributed to an abrupt phase shift equal to π across the resonance and generated by the second-order free waves. Following experiments are conducted to support the findings, revealing the influence of the different behaviours on breaking patterns, and the dependence of the breaking point on the external excitation amplitude.