Theoretical analysis of frequency shift on resonant interaction induced by steady free-surface flow over rippled bottoms

We theoretically investigate the resonant frequency shift behaviors on triad resonance induced by steady free-surface flow over rippled bottoms. This triad resonance involves two free-surface propagating wave components and one bottom fixed topographic profile wave component. The perturbation analysis of the boundary value problems for free-surface steady uniform flow over a finite rippled patch is applied in this study up to the third order by substituting the perturbation expansion series of the frequency term. Then, we derive the high-order terms in the frequency expansion series quantitatively by considering the third-order nonlinear inhomogeneous forcing terms from both free-surface and bottom boundary conditions. For the theoretical frequency shift solution obtained in this study, we compare it with flume experiment results of one representative triad resonant phenomenon, which is induced by a specific triad resonant combination (k1−k2=kb) and also known as upstream-propagating waves. Within the generation parameters (Fr and kbh) domain of upstream-propagating waves, the theoretical solutions reveal that both the resonant frequency upshift and downshift exist, and the free-surface nonlinearity is the dominant factor of the frequency shift terms. This work will provide the theoretical references for the complex free-surface resonant interaction above a rippled patch in ambient flow.