**4. Conclusions**

With considerations of wave forces acting on mooring lines, a new analytic solution is presented for the problem of an underwater moored floating structure with motions of full degrees of freedom subjected to incident waves. A coupling formulation among water waves, underwater floating structure, and mooring lines is presented. Iterations for the drag coe fficients, energy conservation without drag loss, reflection and transmission coe fficients, and comparisons of wave scatterings with a finite elements result, as well as motion amplitudes in comparison with a numerical boundary element model, provide valid validation of the present solution. With additional considerations of wave forces acting on the mooring lines, the drag dampening significantly decreases wave reflections and the motions of the structure at the high-frequency resonance. The magnitudes of the wave forces acting on the mooring lines can reach up to 12% of the incident wave forces. The study of the submerged depth of the structure indicates that the structure deployed nearer the free surface can induce bigger motions. The analytic solution is very much dependent on the geometry of the structure; however, the interaction formulation in this paper can be applied to practical problems for more complete considerations.

**Author Contributions:** Conceptualization, J.-F.L., C.-H.L.; methodology, J.-F.L., C.-T.C.; validation, C.-H.L.; writing—original draft preparation, J.-F.L., C.-H.L.; writing—review and editing, C.-T.C.; funding acquisition, C.-T.C., J.-F.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ministry of Science and Technology, Taiwan, gran<sup>t</sup> number MOST 104-2221-E-006-188.

**Acknowledgments:** Financial support partially by the Yancheng Institute of Technology under Grant Number XJ201750 and partially by the Ministry of Science and Technology, Taiwan, under Grant Number MOST 104-2221-E-006-188 are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Definitions of Coe** ffi**cients**

Definitions of coe fficients *kn*, *k*2*n*, *k*4*n*, γ*<sup>n</sup>*, μ*n*1, μ*n*2, *D*<sup>2</sup> 2*n*, *F*<sup>2</sup> 4*n*, *D*<sup>3</sup> 2*n*, and *F*<sup>3</sup> 4*n*.

$$
\omega^2 = -gk\_n \tan k\_n \text{h. } n = 1, 2, \dots \text{\textquotedbl{}s\textquotedbl{}}, k\_0 = -iK \tag{A1}
$$

$$
\omega^2 = -gk\_{2n} \tan k\_{2n} d\_1, n = 1, 2, \cdots \infty \tag{A2}
$$

$$k\_{4n} = n\pi / d\mathbb{2}, n = 1, \mathbb{2}, \cdots \infty \tag{A3}$$

$$
\gamma\_n = n\pi/2\ell, n = 1, 2, \dots \infty \tag{A4}
$$

$$
\mu\_{n1} = \gamma\_n + \alpha^2 / \,\mathrm{g}, n = 1, 2, \dots \infty \tag{A5}
$$

$$
\mu\_{n2} = \gamma\_n - \omega^2 / \lg, n = 1, 2, \dots \infty \tag{A6}
$$

$$D\_{2n}^2 = \frac{i\alpha s\_2 (1 - \cos 2\gamma\_n \ell)}{\ell \gamma\_n 2 \left(\mu\_{n1} e^{\gamma\_n (-d\_1 - h)} - \mu\_{n2} e^{-\gamma\_n (-d\_1 + h)}\right)}, n = 1, 2, \cdots \infty \tag{A7}$$

$$F\_{4n}^2 = \frac{i\omega\mathfrak{s}\_2(1-\cos 2\gamma\_n\ell)}{\ell\gamma\_n^2 \sinh\gamma\_n d\_2}, n = 1, 2, \cdots \infty \tag{A8}$$

$$D\_{3n}^2 = \frac{-i\imath\kappa\_3(1+\cos 2\gamma\_n\ell)}{\gamma\_n^{\prime}\left(\mu\_{n1}e^{\gamma\_n(-d\_1-h)} - \mu\_{n2}e^{-\gamma\_n(-d\_1+h)}\right)}, n = 1, 2, \cdots \infty \tag{A9}$$

*J. Mar. Sci. Eng.* **2020**, *8*, 146

$$F\_{4n}^3 = \frac{-i\alpha\text{s}\,(1+\cos 2\gamma\_n\ell)}{\gamma\_n\,^2 \sinh \gamma\_n d\_2}, n = 1, 2, \cdots \infty \tag{A10}$$

#### **Appendix B. Wave Forces Acting on the Floating Structure**

Wave forces acting on the floating structure in the direction of each degree of freedom can be calculated as:

$$\begin{split} f\_1 &= -i\omega \rho \cdot e^{-i\omega t} \Big\{ \int\_{-h+d\_2}^{-d\_1} \left[ \left( \phi^l + \phi\_1^D \right) \Big|\_{\mathbf{x}=-\ell} - \phi\_3^D \Big|\_{\mathbf{x}=\ell} \right] d\mathbf{z} \\ &+ s\_1 \cdot \int\_{-h+d\_2}^{-d\_1} \left[ \phi\_1^1 \Big|\_{\mathbf{x}=-\ell} - \phi\_3^1 \Big|\_{\mathbf{x}=\ell} \right] d\mathbf{z} + s\_3 \cdot \int\_{-h+d\_2}^{-d\_1} \left[ \phi\_1^3 \Big|\_{\mathbf{x}=-\ell} - \phi\_3^3 \Big|\_{\mathbf{x}=\ell} \right] d\mathbf{z} \Big\} \end{split} \tag{A11}$$

$$\begin{split} f\_2 &= -i\omega \rho \cdot e^{-i\omega t} \Big[ \int\_{-\ell}^{\ell} \left( \phi\_4^D \Big|\_{z=-h+d\_2} - \phi\_2^D \Big|\_{x=-d\_1} \right) dx \\ &+ s\_2 \cdot \int\_{-\ell}^{\ell} \Big( \phi\_4^2 \Big|\_{z=-h+d\_2} - \phi\_2^2 \Big|\_{z=-d\_1} \Big) dx \Big] \end{split} \tag{A12}$$

$$\begin{array}{ll} f\_{3} = & -i\omega\rho \cdot \epsilon^{-i\omega t} \left\{ \int\_{-h+d\_{2}}^{-d\_{1}} (z-z\_{0}) \left[ \left(\phi^{l} + \phi\_{1}^{D}\right)\big|\_{x=-\ell} - \phi\_{3}^{D}\big|\_{x=\ell} \right] dz \\ & - \int\_{-\ell}^{\ell} \mathbf{x} \left[ \phi\_{4}^{D}\big|\_{z=-h+d\_{2}} - \phi\_{2}^{D}\big|\_{z=-d\_{1}} \right] dx + s\_{1} \cdot \int\_{-h+d\_{2}}^{-d\_{1}} (z-z\_{0}) \left[ \phi\_{1}^{1}\big|\_{x=-\ell} - \phi\_{3}^{1}\big|\_{x=\ell} \right] dz \\ & - s\_{1} \cdot \int\_{-\ell}^{\ell} \mathbf{x} \left[ \phi\_{4}^{1}\big|\_{z=-h+d\_{2}} - \phi\_{2}^{1}\big|\_{z=-d\_{1}} \right] dx + s\_{3} \cdot \int\_{-h+d\_{2}}^{-d\_{1}} (z-z\_{0}) \left[ \phi\_{1}^{3}\big|\_{x=-\ell} - \phi\_{3}^{3}\big|\_{x=\ell} \right] dz \\ & - s\_{3} \cdot \int\_{-\ell}^{\ell} \mathbf{x} \left[ \phi\_{4}^{3}\big|\_{z=-h+d\_{2}} - \phi\_{2}^{3}\big|\_{z=-d\_{1}} \right] dx \end{array} \tag{A13}$$

Integrations shown in Equations (A11)–(A13) can be calculated, and the equations be rewritten, as:

$$
\begin{Bmatrix} f\_1 \\ f\_2 \\ f\_3 \end{Bmatrix} = \begin{bmatrix} f^R \end{bmatrix} \begin{Bmatrix} s\_1 \\ s\_2 \\ s\_3 \end{Bmatrix} + \begin{Bmatrix} f\_1^D \\ f\_3^D \\ f\_3^D \end{Bmatrix} \tag{A14}
$$

where *f R* and "*f D*# can be expressed as:

$$
\begin{bmatrix} f^{\mathbb{R}} \end{bmatrix} = \begin{bmatrix} f\_{11}^{\mathbb{R}} & 0 & f\_{13}^{\mathbb{R}} \\ 0 & f\_{22}^{\mathbb{R}} & 0 \\ f\_{31}^{\mathbb{R}} & 0 & f\_{33}^{\mathbb{R}} \end{bmatrix} \tag{A15}
$$

with the components expressed as:

*<sup>n</sup>*=1

$$f\_{11}^R = \sum\_{n=0}^{\infty} (B\_{1n}^1 - A\_{3n}^1) \frac{\sin k\_{\rm ll}(h - d\_1) - \sin k\_{\rm dl} d\_2}{k\_n} \tag{A16}$$

$$f\_{13}^R = \sum\_{n=0}^{\infty} (B\_{1n}^3 - A\_{3n}^3) \frac{\sin k\_{\rm tr} (h - d\_1) - \sin k\_{\rm tr} d\_2}{k\_{\rm tr}} \tag{A17}$$

$$\begin{split} f\_{22}^{R} &= \left(2\ell B\_{40}^{2}\right) + \sum\_{n=1}^{\infty} \left(A\_{4n}^{2} + B\_{4n}^{2}\right) \cos k\_{4n} d\_{2} \frac{\left(1 - e^{-2k\_{4n}\ell}\right)}{k\_{4n}} \\ &+ \sum\_{n=1}^{\infty} F\_{4n}^{2} \cosh \chi\_{4n} d\_{2} \frac{\left(1 - \cos 2\chi\_{4n}\ell\right)}{\chi\_{4n}} - \sum\_{n=0}^{\infty} \left(A\_{2n}^{2} + B\_{2n}^{2}\right) \frac{\left(1 - e^{-2k\_{2n}\ell}\right)}{k\_{2n}} \\ &- \sum\_{n} D\_{2n}^{2} \left(\mu\_{n1} e^{-\kappa \chi\_{2n}(d\_{1} + h)} + \mu\_{n2} e^{-\kappa \chi\_{2n}(-d\_{1} + h)}\right) \frac{\left(1 - \cos 2\kappa \chi\_{2n}\ell\right)}{\kappa\_{2n}} \end{split} \tag{A18}$$

$$\begin{array}{rcl}f\_{31}^{\mathbb{R}} &=& \sum\_{n=0}^{\infty} \left(B\_{1n}^{1} - A\_{3n}^{1}\right) \mathbb{C}\_{n}^{16} - \frac{2}{3} \mathcal{C}^{3} A\_{40}^{1} \\ &- \sum\_{n=1}^{\infty} \left(A\_{4n}^{1} - B\_{4n}^{1}\right) \cos k\_{4n} d\_{2} \cdot \mathbb{C}\_{4n}^{17} + \sum\_{n=0}^{\infty} \left(A\_{2n}^{1} - B\_{2n}^{1}\right) \mathbb{C}\_{2n}^{17} \end{array} \tag{A19}$$

*J. Mar. Sci. Eng.* **2020**, *8*, 146

$$\begin{array}{rcl} f\_{33}^{\rm R} & = & \sum\_{n=0}^{\infty} \left( B\_{1n}^{3} - A\_{3n}^{3} \right) \mathbf{C}\_{n}^{16} - \frac{2}{3} \mathbf{C}^{3} A\_{40}^{3} \\ & - \sum\_{n=1}^{\infty} \left( A\_{4n}^{3} - B\_{4n}^{3} \right) \cos k\_{4n} d\_{2} \cdot \mathbf{C}\_{4n}^{17} - \sum\_{n=1}^{\infty} F\_{4n}^{3} \cosh \kappa\_{n} d\_{2} \cdot \mathbf{C}\_{n}^{18} \\ & + \sum\_{n=0}^{\infty} \left( A\_{2n}^{3} - B\_{2n}^{3} \right) \mathbf{C}\_{2n}^{17} - \sum\_{n=1}^{\infty} D\_{2n}^{3} \left( \mu\_{n1} e^{-\kappa\_{2n} (d\_{1} + h)} + \mu\_{n2} e^{-\kappa\_{2n} (-d\_{1} + h)} \right) \mathbf{C}\_{n}^{18} \end{array} \tag{A20}$$

$$\begin{array}{rcl}f\_1^D &=& \frac{i\chi A^l e^{-i\mathcal{K}l}}{\stackrel{\text{a}l\text{-}\text{csch}\,Kl}{\text{cs}}} \frac{\sinh\mathcal{K}(l-d\_1) - \sinh\mathcal{K}d\_2}{\mathcal{K}}\\ &+& \sum\_{n=0}^{\infty} \left(B\_{1n}^D - A\_{3n}^D\right) \cdot \frac{\sin k\_n(l-d\_1) - \sin k\_n d\_2}{k\_n}\end{array} \tag{A21}$$

$$\begin{array}{ll} f\_2^D = & 2\ell B\_{40}^D + \sum\_{n=1}^{\infty} \left( A\_{4n}^D + B\_{4n}^D \right) \cos k\_{4n} d\_2 \frac{\left( 1 - e^{-2k\_{4n}\ell} \right)}{k\_{4n}} \\ - & \sum\_{n=0}^{\infty} \left( A\_{2n}^D + B\_{2n}^D \right) \frac{\left( 1 - e^{-2k\_{2n}\ell} \right)}{k\_{2n}} \end{array} \tag{A.22}$$

$$\begin{split} f\_{3}^{D} &= \frac{i\chi A^{1}e^{-i\mathcal{K}}}{\omega \cosh \mathcal{K}t} \mathcal{C}\_{0}^{16} + \sum\_{n=0}^{\infty} \left( B\_{1n}^{D} - A\_{3n}^{D} \right) \mathcal{C}\_{n}^{16} - \frac{2}{3} \mathcal{C}^{3} A\_{30}^{D} \\ &+ \sum\_{n=1}^{\infty} \left( A\_{4n}^{D} - B\_{4n}^{D} \right) \cos k\_{4n} d\_{2} \cdot \mathcal{C}\_{4n}^{17} + \left( A\_{2n}^{3} - B\_{2n}^{3} \right) \mathcal{C}\_{2n}^{17} \end{split} \tag{A23}$$

And the constants *<sup>C</sup>*16*n* , *<sup>C</sup>*172*n*, *<sup>C</sup>*174*n*, and *<sup>C</sup>*18*n* are:

$$\begin{array}{lcl} \mathbb{C}\_{n}^{16} &= \int\_{-d\_{1}}^{0} (z - z\_{0}) \cos[k\_{n}(z + h)] dz \\ &= \frac{(h - d\_{2} - z\_{0}) \sin k\_{n} d\_{2} - (d\_{1} + z\_{0}) \sin k\_{n} (h - d\_{1})}{k\_{n}} \\ &+ \frac{\cos k\_{n} (h - d\_{1}) - \cos k\_{n} d\_{2}}{k\_{n}^{2}} \end{array} \tag{A2.4}$$

$$\begin{split} \mathbb{C}\_{2n,4n}^{17} &= \int\_{-\ell}^{\ell} \mathbf{x} e^{-k\_{2n,4n}(\mathbf{x}+\ell)} d\mathbf{x} = -\int\_{-\ell}^{\ell} \mathbf{x} e^{k\_{2n,4n}(\mathbf{x}-\ell)} d\mathbf{x} \\ &= \frac{1 - e^{-2k\_{2n,4n}\ell}}{k\_{2n,4n}} - \frac{\ell \Big(1 + e^{-2k\_{2n,4n}\ell}\Big)}{k\_{2n,4n}} \end{split} \tag{A25}$$

$$\begin{array}{rcl} \mathsf{C}\_{n}^{18} & = \underset{\substack{\ell \to \ell \ \mathrm{x} \text{sin } \mathrm{y}\_{n} \, (\mathrm{x} + \ell) \, \mathrm{d}\mathbf{x} \\ & = \underset{\substack{\mathrm{y} \text{2} \upprime\_{n} \, \ell}}{\mathrm{sin } 2 \upprime\_{n} \, \ell}}{\mathrm{sin } 2} - \frac{\ell (1 + \cos 2 \upprime\_{n} \, \ell)}{\mathrm{y}^{\prime n^{2}}} \end{array} \tag{A26}$$

#### **Appendix C. Sti**ff**ness Matrix of the Mooring Springs**

The components of the stiffness matrices for the springs *AB* and *CD* are calculated according to the geometrical orientations of the springs and can be expressed as:

$$K\_{11}^{\overline{AB}} = K\_{11}^{\overline{CD}} = K\_{\text{s}} \cos^2 \theta \tag{A27}$$

$$K\_{12}^{\overline{AB}} = K\_{21}^{\overline{AB}} = -K\_{12}^{\overline{CD}} = -K\_{21}^{\overline{CD}} = K\_5 \cos \theta \sin \theta \tag{A28}$$

$$K\_{13}^{\overline{A\overline{B}}} = K\_{31}^{\overline{A\overline{B}}} = K\_{13}^{\overline{C\overline{D}}} = K\_{31}^{\overline{C\overline{D}}} = K\_{\overline{s}} \left[ 0.5 (h - d\_1 - d\_2) \cos^2 \theta - \ell \cos \theta \sin \theta \right] \tag{A29}$$

$$K\_{22}^{\overline{A3}} = K\_{22}^{\overline{CD}} = K\_{\mathbb{S}} \sin^2 \theta \tag{A30}$$

$$K\_{33}^{\overline{AB}} = K\_{33}^{\overline{CD}} = K\_{\mathfrak{s}} [0.5(h - d\_1 - d\_2) \cos \theta - \ell \sin \theta]^2 \tag{A31}$$

#### **Appendix D. Wave Forces Acting on Mooring Lines**

Wave forces acting on the mooring springs are calculated using the linearized Morison equation (Lee, 1994):

$$dF^M = \frac{\rho \mathbb{C}\_{D\ell} D\_S}{2} \Big(\mathcal{U} - \dot{\boldsymbol{\varepsilon}}\Big) dS + \frac{\rho \pi \mathbb{C}\_M D\_S^2}{4} \Big(\dot{\mathcal{U}} - \ddot{\boldsymbol{\varepsilon}}\Big) dS\tag{A32}$$

where ρ is fluid density, *DS* is the diameter of the spring, *U* and *U* are the flow velocity and acceleration in the direction normal to the mooring line, *CM* is the added mass coefficient, .ς and ..ς are the velocity and acceleration of the mooring line, and the linear drag coefficient is expressed as:

$$\mathbf{C}\_{D\ell} = \frac{4\mathbf{C}\_D}{3\pi\alpha} \frac{\int\_{-h}^{-h+d\_2} \left| \boldsymbol{\varPi} - \boldsymbol{\dot{\boldsymbol{\varsigma}}} \right|^3 dz}{\int\_{-h}^{-h+d\_2} \left| \boldsymbol{\varPi} - \boldsymbol{\dot{\boldsymbol{\varsigma}}} \right|^2 dz} \tag{A33}$$

.

Note that motions of the mooring lines are not known in priori until the problem solved: therefore, a complete solution will contain an iteration procedure until a 0.5% convergen<sup>t</sup> criteria is reached. Furthermore, the springs are not subjected to forces in a transverse direction; therefore, the wave forces calculated are transferred to the attached points A and C. Thus,

$$F\_A^M = \frac{\csc\theta}{d\_2} \int\_{-h}^{-h+d\_2} \left[ \frac{\rho \mathbb{C}\_{D\ell} D\_S}{2} \{ \mathcal{U}\_1 - \dot{\boldsymbol{\varsigma}}\_{\overline{AB}} \} + \frac{\rho \pi \mathbb{C}\_M D\_S^2}{4} \{ \dot{\mathcal{U}}\_1 - \ddot{\boldsymbol{\varsigma}}\_{\overline{AB}} \} \right] d\boldsymbol{z} \tag{A34}$$

$$F\_C^M = \frac{\csc\theta}{d\_2} \int\_{-h}^{-h+d\_2} \left[ \frac{\rho \mathbf{C}\_{D\ell} D\_S}{2} \left( \mathcal{U}\_3 - \dot{\boldsymbol{\varsigma}}\_{\overline{\underline{CD}}} \right) + \frac{\rho \pi \mathbf{C}\_M D\_S^2}{4} \left( \dot{\mathcal{U}}\_3 - \ddot{\boldsymbol{\varsigma}}\_{\overline{\underline{CD}}} \right) \right] dz \tag{A35}$$

where the subscripts 1 and 3 stand for regions 1 and 3, while subscripts *AB* and *CD* stand for the spring AB and CD. The corresponding expressions are:

$$\begin{array}{ll} \mathcal{U}\_{1} = & - \begin{pmatrix} \Phi\_{x}^{I} + \Phi\_{1x}^{D} + s\_{1} \cdot \Phi\_{1x}^{1} + s\_{2} \cdot \Phi\_{1x}^{2} + s\_{3} \cdot \Phi\_{1x}^{3} \end{pmatrix} \text{sim } \theta\\ & - \begin{pmatrix} \Phi\_{z}^{I} + \Phi\_{1z}^{D} + s\_{1} \cdot \Phi\_{1z}^{1} + s\_{2} \cdot \Phi\_{1z}^{2} + s\_{3} \cdot \Phi\_{1z}^{3} \end{pmatrix} \text{sim } \theta \end{array} \tag{A36}$$

$$\begin{array}{ll} \mathcal{U}\_3 = & - \begin{pmatrix} \Phi\_{3x}^D + s\_1 \cdot \Phi\_{3x}^1 + s\_2 \cdot \Phi\_{3x}^2 + s\_3 \cdot \Phi\_{3x}^3 \end{pmatrix} \sin \theta \\ - \begin{pmatrix} \Phi\_{3z}^D + s\_1 \cdot \Phi\_{1z}^1 + s\_2 \cdot \Phi\_{3z}^2 + s\_3 \cdot \Phi\_{3z}^3 \end{pmatrix} \cos \theta \end{array} \tag{A37}$$

With substitutions of Equations (A36) and (A37) into Equation (A33) and Equation (A35), one can obtain

$$F\_A^M = F\_A^{Mw} + F\_A^{Ms} \tag{A38}$$

$$F\_{\mathbb{C}}^{M} = F\_{\mathbb{C}}^{Mw} + F\_{\mathbb{C}}^{Ms} \tag{A39}$$

in which

*FMw A* = csc θ *d*2 ρ*CD DS* 2 − *<sup>i</sup>*ωρπ*D*2*SCM* 4 6*KgAIe*−*iK* sin θ ω cosh*Kh Kd*2sinh*Kd*2−cosh *Kd*2+1 *K*<sup>2</sup> +*iKgAIe*−*iK* cos θ ω cosh*Kh Kd*2 cosh *Kd*2−sinh*Kd*<sup>2</sup> *K*<sup>2</sup> + sin θ %∞*<sup>n</sup>*=0 *BD*1*n* + *<sup>s</sup>*1*<sup>B</sup>*11*n* + *<sup>s</sup>*2*<sup>B</sup>*21*n* + *<sup>s</sup>*3*<sup>B</sup>*31*n*−*knd*<sup>2</sup> sin *knd*<sup>2</sup>−cos *knd*<sup>2</sup>−1 *kn* + cos θ %∞*<sup>n</sup>*=0 *BD*1*n* + *<sup>s</sup>*1*<sup>B</sup>*11*n* + *<sup>s</sup>*2*<sup>B</sup>*21*n* + *<sup>s</sup>*3*<sup>B</sup>*31*n*−sin *knd*2+*knd*2 cos *knd*2 *kn* 8 (A40) *FMs* = csc θ + πρ*CM*ω2*D*2*S* 

$$\begin{cases} F\_A^{\rm Ms} = \frac{\rm SG}{d\_2} \Big( \frac{i\omega\rho C\_{D1} D\_S^2}{2} + \frac{\pi\rho C\_{M2}\rho^2 D\_S^2}{4} \Big) \\ \left\{ s\_1 \Big( \frac{\sin\theta}{d\_2} \Big) \frac{d\chi^3}{3} - s\_2 \frac{d\chi^2}{2} \cos\theta - s\_3 \left[ \frac{(h - d\_1 - d\_2)\sin\theta}{2d\_2} \frac{d\chi^3}{3} + \ell \frac{d\chi^2}{2} \cos\theta \right] \right\} \end{cases} \tag{A41}$$

$$\begin{split} &F\_{\mathbb{C}}^{\mathrm{Mw}} = \frac{\mathop{\mathrm{s.c.}}\limits\_{d\_{2}}Q\left(\frac{\rho \mathbb{C}\_{D}d\_{\mathrm{S}}}{2} - \frac{i\omega\rho\pi D\_{\mathrm{S}}^{2}\mathbb{C}\_{M}}{4}\right)}{\Big\{\mathrm{s.c.}}\Big{(}\mathrm{s.c}A\_{\mathrm{3u}}^{\mathrm{D}} + s\_{1}A\_{\mathrm{3u}}^{\mathrm{1}} + s\_{2}A\_{\mathrm{3u}}^{\mathrm{2}} + s\_{3}A\_{\mathrm{3u}}^{\mathrm{3}}\Big{(}\frac{k\_{d}d\_{2}\sin k\_{d}d\_{2} + \cos k\_{d}d\_{2} - 1}{k\_{d}}\Big{)}\Big{]} \\ &+ \cos\theta\Big{\Big{(}\sum\_{\mathrm{tr}=0}^{\infty}\Big{(}A\_{\mathrm{3u}}^{\mathrm{D}} + s\_{1}A\_{\mathrm{3u}}^{\mathrm{1}} + s\_{2}A\_{\mathrm{3u}}^{\mathrm{2}} + s\_{3}A\_{\mathrm{3u}}^{\mathrm{3}}\Big{)}\Big{(}\frac{\sin k\_{d}d\_{2} - k\_{d}d\_{2}\cos k\_{d}d\_{2}}{k\_{u}}\Big{)}\Big{\} \end{split} \tag{A42}$$

$$\begin{cases} F\_C^{\rm Ms} = \frac{\csc\theta}{d\_2} \Big( \frac{i\nu\rho \mathbb{C}\_{Dl} D\_S}{2} + \frac{\pi\rho \mathbb{C}\_{Ma} a^2 D\_S^2}{4} \Big) \\ \left\{ s\_1 \Big( \frac{\sin\theta}{d\_2} \Big) \frac{d\_2^3}{3} + s\_2 \frac{d\_2^2}{2} \cos\theta - s\_3 \left[ \frac{(h - d\_1 - d\_2)\sin\theta}{2d\_2} \frac{d\_2^3}{3} + \ell \frac{d\_2^2}{2} \cos\theta \right] \right\} \end{cases} \tag{A43}$$

Note that the spring *CD* is located at the lee side of the structure; therefore, there is no incident wave in the expression.
