**9. Results and Discussion**

Further hydrodynamic analyses have been carried out to investigate various effects due to drafts, the radii of platforms and polygonal shapes. Floating hexagonal platform and ring structure that oscillate individually are considered. The drafts for two individual floating bodies are respectively taken as: (1) *d*1/*h* = 0.1, *d*2/*h* = 0.1, (2) *d*1/*h* = 0.1, *d*2/*h* = 0.2, (3) *d*1/*h* = 0.2, *d*2/*h* = 0.1 and (4) *d*1/*h* = 0.2, *d*2/*h* = 0.2. The wave exciting forces and RAOs are given in Figure 7 while the added mass and radiation damping are presented in Figure 8. Note that "P" and "R" used in legends stand for "Platform" and "Ring", respectively. Overall, the ring structure shows greater dominance in surge and pitch forces while the platform shows dominance in heave forces as shown in Figure 7 as the ring structure primarily prevents wave propagation. For longer waves (say *kh* < 2), as more waves are transmitted into the inner water basin, they become trapped; thereby, creating a high energetic environment within the ring structure. Accordingly, this aggravates the hydrodynamic interaction on the platform. However, the situations are reversed when considering the RAOs for heave and pitch (see Figure 7e,f). Similar phenomena are observed for added mass and radiation damping in the case of heave (see Figure 8b,f). In general, when the ratio of the draft-to-water depth of the ring structure (i.e., *d*2/*h*) is the same, the

hydrodynamic results such as the wave exciting forces, RAOs, added mass and radiation damping associated with the ring structure are very close to each other regardless of the ratio of the draft-to-water depth of the platform (i.e., *d*1/*h*). Likewise, when *d*1/*h* of the platform is the same, the hydrodynamic results associated with the platform are similarly irrelevant to *d*2/*h*.

**Figure 7.** Normalised wave exciting forces and RAOs for floating hexagonal platforms and ring structures with various draft-to-water depth ratios: (**a**) surge force; (**b**) heave force; (**c**) pitch moment; (**d**) RAO for surge; (**e**) RAO for heave; (**f**) RAO for pitch.

**Figure 8.** Normalised added mass and radiation damping for floating hexagonal platforms and ring structures with various draft-to-water depth ratios: (**a**) added mass for surge; (**b**) added mass for heave; (**c**) added mass for pitch; (**d**) added mass for surge-pitch; (**e**) radiation damping for surge; (**f**) radiation damping for heave; (**g**) radiation damping for pitch; (**h**) radiation damping for surge-pitch.

In Figure 9, the wave exciting forces and RAOs for floating circular platform and ring structure are presented for various radii of the circular platform, i.e., *R*1/*h* = 0.5, 1, 1.5. It can be seen that the wave exciting forces increase with increasing *R*1/*h*. On the other hand, the RAOs for the platform decrease with increasing *R*1/*h*, whereas the RAOs for the ring structure do not show a distinctive relationship. This implies that a large working platform placed within a floating ring breakwater is relatively stable in terms of motions when compared with a small platform. On the other hand, when a small oscillating cylindrical platform is deployed within a floating ring structure, relatively large kinetic energy can be obtained.

**Figure 9.** Normalised wave exciting forces and RAOs for floating circular platform and circular ring structure with *R*1/*h* = 0.5, 1, 1.5: (**a**) surge force; (**b**) heave force; (**c**) pitch moment; (**d**) RAO for surge; (**e**) RAO for heave; (**f**) RAO for pitch.

Next, the free water surface elevation in the inner water basin will be investigated for the various geometries of floating polygonal platform and ring structure. Added mass, radiation damping and RAOs do not represent significant differences for similar sizes of plan shapes as when their geometries are created by using the radius function as given in Equation (1), the plan shapes for every kind of polygons are almost the same as each other. In addition, as far as the wave exciting forces are concerned, the horizontal forces such as surge or sway forces are distinctively changed when polygonal shapes are appropriately used with their orientations. This phenomenon was already reported by Park and Wang [20]. Hence, in this parametric study, the free water surface elevation in the inner water basin will be the focus.

The floating circular platform and circular ring structure are initially considered to investigate the inner wave fields. The draft-to-water depth ratio for the circular floating bodies *d*1/*h* = 0.1, 0.2 and *d*2/*h* = 0.1, 0.2 are combined into 4 cases. The inner wave fields at significant resonance frequencies are presented in Figure 10 by assuming the water depth *h* = 50 m. The maximum value of the scale bar is set to 4; implying wave fields larger than 4 are included in the maximum value (yellow colour). It can be learned that when larger *d*<sup>2</sup> and smaller *d*<sup>1</sup> are considered, there will be more waves trapped in the inner water basin, resulting in higher free water surface elevation (see Figure 10b) and vice versa. Thus, Figure 10b may be used for harvesting more wave energy, whereas Figure 10c is used for creating a calm patch of water space.

**Figure 10.** Normalised inner wave fields for individually oscillating floating circular platforms and ring structures at significant resonance frequencies: (**a**) *d*1/*h* = 0.1, *d*2/*h* = 0.1 (*kh* = 7.5); (**b**) *d*1/*h* = 0.1, *d*2/*h* = 0.2 (*kh* = 5.5); (**c**) *d*1/*h* = 0.2, *d*2/*h* = 0.1 (*kh* = 7.5); (**d**) *d*1/*h* = 0.2, *d*2/*h* = 0.2 (*kh* = 3.75).

In order to investigate the shape effect on the inner wave fields, several combinations of different polygonal shapes such as a square, hexagon and circle are considered for

*d*1/*h* = *d*2/*h* = 0.2 by assuming the water depth *h* = 50 m. Normalised inner wave fields for various floating polygonal platforms and ring structures at significant resonance frequencies are presented in Figure 11. By referring to Figure 10d together with Figure 11, it can be found that the inner wave fields at the significant resonance frequency for circular shapes tend to spread waves to multiple directions (see Figure 10d). For the square shapes, the inner wave fields are mild when compared with other cases (see Figure 11a) and amplified waves appeared to be in one direction (see Figure 11c,e). For the hexagonal shapes, waves are more amplified than the circular shapes and their propagations are in multiple directions. This implies that the square shapes are beneficial in terms of creating a calm patch of water space, whereas the hexagonal shapes are more advantageous for wave energy harvesting.

**Figure 11.** Normalised inner wave fields for various combinations of individually oscillating floating

platforms and ring structures at significant resonance frequencies: (**a**) square platform and square ring (*kh* = 3.75); (**b**) hexagonal platform and hexagonal ring (*kh* = 5.25); (**c**) circular platform and square ring (*kh* = 5.25); (**d**) circular platform and hexagonal ring (*kh* = 4.5); (**e**) square platform and circular ring (*kh* = 4.75); (**f**) hexagonal platform and circular ring (*kh* = 4.75).

#### **10. Concluding Remarks**

A semi-analytical method and computer code have been developed for the hydrodynamic analysis of a floating regular polygonal platform that is centrally placed within a regular polygonal ring structure under wave action. The polygonal shapes for equilateral triangle, square, pentagonal, hexagonal platforms and ring structures can be readily generated by using the cosine-type radial perturbation. This shape function is the key to enabling the problem to be solved semi-analytically. Cases involving the two floating bodies oscillating together or individually were considered with the view to understanding the hydrodynamic interactions among the trapped waves, inner platform and outer ring structure. The method has been shown to be able to furnish accurate hydrodynamic quantities such as wave exciting forces, added mass, radiation damping, RAOs, and wave field. The computational speed has been significantly quickened when compared with numerical methods because of the semi-analytical method.

The effects of several parameters such as drafts, radii of platforms and polygonal shapes on major hydrodynamic quantities are investigated by performing parametric studies. When the draft of the ring structure is larger than the floating platform, trapped waves are amplified more in the inner water basin. Additionally, the wave exciting forces increase with increasing radii of platforms. However, the RAOs decrease with the increasing radii of platforms. The inner wave fields for circular shapes tend to spread waves to multiple directions, whereas those for square shapes are relatively mild and their amplified waves are apt to propagate in one direction. For hexagonal shapes, the wave fields are more amplified than other considered shapes and the waves' propagations are in multiple directions. In sum, floating square platform and square ring structure are beneficial for creating a calm patch of water space, whereas the floating hexagonal platform and the hexagonal ring structure are more advantageous for wave energy harvesting.

**Author Contributions:** Conceptualization, J.C.P. and C.M.W.; methodology, J.C.P.; software, J.C.P.; validation, J.C.P. and C.M.W.; formal analysis, J.C.P.; investigation, J.C.P. and C.M.W.; writing original draft preparation, J.C.P.; writing—review and editing, C.M.W.; visualization, J.C.P.; supervision, C.M.W.; funding acquisition, C.M.W.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (project DP170104546). The views expressed herein are those of the authors and are not necessarily those of the Australian Government or Australian Research Council.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

#### **Appendix A**

The divergences between the generalised motion normal **n***<sup>j</sup>* and unit normal vector **n***sl* (*l* = 1, 2, 3) at *r* = *Rl*(*θ*) are calculated for 6 DOFs in the Cartesian coordinate system as given by

$$\mathbf{n}\_{\mathbf{1}} \cdot \mathbf{n}\_{s\_l} = \mathbf{n}\_{\mathbf{x}} \cdot \mathbf{n}\_{s\_l} = \frac{\cos \theta - \frac{\sin \theta}{R\_l} S\_{l,\theta}}{\sqrt{1 + \frac{S\_{l,\theta}^2}{R\_l^2}}} \tag{A1}$$

$$\mathbf{n\_2} \cdot \mathbf{n\_{s\_l}} = \mathbf{n\_y} \cdot \mathbf{n\_{s\_l}} = \frac{\sin \theta + \frac{\text{QTS} \cdot \theta}{R\_l} S\_{l,\theta}}{\sqrt{1 + \frac{S\_{l,\theta}^2}{R\_l^2}}} \tag{A2}$$

$$\mathbf{n} \mathbf{g} \cdot \mathbf{n}\_{\mathbb{sl}} = \mathbf{n}\_z \cdot \mathbf{n}\_{\mathbb{sl}} = 0 \tag{A3}$$

$$\mathbf{n}\_{\mathsf{F}} \cdot \mathbf{n}\_{\mathsf{S}} = -(z - z\_{\mathsf{G}}) \mathbf{n}\_{\mathsf{Y}} \cdot \mathbf{n}\_{\mathsf{S}\_{l}} + (y - y\_{\mathsf{G}}) \mathbf{n}\_{\mathsf{z}} \cdot \mathbf{n}\_{\mathsf{S}\_{l}} = -\frac{(z - z\_{\mathsf{G}}) \left(\sin \theta + \frac{\cos \theta}{R\_{l}} S\_{l,\theta}\right)}{\sqrt{1 + \frac{S\_{l,\theta}^{2}}{R\_{l}^{2}}}} \tag{A4}$$

$$\mathbf{n}\_{\mathbf{S}} \cdot \mathbf{n}\_{\mathbf{s}} = (z - z\_{\mathbf{G}}) \mathbf{n}\_{\mathbf{x}} \cdot \mathbf{n}\_{\mathbf{s}\_l} - (\mathbf{x} - \mathbf{x}\_{\mathbf{G}}) \mathbf{n}\_{\mathbf{z}} \cdot \mathbf{n}\_{\mathbf{s}\_l} = \frac{(z - z\_{\mathbf{G}}) \left(\cos \theta - \frac{\sin \theta}{R\_l} S\_{I,\theta}\right)}{\sqrt{1 + \frac{S\_{I,\theta}^2}{R\_l^2}}} \tag{A5}$$

$$\mathbf{n}\_{\mathbf{6}} \cdot \mathbf{n}\_{s\_l} = -\left(y - y\_G\right) \mathbf{n}\_x \cdot \mathbf{n}\_{s\_l} + \left(\mathbf{x} - \mathbf{x}\_G\right) \mathbf{n}\_{y'} \cdot \mathbf{n}\_{s\_l} = \frac{S\_{1\theta} + \left(y\mathbf{c} - \mathbf{x}\_G\right)\left(\cos\theta + \sin\theta - \frac{\sin\theta - \cos\theta}{R\_1}S\_{1\theta}\right)}{\sqrt{1 + \frac{S\_{1\theta}}{R\_1^2}}}\tag{A6}$$

If *Sl*,*<sup>θ</sup>* = 0, it can be applied to a circular platform or ring breakwater.

#### **Appendix B**

The reduced expressions *<sup>H</sup>*(*j*) *<sup>l</sup>* , *<sup>P</sup>*(*j*) *<sup>l</sup>* , *<sup>Q</sup>*(*j*) *l* , ∼ *P* (*j*) *<sup>l</sup>* and <sup>∼</sup> *Q* (*j*) *<sup>l</sup>* (*j* = 0, 1, ··· , 6; *l* = 1, 2, 3) introduced in Equations (31)–(39) and Equations (50)–(58) are given by

$$\begin{split} \left. \begin{array}{ccc} H\_{l}^{(0)} & = \frac{igA}{\omega \cosh \Omega} \sum\_{m = -\infty}^{\infty} \mathcal{E}^{(m)} \frac{(\vec{\mathbb{P}} - \boldsymbol{\mathcal{P}})}{m \eta} \mathcal{J}\_{m0}^{\prime} (k R\_{l}) \epsilon^{\mathrm{mid}m} \cosh k \left( z + h \right) & \text{for } \boldsymbol{k} = \boldsymbol{k} \right) \\ & = \frac{igA}{\omega \cosh \Omega} \sum\_{m = -\infty}^{\infty} \mathcal{E}^{(m)} \frac{(\vec{\mathbb{P}} - \boldsymbol{\mathcal{P}})}{q} \sum\_{q = -\infty}^{\infty} \mathcal{E}^{(q)} \frac{\omega}{q + \cos \theta} \sum\_{r = -\infty}^{\infty} \left[ \frac{1}{2} b\_{2,0,r}^{(l)} \left\{ f\_{m - 1, q}^{(l)} - f\_{m + 1, q}^{(l)} \right\} + \text{unu}\_{l} b\_{1,0,u}^{(l)} f\_{m, q}^{(l)} \right] \epsilon^{\mathrm{(m + q + u, q)}} \cosh k \left( z + h \right) \end{split} \tag{A7}$$

$$H\_l^{(1)} = R\_l^2 \cos \theta - R\_l S\_{l,\theta} \sin \theta = \frac{1}{4} \sum\_{n\_r = -\infty}^{\infty} \left\{ (n\_r + 1) b\_{2,0,n\_r - 1}^{(l)} - (n\_r - 1) b\_{2,0,n\_r + 1}^{(l)} \right\} e^{in\_r \theta} \tag{A8}$$

$$H\_l^{(2)} = R\_l^2 \sin \theta + R\_l S\_{l,\theta} \cos \theta = \frac{1}{4i} \sum\_{n\_r = -\infty}^{\infty} \left\{ (n\_r + 1)b\_{2,0,n\_r - 1}^{(l)} + (n\_r - 1)b\_{2,0,n\_r + 1}^{(l)} \right\} e^{in\_r \theta} \tag{A9}$$

$$H\_l^{(3)} = 0\tag{A10}$$

$$\begin{split} H\_{l}^{(4)} &= -\left(z - z\_{G\_{\mathcal{P}}}\right) \left(R\_{l}^{2} \sin \theta + R\_{l} S\_{l, \theta} \cos \theta\right) \\ &= -\frac{\left(z - z\_{G\_{\mathcal{P}}}\right)}{4i} \sum\_{n\_{r} = -\infty}^{\infty} \left\{ (n\_{r} + 1) b\_{2, 0, n\_{r} - 1}^{(I)} + (n\_{r} - 1) b\_{2, 0, n\_{r} + 1}^{(I)} \right\} e^{in\_{r} \theta} \end{split} \tag{A11}$$

$$\begin{array}{l} H\_{l}^{(5)} = \left(z - z\_{G\_{\mathcal{P}}}\right) \left(R\_{l}^{2} \cos\theta - R\_{l} S\_{l,\theta} \sin\theta\right) \\ = \frac{\left(z - z\_{G\_{\mathcal{P}}}\right)}{4} \sum\_{n\_{r} = -\infty}^{\infty} \left\{ (n\_{r} + 1) b\_{2,0,n\_{r}-1}^{(l)} - (n\_{r} - 1) b\_{2,0,n\_{r}+1}^{(l)} \right\} e^{in\_{r}\theta} \end{array} \tag{A12}$$

$$H\_l^{(6)} = R\_l^2 S\_{l, \theta} = -\frac{i n\_r}{3} \sum\_{n\_r = -\infty}^{\infty} b\_{3, 0, n\_r}^{(l)} e^{i n\_r \theta} \tag{A13}$$

where *Sl*,*<sup>θ</sup>* denotes *<sup>∂</sup>Sl ∂θ* ( ( ( *r*=*Rl*(*θ*) and *zGp* the z-coordinate of the centre of gravity for the oscillating body *p*.

$$\begin{split} & \overset{(b)}{P}\_{l}^{(\beta)} - \frac{igA}{\omega \cosh \hbar k} \sum\_{m=-\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} \mathcal{J}\_{m0}^{\prime} (k R\_{l}) e^{im\theta} \cosh k (z + h) \\ & - \frac{igA}{\omega \cosh \hbar k} \sum\_{m=-\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} \sum\_{q=-\infty}^{\infty} \sum\_{m=-\infty}^{\infty} \left[ \frac{k}{2} b\_{2, \hbar v\_{\neq}}^{(l)} \left\{ f\_{m-1, q}^{(l)} - f\_{m+1, q}^{(l)} \right\} + m n\_{1} b\_{1, 0, \omega\_{\neq}}^{(l)} f\_{m, q}^{(l)} \right] e^{i (m + q + n\_{\theta})t} \cosh k (z + h) \end{split} \tag{A14}$$

$$\stackrel{\sim}{P}\_l^{(3)} = -\frac{R\_l^3}{2(h - d\_1)} = -\frac{1}{2(h - d\_1)} \sum\_{n\_r = -\infty}^{\infty} b\_{3,0,n\_r}^{(l)} e^{in\_r \theta} \tag{A15}$$

$$\begin{split} \widehat{P}\_{l}^{(4)} &= \frac{1}{8(h-d)} \left\{ 4(z+h)^{2} \left( R\_{l}^{2} \sin \theta + R\_{l} S\_{l,\theta} \cos \theta \right) - \left( 3R\_{l}^{4} \sin \theta + R\_{l}^{3} \cos \theta S\_{l,\theta} \right) \right\} \\ &= -\frac{i}{64(h-d)} \sum\_{n\_{r}=-\infty}^{\infty} \left[ 8(z+h)^{2} \Big\{ (n\_{r}+1)b\_{2,0,r-1}^{(l)} + (n\_{r}-1)b\_{2,0,r+1}^{(l)} \right\} \\ &- \left\{ (n\_{r}+11)b\_{4,0,n\_{r}-1}^{(l)} + (n\_{r}-11)b\_{4,0,n\_{r}+1}^{(l)} \right\} e^{in\theta} \\ &\overset{(5)}{P}\_{l}^{(5)} = -\frac{1}{8(h-d)} \left\{ 4(z+h)^{2} \Big( R\_{l}^{2} \cos \theta - R\_{l} S\_{l,0} \sin \theta \right) - \left( 3R\_{l}^{4} \cos \theta - R\_{l}^{3} \sin \theta S\_{l,\theta} \right) \right\} \\ &= -\frac{1}{64(h-d\_{1})} \sum\_{n\_{r}=-\infty}^{\infty} \left[ 8(z+h)^{2} \Big\{ (n\_{r}+1)b\_{2,0,n\_{r}-1}^{(l)} - (n\_{r}-1)b\_{2,0,n\_{r}+1}^{(l)} \right\} \\ &- \Big\{ (n\_{r}+11)b\_{4,0,n\_{r}-1}^{(l)} - (n\_{r}-11)b\_{4,0,n\_{r}+1}^{(l)} \Big\} e^{in\theta} \end{split} \tag{A17}$$

$$
\stackrel{\frown}{P}\_I^{(1)} = \stackrel{\frown}{P}\_I^{(2)} = \stackrel{\frown}{P}\_I^{(6)} = 0 \tag{A18}
$$

$$\begin{split} P\_l^{(0)} &= \frac{i\chi A}{\omega \cosh kl} \sum\_{m = -\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} f\_m(kR\_l) e^{im\theta} \cosh k(z + h) \\ &= \frac{i\chi A}{\omega \cosh kl} \sum\_{m = -\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} \sum\_{q = -\infty}^{\infty} f\_{m,q}^{(l)} e^{i(m+q)\theta} \cosh k(z + h) \end{split} \tag{A19}$$

$$P\_l^{(3)} = \frac{2\left(z+h\right)^2 - R\_l^2}{4\left(h-d\_1\right)} = \frac{\left(z+h\right)^2}{2\left(h-d\right)} - \frac{1}{4\left(h-d\right)}\sum\_{n\_l=-\infty}^{\infty} b\_{2,0,n\_l}^{(l)} e^{in\_l\theta} \tag{A20}$$

$$\begin{split} P\_l^{(4)} &= \frac{4(z+h)^2 - R\_l^2}{8(h-d\_1)} R\_l \sin \theta \\ &= -\frac{i}{16(h-d\_1)} \sum\_{n\_l=-\infty}^{\infty} \left[ 4(z+h)^2 \left\{ b\_{1,0,n\_l-1}^{(l)} - b\_{1,0,n\_l+1}^{(l)} \right\} - \left\{ b\_{3,0,n\_l-1}^{(l)} - b\_{3,0,n\_l+1}^{(l)} \right\} \right] e^{in\vartheta} \end{split} \tag{A21}$$

$$\begin{split} P\_l^{(5)} &= -\frac{4(z+h)^2 - R\_l^2}{8(h-d\_1)} R\_l \cos\theta \\ &= -\frac{1}{16(h-d\_1)} \sum\_{n\_\ell=-\infty}^{\infty} \left[ 4(z+h)^2 \left\{ b\_{1,0,n\_\ell-1}^{(l)} + b\_{1,0,n\_\ell+1}^{(l)} \right\} - \left\{ b\_{3,0,n\_\ell-1}^{(l)} + b\_{3,0,n\_\ell+1}^{(l)} \right\} \right] e^{in\eta\theta} \end{split} \tag{A.22}$$

$$P\_l^{(1)} = P\_l^{(2)} = P\_l^{(6)} = 0 \tag{A23}$$

$$\begin{cases} \stackrel{\star}{Q}\_{l}^{(0)} = \frac{igA}{\omega \cosh kh} \sum\_{m=-\infty}^{\infty} \epsilon^{jm(\frac{\pi}{2} - \theta)} \mathcal{J}\_{m0}^{\ell}(k\ell\_{l}) \epsilon^{im\theta} \cosh k(z + h) \\\ \stackrel{\star}{Q}\_{l}^{(1)} = \frac{igA}{\omega \cosh kh} \sum\_{m=-\infty}^{\infty} \epsilon^{jm(\frac{\pi}{2} - \theta)} \sum\_{q=-\infty}^{\infty} \sum\_{n=-\infty}^{\infty} \left[ \frac{h}{2} b\_{2,0n,q}^{(l)} \left\{ f\_{m-1,q}^{(l)} - f\_{m+1,q}^{(l)} \right\} + mn\_{l} b\_{1,0,n\_{r}}^{(l)} f\_{m,q}^{(l)} \right] \epsilon^{j(m+q+n\_{r})t} \cosh k(z + h) \end{cases} \tag{A.24}$$

$$\stackrel{\sim}{Q}\_{l}^{(3)} = -\frac{R\_{l}^{3}}{2(h - d\_{2})} = -\frac{1}{2(h - d\_{2})} \sum\_{n\_{r} = -\infty}^{\infty} b\_{3,0,n\_{r}}^{(l)} e^{in\_{r}\theta} \tag{A25}$$

$$\begin{split} \overset{\star}{Q}\_{l}^{(4)} &= \frac{1}{6(h-d\_{2})} \Big\{ 4(z+h)^{2} \Big( R\_{l}^{2} \sin \theta + R\_{l} S\_{l,0} \cos \theta \Big) - \left( 3R\_{l}^{4} \sin \theta + R\_{l}^{3} \cos \theta \mathbf{S}\_{l,0} \right) \Big\} \\ &- \Big\{ -\epsilon \frac{i}{6h(h-d\_{2})} \sum\_{n\_{r}=-\infty}^{\infty} \Big[ 8(z+h)^{2} \Big\{ (n\_{r}+1)b\_{2,0,n\_{r}-1}^{(l)} + (n\_{r}-1)b\_{2,0,n\_{r}+1}^{(l)} \Big\} - \left\{ (n\_{r}+11)b\_{4,0,n\_{r}-1}^{(l)} + (n\_{r}-11)b\_{4,0,n\_{r}+1}^{(l)} \right\} \Big] e^{ic\_{0}\theta} \end{split} \tag{A.26}$$

$$\begin{cases} \overset{\sim}{Q}\_{l}^{(5)} = -\frac{1}{8(h-d\_{2})} \left\{ 4(z+h)^{2} \left( R\_{l}^{2} \cos \theta - R\_{l} S\_{l,\theta} \sin \theta \right) - \left( 3R\_{l}^{4} \cos \theta - R\_{l}^{3} \sin \theta S\_{l,\theta} \right) \right\} \\ = -\frac{1}{64(h-d\_{2})} \sum\_{n\_{r}=-\infty}^{\infty} \left[ 8(z+h)^{2} \left\{ (n\_{r}+1)b\_{2,0,n\_{r}-1}^{(l)} - (n\_{r}-1)b\_{2,0,n\_{r}+1}^{(l)} \right\} \right. \\ \left. - \left\{ (n\_{l}+11)b\_{4,0,n\_{r}-1}^{(l)} - (n\_{r}-11)b\_{4,0,n\_{r}+1}^{(l)} \right\} \right] e^{in\_{l}\theta} \end{cases} \tag{A27}$$

$$
\stackrel{\curvearrowleft}{Q}\_{l}^{(1)} = \stackrel{\curvearrowleft}{Q}\_{l}^{(2)} = \stackrel{\curvearrowleft}{Q}\_{l}^{(6)} = 0 \tag{A28}
$$

$$\begin{split} Q\_{l}^{(0)} &= \frac{i\underline{\chi}A}{\omega \cosh kl} \sum\_{m = -\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} J\_{m}(kR\_{l}) e^{im\theta} \cosh k(z + h) \\ &= \frac{i\underline{\chi}A}{\omega \cosh kl} \sum\_{m = -\infty}^{\infty} e^{im(\frac{\pi}{2} - \beta)} \sum\_{q = -\infty}^{\infty} f\_{m,q}^{(l)} e^{i(m + q)\theta} \cosh k(z + h) \end{split} \tag{A29}$$

$$Q\_l^{(3)} = \frac{2(z+h)^2 - R\_l^2}{4(h-d\_2)} = \frac{(z+h)^2}{2(h-d\_2)} - \frac{1}{4(h-d\_2)} \sum\_{n\_r=-\infty}^{\infty} b\_{2,0,n\_r}^{(l)} e^{in\_r \theta} \tag{A30}$$

$$\begin{split} Q\_{l}^{(4)} &= \frac{4(z+h)^{2} - R\_{l}^{2}}{8(h-d\_{2})} R\_{l} \sin \theta \\ &= -\frac{i}{16(h-d\_{2})} \sum\_{n\_{l}=-\infty}^{\infty} \left[ 4(z+h)^{2} \left\{ b\_{1,0,n\_{l}-1}^{(l)} - b\_{1,0,n\_{l}+1}^{(l)} \right\} - \left\{ b\_{3,0,n\_{l}-1}^{(l)} - b\_{3,0,n\_{l}+1}^{(l)} \right\} \right] e^{in\_{l}\theta} \end{split} \tag{A31}$$

$$\begin{split} \mathbf{Q}\_{l}^{(5)} &= -\frac{4(z+h)^{2} - \mathbf{R}\_{l}^{2}}{\delta(h-d\_{2})} \mathbf{R}\_{l} \cos\theta \\ &= -\frac{1}{16(h-d\_{2})} \sum\_{u\_{l}=-\infty}^{\infty} \left[ 4(z+h)^{2} \left\{ b\_{1,0,n\_{l}-1}^{(l)} + b\_{1,0,n\_{l}+1}^{(l)} \right\} - \left\{ b\_{3,0,n\_{l}-1}^{(l)} + b\_{3,0,n\_{l}+1}^{(l)} \right\} \right] e^{in\eta\theta} \end{split} \tag{A32}$$

$$\mathbf{Q}\_{l}^{(1)} = \mathbf{Q}\_{l}^{(2)} = \mathbf{Q}\_{l}^{(6)} = \mathbf{0} \tag{A33}$$

#### **Appendix C**

The reduced expressions <sup>∼</sup> *a* (1) *<sup>m</sup>*,*n*,*q*, ∼ *b* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *c* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *d* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *e* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *f* (3) *<sup>m</sup>*,*n*,*q*, ∼ *a* (1) *<sup>m</sup>*,*n*,*q*, ∼ *b* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *c* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *d* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *e* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* and <sup>∼</sup> *f* (3) *<sup>m</sup>*,*n*,*<sup>q</sup>* introduced in Equations (50)–(58) are given by

$$\stackrel{\sim}{a}\_{m,0,\emptyset}^{(1)} = \frac{a\_{m,0,\emptyset}^{(1)}}{b\_1^{|m|}} \ (n=0), \quad \stackrel{\sim}{a}\_{m,n,\emptyset}^{(1)} = \frac{a\_{m,n,\emptyset}^{(1)}}{I\_m(p\_n b\_1)} \ (n>0) \tag{A34}$$

$$\widetilde{a}\_{m,0,q}^{'(1)} = \frac{1}{b\_1^{|m|}} \left( |m| a\_{1,0,n\_r}^{(1)} a\_{m,0,q}^{(1)} + m n\_r a\_{1,0,n\_r}^{(1)} a\_{m,0,q}^{(1)} \right) (n=0) \tag{A35}$$

$$\widehat{a}\_{m,n,\boldsymbol{\eta}}^{'(1)} = \frac{1}{I\_m(p\_n b\_1)} \left\{ \frac{p\_n}{2} a\_{2,0,n\_\ell}^{(1)} \left( a\_{m-1,n,\boldsymbol{\eta}}^{(1)} + a\_{m+1,n,\boldsymbol{\eta}}^{(1)} \right) + m m\_\ell a\_{1,0,n\_\ell}^{(1)} a\_{m,n,\boldsymbol{\eta}}^{(1)} \right\} (n > 0) \tag{A36}$$

$$\stackrel{\sim}{b}\_{m,0,q}^{(1,2)} = b\_{m,0,q}^{(1,2)} \ (n=0), \quad \stackrel{\sim}{b}\_{m,n,q}^{(1,2)} = \frac{b\_{m,n,q}^{(1,2)}}{I\_m(k\_nb\_2)} \ (n>0) \tag{A37}$$

$$
\widetilde{b}\_{m,0,q}^{'(1)} = \frac{k}{2} a\_{2,0,n\_r}^{(1)} \left( b\_{m-1,0,q}^{(1)} - b\_{m+1,0,q}^{(1)} \right) + m n\_r a\_{1,0,n\_r}^{(1)} b\_{m,0,q}^{(1)} \quad (n=0) \tag{A38}
$$

$$
\stackrel{\sim}{b}\_{m,0,q}^{\prime\prime(2)} = \frac{k}{2} d\_{2,0,n\_r}^{(2)} \left( b\_{m-1,0,q}^{(2)} - b\_{m+1,0,q}^{(2)} \right) + m n\_r d\_{1,0,n\_r}^{(2)} b\_{m,0,q}^{(2)} \quad (n=0) \tag{A39}
$$

$$\stackrel{\sim}{b}\_{m,n,\boldsymbol{\eta}}^{\prime(1)} = \frac{1}{I\_m(k\_nb\_2)} \left\{ \frac{k\_\Pi}{2} a\_{2,0,n\_\boldsymbol{\eta}}^{(1)} \left( b\_{m-1,n,\boldsymbol{\eta}}^{(1)} + b\_{m+1,n,\boldsymbol{\eta}}^{(1)} \right) + m n\_\boldsymbol{\eta} a\_{1,0,n\_\boldsymbol}^{(1)} b\_{m,n,\boldsymbol{\eta}}^{(1)} \right\} \quad (\boldsymbol{n} > 0) \tag{A40}$$

$$\widetilde{\boldsymbol{b}}\_{m,n,\boldsymbol{\eta}}^{\prime\prime} = \frac{1}{I\_m(k\_n b\_2)} \left\{ \frac{k\_n}{2} d\_{2,0,n\_r}^{(2)} \left( b\_{m-1,n,\boldsymbol{\eta}}^{(2)} + b\_{m+1,n,\boldsymbol{\eta}}^{(2)} \right) + m n\_r d\_{1,0,n\_r}^{(2)} b\_{m,n,\boldsymbol{\eta}}^{(2)} \right\} \tag{A41}$$

$$
\stackrel{\sim}{\hat{c}}\_{m,0,\emptyset}^{(1,2)} = \frac{c\_{m,0,\emptyset}^{(1,2)}}{H\_m(ka\_1)} \ (n=0), \quad \stackrel{\sim}{\hat{c}}\_{m,n,\emptyset}^{(1,2)} = \frac{c\_{m,n,\emptyset}^{(1,2)}}{K\_m(k\_na\_1)} \ (n>0) \tag{A42}
$$

$$\mathcal{L}\_{m,0,q}^{'(1)} = \frac{1}{H\_m(ka\_1)} \left\{ \frac{k}{2} a\_{2,0,n\_r}^{(1)} \left( c\_{m-1,0,q}^{(1)} - c\_{m+1,0,q}^{(1)} \right) + m n\_r a\_{1,0,n\_r}^{(1)} c\_{m,0,q}^{(1)} \right\} \tag{A43}$$

$$c\_{m,0,q}^{'(2)} = \frac{1}{H\_m(ka\_1)} \left\{ \frac{k}{2} d\_{2,0,n\_r}^{(2)} \left( c\_{m-1,0,q}^{(2)} - c\_{m+1,0,q}^{(2)} \right) + m m\_r a\_{1,0,n\_r}^{(2)} c\_{m,0,q}^{(2)} \right\} \tag{A44}$$

$$\boldsymbol{\hat{\mathcal{L}}}\_{m,n,\boldsymbol{\eta}}^{\prime} = \frac{1}{K\_{\boldsymbol{m}}(k\_{\boldsymbol{n}}\boldsymbol{a}\_{1})} \left\{ -\frac{k\_{\boldsymbol{n}}}{2} a\_{2,0,\boldsymbol{n}\_{\boldsymbol{r}}}^{(1)} \left( \boldsymbol{c}\_{m-1,\boldsymbol{n},\boldsymbol{\eta}}^{(1)} + \boldsymbol{c}\_{m+1,\boldsymbol{n},\boldsymbol{\eta}}^{(1)} \right) + m n\_{\boldsymbol{r}} a\_{1,0,\boldsymbol{n}\_{\boldsymbol{r}}}^{(1)} \boldsymbol{c}\_{m,\boldsymbol{n},\boldsymbol{\eta}}^{(1)} \right\} \tag{A45}$$

$$\widetilde{\boldsymbol{c}}\_{m,n,\boldsymbol{\eta}}^{\prime\prime} = \frac{1}{K\_m(k\_n a\_1)} \left\{ -\frac{k\_n}{2} d\_{2,0,n\_r}^{(2)} \left( \boldsymbol{c}\_{m-1,n,\boldsymbol{\eta}}^{(2)} + \boldsymbol{c}\_{m+1,n,\boldsymbol{\eta}}^{(2)} \right) + m n\_r d\_{1,0,n\_r}^{(2)} \boldsymbol{c}\_{m,n,\boldsymbol{\eta}}^{(2)} \right\} \tag{A46}$$

$$\stackrel{\sim}{d}\_{0,0,\emptyset}^{(2,3)} = d\_{0,0,\emptyset}^{(2,3)} \left( m = 0, n = 0 \right), \quad \stackrel{\sim}{d}\_{m,0,\emptyset}^{(2,3)} = \frac{d\_{m,0,\emptyset}^{(2,3)}}{b\_3^{|m|}} \left( m \neq 0, n = 0 \right) \tag{A47}$$

$$\stackrel{\sim}{d}\_{m,n,q}^{(2,3)} = \frac{d\_{m,n,q}^{(2,3)}}{I\_m(q\_n b\_3)} \text{ ( $n > 0$ )}\tag{A48}$$

$$\stackrel{\sim}{d}\_{0,0,\emptyset}^{(2,3)} = d\_{2,0,n\_r}^{(2,3)} e\_{1,0,\emptyset}^{(2,3)} \quad (m=0, n=0) \tag{A49}$$

$$\widetilde{d}\_{m,0,q}^{'(2,3)} = \frac{1}{b\_3^{|m|}} \left\{ |m| d\_{1,0,n\_r}^{(2,3)} d\_{m,0,q}^{(2,3)} + m n\_r d\_{1,0,n\_r}^{(2,3)} d\_{m,0,q}^{(2,3)} \right\} (m \neq 0, n = 0) \tag{A50}$$

$$\hat{d}\_{m,n,q}^{'(2,3)} = \frac{1}{I\_m(q\_n b\_3)} \left\{ \frac{q\_n}{2} d\_{2,0,n\_r}^{(2,3)} \left( d\_{m-1,n,q}^{(2,3)} + d\_{m+1,n,q}^{(2,3)} \right) + m n\_r d\_{1,0,n\_r}^{(2,3)} d\_{m,n,q}^{(2,3)} \right\} (n > 0) \tag{A51}$$

$$\stackrel{\sim}{e}\_{0,0,\emptyset}^{(2,3)} = e\_{0,0,\emptyset}^{(2,3)}(m=0, n=0), \quad \stackrel{\sim}{e}\_{m,0,\emptyset}^{(2,3)} = \frac{e\_{m,0,\emptyset}^{(2,3)}}{a\_2^{-|m|}} \text{ (\$m \neq 0, n=0)}\tag{A52}$$

$$\widetilde{\mathfrak{e}}\_{m,n,q}^{(2,3)} = \frac{e\_{m,n,q}^{(2,3)}}{\mathcal{K}\_m(q\_n a\_2)} \begin{pmatrix} n > 0 \end{pmatrix} \tag{A53}$$

$$\stackrel{\sim}{\hat{\epsilon}}\_{0,0,q}^{'(2,3)} = -d\_{2,0,n\_r}^{(2,3)} \epsilon\_{1,0,q}^{(2,3)} \ (m=0, n=0) \tag{A54}$$

$$\widetilde{\boldsymbol{e}}\_{m,0,\boldsymbol{\eta}}^{'(2,3)} = \frac{1}{a\_2^{-|m|}} \left( -|m| d\_{1,0,\boldsymbol{\eta}\_r}^{(2,3)} \boldsymbol{e}\_{m,0,\boldsymbol{\eta}}^{(2,3)} + m \boldsymbol{n}\_r d\_{1,0,\boldsymbol{\eta}\_r}^{(2,3)} \boldsymbol{e}\_{m,0,\boldsymbol{\eta}}^{(2,3)} \right) (m \neq 0, n = 0) \tag{A55}$$

$$\widetilde{\boldsymbol{e}}\_{m,n,\boldsymbol{\eta}}^{'(2,3)} = \frac{1}{K\_m(q\_n a\_2)} \left\{ -\frac{q\_n}{2} d\_{2,0,n\_r}^{(2,3)} \left( e\_{m-1,n,\boldsymbol{\eta}}^{(2,3)} + e\_{m+1,n,\boldsymbol{\eta}}^{(2,3)} \right) + m m\_r d\_{1,0,n\_r}^{(2,3)} e\_{m,n,\boldsymbol{\eta}}^{(2,3)} \right\} (n > 0) \tag{A56}$$

$$\stackrel{\sim (3)}{f}\_{m,0,q}^{(3)} = \frac{f\_{m,0,q}^{(3)}}{H\_{\text{m}}(ka\_3)} \ (n=0), \quad \stackrel{\sim (3)}{f}\_{m,n,q}^{(3)} = \frac{f\_{m,n,q}^{(3)}}{K\_{\text{m}}(k\_{\text{n}}a\_3)} \ (n>0) \tag{A57}$$

$$\stackrel{\sim}{f}\_{m,0,q}^{'(3)} = \frac{1}{H\_m(ka\_3)} \left\{ \frac{k}{2} d\_{2,0,n\_r}^{(3)} \left( f\_{m-1,0,q}^{(3)} - f\_{m+1,0,q}^{(3)} \right) + m n\_r d\_{1,0,n\_r}^{(3)} f\_{m,0,q}^{(3)} \right\} \tag{A58}$$

$$\stackrel{\sim}{f}\_{m,n,\eta}^{(\ \ \ \ \ \ \ )} = \frac{1}{K\_m(k\_n a\_3)} \left\{ -\frac{k\_n}{2} d\_{2,n,n\_r}^{(\ \ \ \ \ \ \ )}\_{m-1,n,\eta} + f\_{m+1,n,\eta}^{(\ \ \ \ \ \ \ \ \ \ \ \mu \rightarrow \ \ n \ \ n\_r} d\_{1,n,n\_r}^{(\ \ \ \ \ \ \ \ \ \ \ \ \mu \rightarrow \ \ n \ \ \ \ \mu \rightarrow \ \ \ \nu \rightarrow \ \ \ \nu \rightarrow \ \ \ \ \nu \rightarrow \ \ \ \ \ \nu \rightarrow \ \ \ \ \ \ \ \text{(A.59)}\_{m,n,m} \right\} \quad (\ \ \nu > 0) \quad \text{(A.59)}$$

#### **References**

