*3.1. BW Model of MIKE 21 (DHI)*

The Boussinesq model is capable of reproducing the combined effects of important wave phenomena, such as diffraction and refraction. Therefore, it has been used in numerical experimentation to obtain diffracted wave fronts generated by headland.

MIKE 21 BW includes two modules, the 2DH wave model and 1DH wave model, both based on the numerical solution of time domain formulations of Boussinesq-type equations, which are solved using a flux-formulation with improved linear dispersion characteristics. The enhanced Boussinesq equations were originally derived by [30,31], making the modules suitable for simulation of the propagation of directional wave trains travelling from deep to shallow water. Moreover, it contains wave breaking and moving shorelines, as described in [32–34]. The 2DH BW model has been used in this work.

The enhanced Boussinesq equations are expressed in terms of the free surface elevation, *ξ*, and the depth-integrated velocity-components, *P* and *Q*. The continuity and the momentum-conservation equations read:

$$m\frac{\partial \xi}{\partial t} + \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0 \tag{10}$$

$$n\frac{\partial P}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \left(\frac{P^2}{h}\right) + \frac{\partial}{\partial y} \left(\frac{PQ}{h}\right) + \frac{\partial R\_{xx}}{\partial \mathbf{x}} + \frac{\partial R\_{xy}}{\partial y} + n^2 g h \frac{\partial \mathbf{j}^x}{\partial \mathbf{x}} + n^2 \mathbf{P} \left[a + \beta \frac{\sqrt{P^2 + Q^2}}{h}\right] + \frac{\mathbf{g}P\sqrt{P^2 + Q^2}}{h^2 C^2} + n\psi\_1 = 0 \tag{11}$$

$$n\frac{\partial Q}{\partial t} + \frac{\partial}{\partial y} \left(\frac{Q^2}{h}\right) + \frac{\partial}{\partial x} \left(\frac{QP}{h}\right) + \frac{\partial R\_{yy}}{\partial y} + \frac{\partial R\_{yx}}{\partial x} + n^2 g h \frac{\partial \xi}{\partial y} + n^2 Q \left[a + \beta \frac{\sqrt{P^2 + Q^2}}{h}\right] + \frac{gQ\sqrt{P^2 + Q^2}}{h^2 C^2} + n\psi\_2 = 0 \tag{12}$$

with the dispersive Boussinesq terms *ψ*<sup>1</sup> and *ψ*2, defined by:

$$\Psi\_1 = -\left(B + \frac{1}{3}\right)d^2 \left(P\_{\rm xxt} + Q\_{\rm xyt}\right) - nB\mathfrak{gl}^3\left(\mathfrak{k}\_{\rm xxx} + \mathfrak{k}\_{\rm tyy}\right) \\ -dd\_x\left(\frac{1}{3}P\_{\rm xtt} + \frac{1}{6}Q\_{\rm xtt} + n\mathfrak{g}Bd\left(2\mathfrak{k}\_{\rm xxt} + \mathfrak{k}\_{\rm yyy}\right)\right) \\ -dd\_x\left(\frac{1}{6}Q\_{\rm xtt} + n\mathfrak{g}Bd\mathfrak{k}\_{\rm tyy}\right) \\ \tag{13}$$

$$d\varphi\_2 = -\left(B + \frac{1}{3}\right)d^2\left(P\_{x\sharp t} + Q\_{y\sharp t}\right) - nBgd^3\left(\xi\_{xxy} + \xi\_{yyy}\right) - d d\_x\left(\frac{1}{6}P\_{x\sharp t} + \frac{1}{3}Q\_{x\sharp t} + ngBd\left(2\xi\_{yy} + \xi\_{xx}\right)\right) - d d\_x\left(\frac{1}{6}P\_{y\sharp t} + ngBd\xi\_{xy}\right) \tag{14}$$

where *d* is the still water depth; *g* is the gravitational acceleration; *n* is the porosity; *C* is the Chezy resistence number; *α* is the resistance coefficient for laminar flow in porous media; *β* is the resistance coefficient for turbulent flow in porous media; *B* is the Boussinesq dispersion coefficient; the terms *Rxx*, *Rxy* and *Ryy* indicate the incorporation of the wave breaking by means of the surface roller model [35].

In MIKE 21 BW, the waves may either be specified along open boundaries or be generated internally within the model through the generation line; the latter must be placed in front of a sponge layer absorbing all outgoing waves. Moreover, porosity (e.g., to model partial transmission through porous structures) and sponge layers can be used on an ad hoc basis. At open boundaries, either a level boundary, namely wave energy given as time series of surface elevation, or flux boundary, where flux density is perpendicular to the boundary, can be set.
