*2.3. Model Establishment*

The V-shaped undulating bottom is shown in Figure 1. It shows that the two continuous undulating bottoms of a horizontal V-shaped pattern, a set of oblique undulations *k*b1 (the upper part of the V-shaped pattern). To match with *k*b1, the other set of oblique undulations *k*b2 (the lower part of the V-shaped pattern matched) is given. The values of *k*b1 and *k*b2 are the same for a given V-shaped undulating bottom, while the difference of the *k*b1 and *k*b2 is in the directions of the two vectors. The vectors of *k*b1 and *k*b2 are symmetric about the *x*-axis. The color vectors in Figure 1 describe simply the V-shaped undulating bottom configuration, where the number of the continuous sinusoidal undulations *N*b = 10, an incident wave wavenumber *k*1 (the direction of *k*1 along the *x*-axis); two bottom undulations wavenumber *k*b1 and *k*b2 are perpendicular to crest lines of the each

side of V-shaped undulating bottom respectively; the reflected waves *k*r1 and *k*r2 due to the two parts of bottom superpose each other ahead of the bottom. In addition, the relationship of the angle *θ* between *k*b1/*k*b2 and *x*-axis and the angle *α* between the two sets of oblique crest lines satisfy:

$$
\mathfrak{\alpha} = 180^{\circ} - 2\theta \tag{4}
$$

**Figure 1.** Schematic diagram of the relationship of the angle *θ* between *k*b1/*k*b2 and *x*-axis and the angle *α* between the two sets of oblique crest lines: (**a**) Spatial graph of V-shaped undulating bottom (α < 180◦); (**b**) Plane graph of V-shaped undulating bottom (α < 180◦).

The following parameters for the HOS numerical model are given referring to reference [20]: the undulating bottom *k*b*d =* 0.31, where *d* and *k*b (*k*b refers to *k*b1 and *k*b2, and below is the same) are the amplitude and wavenumber of bottom undulations; the incident waves *k*1*A =* 0.05, where 2*A* ≡ *η*max − *η*min, where *A*, *k*1, *η*max and *η*min are the amplitude, wavenumber, maximum elevation, and minimum elevation for surface wave of the incident regular waves respectively; relative water depth *d*/*h* = 0.16, where *d* and *h* are the amplitude of bottom and water depth; the total numbers of bottom undulations *N*b = *L*0/*λ*b = 10, where *N*b, *L*0, and *λ*b are the number, total length, and wavelength of the sinusoidal undulating bottom respectively; the time step for the fourth-order Runge–Kutta integration Δ*t*; the total simulation time *T*s; running steps per period *T*/Δ*t* = 64, where *T* is the period of the incident waves; the duration of the simulation periods *T*s/*T =* 20; node numbers *Nx* × *Ny* = 512 × 512; nonlinear order *M* = 3; V-shaped undulating bottom angle *α* = 90◦–180◦, where 24 values between 90◦ and 180◦ are distributed unevenly to the angle α of inclination for the V-shaped undulating bottom, as shown in Table 1.

**Table 1.** The simulation cases of V-shaped undulating bottom angles *α*.

