**Abbreviations**



#### **Appendix A**

The calculation of wave excitation forces in the vertical direction of multi-type floating bodies.

#### *Appendix A.1 Horizontal Rectangular Floating Body*

The wave excitation forces in the vertical direction of the horizontal rectangular floating body is

$$\begin{split} F^{z}\_{\rm ex} &= \mathbb{C}\_{\upsilon} \iint p\_{z} ds = \mathbb{C}\_{\upsilon} a \int\_{\underline{x}\_{1}}^{\underline{x}\_{1} + b} \frac{\rho gH}{2} \frac{c \mathrm{ch}(d - \lambda(t))}{c \mathrm{ch} d \overline{t}} \cos(k \underline{x} - \omega t) d \underline{x} \\ &= \mathbb{C}\_{\upsilon} a \frac{\rho gH}{2} \frac{c \mathrm{ch}(d - \lambda(t))}{c \mathrm{ch} d} \int\_{\underline{x}\_{1}}^{\underline{x}\_{1} + b} \cos(k \underline{x} - \omega t) d \underline{x} \\ &= \mathbb{C}\_{\upsilon} a \frac{\rho gH}{2k} \frac{c \mathrm{ch}(d - \lambda(t))}{c \mathrm{ch} d} [\sin(k(\underline{x}\_{1} + b) - \omega t) - \sin(k \underline{x}\_{1} - \omega t)] \\ &= \mathbb{C}\_{\upsilon} \frac{\rho gH}{k} \frac{c \mathrm{ch}(d - \lambda(t))}{c \mathrm{ch} d} \cos(k \underline{x}\_{1} - \omega t + \frac{k b}{2}) \sin\frac{k b}{2} \end{split} \tag{A1}$$

#### *Appendix A.2 Vertical Cylinder Floating Body*

The relationship between any point (*x*, *y*, *z*) on the curved surface element *ds* and the cylinder coordinate [54] (*r*, *α*, *z*) is

$$\begin{cases} \ \mathfrak{T} = \mathbb{R} \cos \alpha \\ \ \mathfrak{Y} = \mathbb{R} \sin \alpha \\ \ \mathfrak{T} = \mathfrak{T} \end{cases} \tag{A2}$$

Then, the above equation can be written in the global coordinate system is

$$\begin{cases} \begin{array}{l} x = R\cos\alpha + x\_1 + R\\ y = R\sin\alpha + y\_1 \end{array} \\\ z = z \end{cases} \tag{A3}$$

Let *α* = *<sup>π</sup>* <sup>2</sup> + *ϕ*, the dynamic pressure at any point on the cylinder surface at this time in the global coordinate system can be written as:

$$\begin{array}{l} P\_{\rm dy} = \frac{\rho gH}{2} \frac{c\hbar k \tau}{c\hbar kl} \cos(k(R\cos\omega + \mathbf{x}\_1 + R) - \omega t) \\ = \frac{\rho gH}{2} \frac{c\hbar k \tau}{c\hbar kl} [\cos(kR\sin\phi)\cos(k\mathbf{x}\_1 + kR - \omega t) \\ + \sin(kR\sin\phi)\sin(k\mathbf{x}\_1 + kR - \omega t)] \end{array} \tag{A4}$$

According to the properties of the Bessel function [55] is as follows:

$$\begin{array}{l} \cos(kR\sin\varphi) = I\_0(kR) + 2\sum\_{m=1}^{m=\infty} I\_{2m}(kR)\cos(2m\varphi) \\ \sin(kR\sin\varphi) = 2\sum\_{m=1}^{m=\infty} I\_{2m-1}(kR)\sin(2m-1)\varphi \end{array} \tag{A5}$$

*Energies* **2018**, *11*, 3282

When *m* = 1, and then:

$$\begin{Bmatrix} \cos(kR\sin\varphi) = J\_0(kR) + 2J\_2(kR)\cos 2\varphi\\ \sin(kR\sin\varphi) = 2J\_1(kR)\sin\varphi \end{Bmatrix} \tag{A6}$$

Combining above equation, we can obtain the following equation:

$$\begin{array}{l} P\_{\rm dy} = \frac{\rho\_3 \eta H}{2} \frac{\epsilon \hbar k\_1}{\epsilon \hbar k} \left[ \left( \left( \rho\_0 (kR) + 2f\_2 (kR) \cos 2\rho \right) \cos \left( kx\_1 + kR - \omega t \right) \right. \\\ \left. + \left( 2f\_1 (kR) \sin \rho \right) \sin \left( kx\_1 + kR - \omega t \right) \right] \end{array} \tag{A7}$$

where, *J*0(*kR*) is the first kind zero-order Bessel function, *J*1(*kR*) is the first kind one-order Bessel function, *J*2(*kR*) is the first kind two-order Bessel function. In order to convenient calculation, let *Jm*(*kR*) = *Jm*.

The wave excitation forces in the vertical direction of the vertical cylinder floating body is

$$\begin{split} F^{\mathbb{Z}}\_{\alpha\mathbb{x}} &= \mathbb{C}\_{\mathbb{V}} \iint p\_{2} ds = 2 \mathbb{C}\_{\mathbb{V}} \int\_{d-\lambda(t)}^{d} \int\_{0}^{\pi} p\_{2} R d\omega dz = 2 \mathbb{C}\_{\mathbb{V}} \int\_{d-\lambda(t)}^{d} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} p\_{2} R d\rho dz \\ &= \mathbb{C}\_{\mathbb{V}} \cdot \frac{R\rho H}{c\hbar kd} \cdot \int\_{d-\lambda(t)}^{d} c\hbar kz d\mathbb{z} \cdot \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left\{ \begin{array}{l} (l\_{0} + 2l\_{2}\cos 2\rho)\cos(k\mathbf{x}\_{1} + kR - \omega t) \\ + (2l\_{1}\sin \rho)\sin(k\mathbf{x}\_{1} + kR - \omega t) \end{array} \right\} d\rho \\ &= \mathbb{C}\_{\mathbb{V}} \frac{k\rho\chi\pi H \vert h\_{0}(k\mathbf{k}) \right\} \frac{\sinh d - \operatorname{sph}(d-\lambda(t))}{\cosh k} \cos(k\mathbf{x}\_{1} + kR - \omega t) \end{split} \tag{A8}$$

#### *Appendix A.3 Sphere Floating Body*

The relationship between any point (*x*, *y*, *z*) on the curved surface element *ds* and the spherical coordinate [54] (*r*, *θ*, *α*) is

$$\begin{cases} \begin{array}{l} \mathfrak{T} = R \sin \theta \sin \alpha \\ \mathfrak{Y} = R \sin \theta \cos \alpha \\ \mathfrak{T} = R \cos \theta \end{array} \end{cases} \tag{A9}$$

Then, the above equation can be written in the global coordinate system is

$$\begin{cases} \begin{array}{l} \boldsymbol{x} = R\sin\theta\sin\alpha + \boldsymbol{x}\_1 + \boldsymbol{R} \\ \boldsymbol{y} = R\sin\theta\cos\alpha + \boldsymbol{y}\_1 + \boldsymbol{R} \\ \boldsymbol{z} = R\cos\theta + \boldsymbol{d} + \boldsymbol{R} - \lambda(t) \end{array} \tag{A10}$$

Let *α* = *<sup>π</sup>* <sup>2</sup> + *ϕ*, the dynamic pressure at any point on the spherical surface at this time in the global coordinate system can be written as:

$$\begin{split} P\_{\rm{dy}} &= \frac{\rho\_{\rm{R}}H}{2} \frac{\mathrm{chk}(R\cos\theta + d + R - \lambda(t))}{\mathrm{chk}d} \cos(k(R\sin\theta\sin\alpha + \mathbf{x}\_{1} + R) - \omega t) \\ &= \frac{\rho\_{\rm{R}}H}{2} \frac{\mathrm{chk}(R\cos\theta + d + R - \lambda(t))}{\mathrm{chk}d} \left\{ \begin{array}{l} \cos(kR\sin\theta\sin\phi)\cos(k\mathbf{x}\_{1} + kR - \omega t) \\ + \sin(kR\sin\theta\sin\phi)\sin(k\mathbf{x}\_{1} + kR - \omega t) \end{array} \right\} \end{split} \tag{A11}$$

According to the properties of the Bessel function [55] is as follows:

$$\begin{array}{l} \cos(kR\sin\theta\sin\varphi) = J\_0(kR\sin\theta) + 2\sum\_{m=1}^{m=-\infty} I\_{2m}(kR\sin\theta)\cos(2m\varphi) \\\sin(kR\sin\theta\sin\varphi) = 2\sum\_{m=1}^{m=-\infty} I\_{2m-1}(kR\sin\theta)\sin(2m-1)\varphi \end{array} \tag{A12}$$

When *m* = 1, and then:

$$\begin{aligned} \cos(kR\sin\theta\sin\varphi) &= J\_0(kR\sin\theta) + 2J\_2(kR\sin\theta)\cos 2\varphi\\ \sin(kR\sin\theta\sin\varphi) &= 2J\_1(kR\sin\theta)\sin\varphi \end{aligned} \tag{A13}$$

Combining above equation, we can obtain the following equation

$$P\_{\rm dy} = \frac{\rho gH}{2} \frac{c\hbar k (R\cos\theta + d + R - \lambda(t))}{c\hbar k d} \left\{ \begin{array}{c} \left( J\_0 (kR\sin\theta) + 2J\_2 (kR\sin\theta)\cos 2\rho \right) \cos(k x\_1 + kR - \omega t) \\ + (2J\_1 (kR\sin\theta)\sin\rho) \sin(k x\_1 + kR - \omega t) \end{array} \right\} \tag{A14}$$

where, *J*0(*kR* sin *θ*) is the first kind zero-order Bessel function, *J*1(*kR* sin *θ*) is the first kind one-order Bessel function, *J*2(*kR* sin *θ*) is the first kind two-order Bessel function. In order to convenient calculation, let *Jm*(*kR* sin *θ*) = *Jm*.

The wave excitation forces in the vertical direction of the sphere floating body is

$$\begin{split} F\_{\mathrm{w}}^{z} &= 2\mathsf{C}\_{\mathrm{J}} \int\_{\pi-\mathrm{arcc}\cos\left(\frac{(R-\lambda)\ell}{\hbar}\right)}^{\pi} \int\_{0}^{\pi} p\_{z} \cos\theta R^{2} \sin\theta dad\theta\\ &= \mathsf{C}\_{\mathrm{w}} \cdot \frac{R^{2}\rho\_{\mathrm{g}}H}{\mathrm{cch}\hbar} \cdot \int\_{\pi-\mathrm{arcc}\cos\left(\frac{R-\lambda}{\hbar}\right)}^{\pi} \int\_{-\frac{\pi}{\sqrt{2}}}^{\frac{\pi}{\sqrt{2}}} \mathrm{cch}(R\cos\theta + d + R - \lambda(t))\\ &\cdot \left\{ \left(I\_{0} + 2I\_{2}\cos 2\varrho\right) \cos(k\mathbf{x}\_{1} + kR - \omega t) \right.\\ &\left. + 2I\_{1}\sin\varrho\sin(k\mathbf{x}\_{1} + kR - \omega t) \right.\\ &= \mathsf{C}\_{\mathrm{w}} \cdot \frac{\pi R^{2}\rho\_{\mathrm{g}}H}{\mathrm{cch}\hbar} \cdot \left(I\_{1}(kR) - I\_{2}(kR)\right) \cdot \cos(k\mathbf{x}\_{1} + kR - \omega t) \end{split} \tag{A15}$$

where, *<sup>I</sup>*1(*kR*) = *<sup>π</sup> <sup>π</sup>*−arccos( *<sup>R</sup>*−*λ*(*t*) *<sup>R</sup>* ) *ch*(*kR* cos *<sup>θ</sup>*)*chk*(*<sup>d</sup>* <sup>+</sup> *<sup>R</sup>* <sup>−</sup> *<sup>λ</sup>*(*t*)) · *<sup>J</sup>*0(*kR* sin *<sup>θ</sup>*) · cos *<sup>θ</sup>* sin *<sup>θ</sup>dθ*, *<sup>I</sup>*2(*kR*) = *<sup>π</sup> <sup>π</sup>*−arccos( *<sup>R</sup>*−*λ*(*t*) *<sup>R</sup>* ) *sh*(*kR* cos *<sup>θ</sup>*)*shk*(*<sup>d</sup>* <sup>+</sup> *<sup>R</sup>* <sup>−</sup> *<sup>λ</sup>*(*t*)) · *<sup>J</sup>*0(*kR* sin *<sup>θ</sup>*) · cos *<sup>θ</sup>* sin *<sup>θ</sup>dθ*.

#### **References**


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