Influence of ocean currents and surface tension on class II Bragg resonance using multi-scale analysis

This paper addresses how the ocean currents can influence the class II Bragg resonance of surface waves with non-zero surface tension interacting with bottom topography comprising of two different wavenumbers under the assumption of a small-amplitude wave theory. A multi-scale analysis technique is applied up to the third-order to obtain the analytical expressions for reflection and transmission coefficients by solving the coupled evolution equations involving the amplitudes of both reflected and transmitted waves. In the absence of current and surface tension, the findings are validated with the existing literature results. The Bragg peak decreases with increasing U and reaches closely to zero for U lying between 2.75 and 2.81 cm/s and further peak started to increase with increasing U. Phase shifting in the Bragg peak is observed in a non-monotonic manner with an increase in U as there is upshift when U changes from 0 to 2.75 cm/s, with sudden downshift at U=2.77 cm/s and followed again by upshift for U=2.81 cm/s. There is asymmetrical behavior in the amplitude of left and right subharmonic peaks that are observed for U lying between 2.75 and 2.81 cm/s and also, in the same range of U, the resonance bandwidth B is too short. These different qualitative behavior in the Bragg resonance is contingent upon the role played by the both positive and negative group velocities. The role of surface tension parameter T is also not trivial as the Bragg peak increases when T increases up to U=2.81 cm/s but beyond this range, the effect of T is opposite which means the Bragg peak decreases when T increases. In addition to this, this study predicted that the choice of number of ripples M to achieve full reflection is dependent upon the value of U, and it is possible to accurately capture the effect of reflected energy for large values of M. Moreover, the subcritical detuning is observed in proximity to the Bragg resonance. This study has potential to build an understanding about the role of currents on higher-order Bragg resonance on the coastal bathymetries and extremely important for coastal rehabilitation.