Waves of intermediate length through an array of vertical cylinders

Abstract

We report a semi-analytical theory of wave propagation through a vegetated water. Our aim is to construct a mathematical model for waves propagating through a lattice-like array of vertical cylinders, where the macro-scale variation of waves is derived from the dynamics in the micro-scale cells. Assuming infinitesimal waves, periodic lattice configuration, and strong contrast between the lattice spacing and the typical wavelength, the perturbation theory of homogenization (multiple scales) is used to derive the effective equations governing the macro-scale wave dynamics. The constitutive coefficients are computed from the solution of micro-scale boundary-value problem for a finite number of unit cells. Eddy viscosity in a unit cell is determined by balancing the time-averaged rate of dissipation and the rate of work done by wave force on the forest at a finite number of macro stations. While the spirit is similar to RANS scheme, less computational effort is needed. Using one fitting parameter, the theory is used to simulate three existing experiments with encouraging results. Limitations of the present theory are also pointed out.

Appendix: A constant eddy viscosity model for wave scattering by a finite forest belt

For checking the numerical solution for variable $M$ and $N$ (see §7.1), we give the analytical solution for constant eddy viscosity.

The formal solutions in incidence zone and transmission zone are given by (7.2) and (7.3), respectively. For constant eddy viscosity in the forest region, $M$ and $N$ are complex constants. By separation of variables as for the open waters we obtain

$$\begin{aligned} \phi ^{\text {F}}(X,Z)=\sum _{q=0}^\infty \left[ C_q\cos (\gamma \hat{k}_q x)+D_q\sin (\gamma \hat{k}_q x)\right] \frac{\cosh \hat{k}_q(Z+h)}{\cosh \hat{k}_qh}, \end{aligned}$$

(A.1)

where $\gamma =\sqrt{(n+N)/(n+M)}$ and $\hat{k}_q$ are complex roots of the transcendental equation

$$\begin{aligned} 1={\left( 1+\frac{N}{n}\right) } \hat{k}_q\tanh \hat{k}_q h. \end{aligned}$$

(A.2)

Note that boundary conditions (6.14) and (6.15) have been used. $C_q$ and $D_q$ are coefficients to be determined.

Coefficients $B_p$, $C_q$, $D_q$, and $E_p$ are to be determined by matching the pressure and the horizontal velocity component at the interfaces between two adjacent regions. At $X=0$, the interfacial conditions are

$$\begin{aligned} \phi ^{\text {I}}=\phi ^{\text {F}},\quad -h<Z<0, \end{aligned}$$

(A.3)

$$\begin{aligned} \frac{\partial \phi ^{\text {I}}}{\partial X}=(n+M)\frac{\partial \phi ^{\text {F}}}{\partial X},\quad -h<Z<0. \end{aligned}$$

(A.4)

At $X=L_B$,

$$\begin{aligned}&\displaystyle \phi ^{\text {F}}=\phi ^{\text {T}},\quad -h<Z<0, \end{aligned}$$

(A.5)

$$\begin{aligned} (n+M)\frac{\partial \phi ^{\text {F}}}{\partial X}=\frac{\partial \phi ^{\text {T}}}{\partial X},\quad -h<Z<0. \end{aligned}$$

(A.6)

The following condition of orthogonality applies to both $k_p$ and $\hat{k}_q$:

$$\begin{aligned} \int _{-h}^0 \cosh {\kappa }_m (Z+h) \cosh {\kappa }_n (Z+h) dZ = \left\{ \begin{array}{ll} 0, &{} \quad m\ne n\\ \frac{2{\kappa }_m h+\sinh {2\kappa }_m h}{4{\kappa }_m}, &{} \quad m=n \end{array} \right. , \end{aligned}$$

(A.7)

where $\kappa =k$ or $\hat{k}$. By truncating the series and using (A.7) in (A.3) and (A.6), we obtain

$$\begin{aligned}&\displaystyle -\text {i} \frac{\Gamma _{0q}}{\cosh {k}_0 h} + \sum _{p=0}^{P_r}B_p \frac{\Gamma _{pq}}{\cosh {k}_p h} =C_q \frac{\Pi \left( \hat{k}_q\right) }{\cosh \hat{k}_qh},\quad q=0,1,2,\ldots ,Q, \end{aligned}$$

(A.8)

$$\begin{aligned} \frac{{k}_0 \Pi \left( {k}_0\right) }{\cosh {k}_0 h}\delta _{0p}-\text {i} B_p \frac{{k}_p \Pi \left( {k}_p\right) }{\cosh {k}_p h} =\gamma (n+M) \sum _{q=0}^Q D_q \frac{\hat{k}_q \Gamma _{pq}}{\cosh \hat{k}_qh},\quad p=0,1,2,\ldots ,P_r, \nonumber \\ \end{aligned}$$

(A.9)

at $X=0$, and

$$\begin{aligned} \left[ C_q\cos (\gamma \hat{k}_q L_B)+D_q\sin (\gamma \hat{k}_q L_B)\right] \frac{\Pi \left( \hat{k}_q\right) }{\cosh \hat{k}_qh} \nonumber \\ =\sum _{p=0}^{P_t} E_p e^{\text {i} {k}_p L_B} \frac{\Gamma _{pq}}{\cosh {k}_p h},\quad q=0,1,2,\ldots ,Q, \end{aligned}$$

(A.10)

$$\begin{aligned} \gamma (n+&M) \sum _{q=0}^Q \left[ -C_q\sin (\gamma \hat{k}_q L_B)+D_q\cos (\gamma \hat{k}_q L_B)\right] \frac{\hat{k}_q \Gamma _{pq}}{\cosh \hat{k}_qh} \nonumber \\&=\text {i} E_p e^{\text {i} {k}_p L_B} \frac{{k}_p \Pi \left( {k}_p\right) }{\cosh {k}_p h},\quad p=0,1,2,\ldots ,P_t, \end{aligned}$$

(A.11)

at $X=L_B$, where

$$\begin{aligned} \Gamma _{pq}&=\int _{-h}^0\cosh k_p(Z+h) \cosh \hat{k}_q(Z+h) dZ \nonumber \\&=\frac{k_p\sinh k_p h \cosh \hat{k}_q h-\hat{k}_q\cosh k_p h \sinh \hat{k}_q h}{k_p^2-\hat{k}_q^2}, \end{aligned}$$

(A.12)

$$\begin{aligned} \Pi (\kappa )=\frac{2{\kappa } h+\sinh {2\kappa } h}{4{\kappa }}, \quad \kappa =k_p, {\hat{k}}_q. \end{aligned}$$

(A.13)

In addition, $P_r$, $Q$, and, $P_t$ are the finite terms after truncation in zones (I), (F), and (T), respectively. The expansion coefficients $\left( B_p,C_q,D_q,E_p\right)$ can now be obtained by solving numerically the above algebraic equations. Afterwards, $|R|=|B_0|$ and $|T|=|E_0|$ are the reflection and transmission coefficients respectively.