*2.1. Generalized Bragg Conditions*

The mechanism for Bragg resonances is analogous to that for nonlinear (surface) wave-wave resonant interactions in the absence of bottom undulations. Thus, general Bragg conditions can be deduced from the well-known resonance condition for nonlinear wave-wave interactions [45]. For a wave field over uniform depth *h*, interactions among different wave components become resonant at order *m* (in wave steepness) if the wave numbers *kj* and the corresponding frequencies *<sup>ω</sup>j* satisfy:

$$\begin{aligned} k\_1 \pm k\_2 \pm \cdots \pm k\_{m+1} &= 0\\ \omega\_1 \pm \omega\_2 \pm \cdots \pm \omega\_{m+1} &= 0 \end{aligned} \quad \begin{cases} (m \ge 2) \\ \end{cases} \tag{1}$$

where the same combination of signs is to be taken in both equations, and *kj* and *<sup>ω</sup>j* satisfy the linear dispersion relation:

$$
\omega\_j^2 = \gcd|k\_j| \tanh|k\_j| h \tag{2}
$$

Generalized Bragg resonance conditions in the presence of bottom ripples are obtained by replacing one or more of the free-surface wave components in Equation (1) by periodic bottom ripple components of corresponding wavenumbers *k*bj but with zero frequencies (since the ripples are fixed) [20]. Thus, by combining wavenumbers and frequencies of surface waves and bottom ripples, we obtain general conditions for Bragg resonances at each order, *m* = 2, 3 . . .

Here, the Bragg resonance between free wave surface and V-shaped undulating bottom originated satisfies generalized Bragg conditions. In addition, we propose that the first case considers two surface wave components of an incident wavenumber *k*1 and a reflected wavenumber *k*2, propagating over a V-shaped undulating horizontal bottom containing a single wavenumber *k*b (analogously to surface waves, which refers to a fixed sinusoidally varying bottom with crest lines normal to *k*b). The first case belongs to Class I Bragg resonance. The second case considers random surface wave generated by the Gaussian spectrum, propagating over a V-shaped undulating horizontal bottom containing a single wavenumber *k*b (analogously to surface waves, which refers to a fixed sinusoidally varying bottom with crest lines normal to *k*b). In the paper, Class I Bragg resonance is considered, and numerical simulations all adopt the high-order spectral (HOS) method.
