Figure 1 depicts a simple description of the problem, where $R$ is the radius of the cylinder, $h$ is the water depth, and $d$ is the height of the submerging part of the cylinder. The origin of the coordinate system is located at the still water line (SWL), with z-axis coinciding with the symmetry axis of the cylinder. The incident wave propagates along the x+ direction. To simplify the problem, the entire computational domain is divided into two sub-domains, i.e., $\varOmega_1$ and $\varOmega_2$. $\varOmega_1$ stands for the region $ - h < z < 0$ and $r > R$, and $\varOmega_2$ $ - h < z < - d$ and $r < R$. Here $r$ denotes the distance from the cylinder symmetry axis.

By ignoring the viscosity and surface tension of water, the problem can be solved using the potential theory. Let ${\varPhi _1}{{\rm{e}}^{ - {\rm{i}}\left({k{x_0} + \omega t} \right)}}$ and ${\varPhi _2}{{\rm{e}}^{ - {\rm{i}}\left({k{x_0} + \omega t} \right)}}$ be the velocity potentials of $\varOmega_1$ and $\varOmega_2$, where ${\rm{i}}$ is imaginary unit, $k$ and $\omega $ are the wave number and angular frequency of the incident wave, respectively. ${x_0}$ stands for the focusing position of the rogue wave, which will be used later in Section 3. The governing equations are written as

The two velocity potentials are connected by

The velocity potential ${\varPhi _1}$ can be written as the summation of the incident-wave potential ${\varPhi _{1, \,\, {\rm{i}}}}$ and the diffraction potential ${\varPhi _{1, \,\, {\rm{d}}}}$. The incident potential (namely, the linear-wave potential), formulated under the cylindrical coordinate, can be written as

where ${{\rm{J}}_n}$ is the first-kind Bessel function. Equation (5) is deduced by using the Bessel-function expansion on term ${{\rm{e}}^{{\rm{i}}kx}}$.

The diffraction potential can be also written in the expansion form, as

where ${\rm{H}}_n^{\left(1 \right)}$ and ${\rm{H}}_n^{\left(2 \right)}$ are the first- and second-kind Hankel functions, and ${{\rm{I}}_n}$ and ${{\rm{K}}_n}$ are the first- and second-kind modified Bessel functions, respectively. Since ${\rm{H}}_n^{\left(2 \right)}\left({kr} \right)$ and ${{\rm{I}}_n}\left({{k_l}r} \right)$ in Eq. (6) contradicts against the radiation condition when $r \to \infty $, ${\beta _{n, \;\, 0}}$ and ${\beta _{n, \;\, l}}$ should be 0, i.e.,

where ${{\rm{H}}_n}$ is short for ${\rm{H}}_n^{\left(1 \right)}$.

Similarly, the series expansion on the velocity potential ${\varPhi _2}$ is written as

noted that ${\varPhi _2}$ should be bounded at $r = 0$. It can be easily verified that Eqs (9) and (10) satisfy Eqs (1) and (2), respectively. The unknown ${\alpha _{n, \,\, l}}$ and ${\gamma _{n, \,\, j}}$ can be determined by using Eqs (3) and (4), and the detailed derivation is listed in Appendix A.

After the unknown ${\alpha _{n, \,\, l}}$ and ${\gamma _{n, \,\, j}}$ are obtained, the sectional force is formulated as

The horizontal force and the bending moment are

It should be noted that only a finite number of expansion terms are kept in Eqs (13) and (14). To verify the above equations, the results predicted by Eqs (13) and (14) are compared with those calculated using a commercial potential-theory solver–WADAM. The WADAM is a general analysis program for calculation of the wave-structure interaction for the fixed and floating structures of arbitrary shape, e.g., semi-submersible platforms, tension-leg platforms, gravity-base structures and ship hulls. It solves diffraction issues of zero-speed floating bodies using a 3–D boundary element method (BEM) with free surface Green functions. The analysis capabilities in the WADAM comprise calculation of global responses of loads and motions including first- and second-order components. The Morison’s equation can be also specified if slender structures are considered. The 3–D potential theory in WADAM is based directly on the Wamit program developed by Massachusetts Institute of Technology, which has been thoroughly verified.

Four test cases are performed, which are listed in Table 1. In Table 1, Case b is identical to Case a, except that a dense grid setting is used. Therefore, the results of Case a and Case b are displayed together in Fig. 2. The results of Case c and Case d are shown in Figs 3 and 4, respectively. As shown in Fig. 2, the results obtained with two sets of grid accord with each other, indicating that the grid setting is appropriate, and that the coarse grid set is enough for this problem. In Figs 2–4, the theoretical results agree well with the numerical ones, under various water depths, cylinder lengths and wavelengths. This demonstrates the reliability of the theoretical formulas.

Although Eqs (13) and (14) give exact formulation of the wave force and moment, it is not convenient to use them in realistic engineering applications. For instance, parameter ${\alpha _{1, \, \, l}}$ and ${\gamma _{1, \,\, j}}$ have to be predetermined by solving linear algebraic equations, and the results are written as the summation of series expansions rather than exact formulas. Therefore, simplification is made on Eqs (13) and (14) before further discussion.

First, the influence of the series expansion is studied. By computing the wave force and moment with various values of L and J, the influence of the summation terms can be estimated. Here, for most cases listed in Table 1, L = J = 5 is considered to be large enough to give precise results, noticing that the theoretical curves in Figs 2–4 are obtained with L = J = 5. Figure 5 shows the wave force and moment computed with L and J changing from 0 to 5, using the parameters of Case a. Here, that L or J equals 0 means the summation of the series expansion vanishes. In Fig. 5, computation with various L and J almost gives the identical results, even when L = J = 0. This reveals that the summation terms of Eqs (13) and (14) are far smaller than the terms with respect to ${\alpha _{1, \,\, 0}}$ and ${\gamma _{1, \,\, 0}}$, for Case a.

Similar to Fig. 5, Figs 6 and 7 give the RAOs calculated with various L and J, using parameters of Case c and Case d, respectively. As shown in Fig. 7, L = J = 0 gives acceptable prediction when $h = 20$ m and $d = 15$ m. However, it causes non-ignorable deviation when $h = 20$ m and $d = 3$ m, which can be seen in Fig. 6. This is due to the fact that the terms related to ${\gamma _{n, \,\, j}}$ can no longer be omitted. This fact causes the inherent difference between cylinders in the finite-depth water and in the deep water. Here, the term “finite-depth” means the water depth has the same order with the submerging length of the cylinder, and has a little difference between its common meaning in wave mechanics. To be specific, even if the water depth is fairly large, the term “finite-depth” still applies provided the submerging height of the cylinder is large enough such that $d/h \sim O\left(1 \right)$. The term “deep” has its common meaning, i.e., $h \to \infty $, making both submerging height and wavelength far smaller by comparison. This subsection mainly deals with formulas in the finite-depth water, and the deep water condition will be discussed in Subsection 2.3.

As mentioned above, the summation terms of Eqs (13) and (14) can be omitted under the finite-depth water, and therefore the key is the determination of parameters ${\alpha _{1, \,\, 0}}$ and ${\gamma _{1, \,\, 0}}$. Let $\alpha _{1, \,\, 0}^* = {\alpha _{1, \,\, 0}} + \frac{{{\rm i}g{\beta _1}{{\rm{J}}_1}\left({kR} \right)}}{\omega }$, and parameters ${\gamma _{1, \,\, 0}}$ and $\alpha _{1, \,\, 0}^*$ are governed by equations as follows (Eqs (A3) and (A5)):

Equations (15) and (16) are derived using the identity ${{\rm{J}}_n}\left({kR} \right) - \frac{{{{{\rm{J'}}}_n}\left({kR} \right)}}{{{{{\rm{H'}}}_n}\left({kR} \right)}}{{\rm{H}}_n}\left({kR} \right){\rm{ = }}\frac{{2{\rm{i}}}}{{{\text{π}}kR{{{\rm{H'}}}_n}\left({kR} \right)}}$. Substituting Eq. (16) into Eq. (15), parameter $\alpha _{1,\,\, 0}^*$ is written as

where ${\text{λ} _1} = 1 - d/h{\rm{i}}$ and ${\text{λ} _2} = R/h{\rm{i}}$ and the term $ - \frac{{4{\rm{i}}g}}{{{\text{π}}\omega kR{{{\rm{H'}}}_1}\left({kR} \right)}}$, which can be found in may textbooks, corresponds to the basic solution when $d = h$, namely, when the cylinder extends from the still water surface to the bottom. Hence, Eq. (17) can be treated as the basic solution multiplied by a correction, and the correction is related to the relative extension (${\text{λ} _1}$) and relative radius (${\text{λ} _2}$) of the cylinder. The horizontal force and the bending moment are finally written as

It is found during application that the finite-depth approximation shows a distinctive deviation against the exact solution when $d/h < 0.25$. Under such condition, the deep-water solution applies. The deep-water condition is much more complex than the case under the finite water depth. Not only the summation terms of Eqs (13) and (14) affect the calculation results, but also the series expansions own a bad convergence rate, making it difficult to compute the wave force and moment precisely. Here we give a brief analysis of the phenomenon qualitatively. To simplify the discussion, let $h \to \infty $ and $d \to 0$ simultaneously, and the unknown ${\alpha _{n, \,\, l}}$ and ${\gamma _{n, \,\, j}}$ of Eqs (9) and (10) are decided by the equations as (see Appendix B for details)

Let ${\alpha _{n, \,\, 0}} \approx O\left({{h^a}} \right)$, ${\alpha _{n, \,\, l}} \approx O\left({{h^b}} \right)$, ${\gamma _{n, \,\, 0}} \approx O\left({{h^c}} \right)$ and ${\gamma _{n, \,\, j}} \approx $ O(h^d) and thereby $a$, $b$, $c$ and $d$ have the relationship as follows:

From Eq. (26), one can easily obtain that $c = d = b + 1$ and $a = b + 1$ or $a = b + 2$. If $a = b + 1$, then $c \geqslant 0$ and $d \geqslant 0$. This indicates that the “constant” term - $\frac{{{\rm{i}}g{\beta _n}{{\rm{J}}_n}\left({kR} \right)}}{{\omega kh}}$ in Eqs (24) and (25) is of higher order and should be neglected. However, this will make the entire linear equation system has infinite number of solutions or trivial solution. To make the solution meaningful (i.e., the “constant” term should be reserved), $c$ and $d$ must equal $ - 1$. Therefore, $a = 0$, $b = - 2$ and $c = d = - 1$. Although Eqs (22)–(25) are derived under the condition that $h \to \infty $ and $d \to 0$, it should be noted that the value of $d$ does not affect the relationship of orders listed in Eq. (26), and it only makes the terms of Eqs (22)–(25) more complex and tedious to formulate. Here, $d \to 0$ is only used to make the equations simpler, and it is not an indispensable requirement of this subsection.

From the above discussion, by omitting higher-order components, the horizontal force and the bending moment can be formulated as

where

Figure 8 shows the outline of function $C\left(x \right) = \frac{{{{\rm{I}}_2}\left(x \right)}}{{x{{\rm{I}}_1}\left(x \right)}}$. Usually, when h is not very large, namely ${m_j}R$ is not a tiny number, ${C_j}$ tends to 0 rapidly and the summation converges fast. However, when $h \to \infty $, ${C_j}$ always tends to a constant (0.25 from Fig. 8) unless J is large enough. This means a large number of terms are required to reach convergence.

Figure 9 displays the RAOs of the bending moment, computed with various J and L. The parameters used in Fig. 9 are identical to those listed in test Case c except that h is 3 000 m here, i.e., the deep-water condition is adopted. As is shown in Fig. 9, the accurate solution can be obtained only when L and J are large enough. Hence, Eq. (28) has little value for realistic applications, if no simplification is made. From Eqs (24) and (25), one can find that ${\gamma _{1,\,\, j}} \propto {a'_{1, \,\, 0}}$, and therefore we assume that Eq. (28) can be rewritten as

By conducting a series of simulations with various $R/d$, we give the approximation of $\,\,{f_1}$ and $\,\,{f_2}$ as follows:

In order to validate Eqs (31) and (32), Fig. 10 compares the moment-RAOs computed using Eq. (21), Eq. (31) and WADAM, with a series of $R/d$-values listed in Table 2.

In Fig. 10, Eq. (31) gives the results that well agree with the numerical ones of the WADAM, yet the finite-water formula – Eq. (21) fails when $R/d$ is large. It is worth noting that Eq. (21) can still give reliable prediction of the bending moment under the condition that $R/d$ is small enough. This is due to the fact that the second term of Eq. (31) (i.e., the contribution from the tank bottom) becomes unimportant either when $R$ becomes small or when the submerging length $d$ becomes large. Another phenomenon which should be mentioned is that two peaks occur in Fig. 10–6. Figure 11 enlarges the plots of Fig. 10-6, where the double peaks can be observed more clearly. The double peaks are induced by the two terms, which correspond to the contribution from the upright wall and the bottom of the cylinder respectively, reaching their maximum crests at distinct frequencies. Moreover, when the contributions from the two terms have the same magnitude but opposite directions, a zero bending moment is reached! Since the finite-depth approximation wrongly estimates the influence of the bottom under the deep-water condition, Eq. (21) cannot reflect the double-peak phenomenon.

Hu et al. (2014) discussed the evolution of Gauss-envelope-based rogue waves in detail, from the point of view of the inversion problem of initial disturbances. Here, we directly list the corresponding equations of the Gauss envelope, without going into the detail of derivation. The free surface elevation and velocity potential of the Gauss envelope are written as

where ${\omega _0}$ and ${k_0}$ are the dominant angular frequency and wave number of the envelope. The rogue wave formulated in Eqs (33) and (34) reaches its maximum crest at $t = 0$. This can be achieved by selecting the origin of the time axis such that the maximum wave crest occurs at $t = 0$. Parameter $\sigma $ controls the bandwidth of the amplitude spectrum, which is formulated as

Noticing that Section 2 gives the wave-induced load with respect to unit wave component ${{\rm{e}} ^{{\rm{i}} \left[ {k\left({x - {x_0}} \right) - \omega t} \right]}}$, it will make the derivation intuitive and concise if the Gauss envelope is transformed into the superposition of unit waves. Using Eq. (35), the Gauss envelope can be rewritten as

Hence, the Gauss-envelope-induced load under the finite-water depth can be written in integral form as

where

And the Gauss-envelope-induced load under the deep water depth can be written as

where

Since the amplitude spectrum of Gauss envelope owns a narrow bandwidth, the contribution from wave components whose wave numbers differ from ${k_0}$ is little. Therefore, $\omega $ is expanded into Taylor series around ${k_0}$, as $\omega = {\omega _0} + {\omega '_0}\left({k - {k_0}} \right) + \frac{1}{2}{\omega ''_0}{\left({k - {k_0}} \right)^2} $ + .... Using this expansion, the integrals of Eqs (37)–(38) and (41)–(42) can be approximated as

where

note that the expressions of ${f_1}$ and ${f_2}$ are given by Eq. (32).

From Eqs (59)–(62), one can find that the focusing degree parameter $\bar \sigma $ and the focusing position ${\bar x_0}$ influence various sorts of wave load through the same term $\bar \sigma \bar X$, and that ${\bar x_0} + \bar R$ can be treated as one variable in affecting the maximum wave load. Figure 14 investigates the influence of $\bar \sigma $ and ${\bar x_0}$ on term $\bar \sigma \bar X$. Although the focusing degree parameter $\bar \sigma $, when no cylinder exists within the flow field, does not affect the maximum crest height of the focusing wave, it can affect the wave load to some extent. In Fig. 14, the wave load increases with the growth of the energy focusing degree, although the rate of increasement seems to be unobvious if the focusing degree is already high enough. It is also found in Fig. 14 that the largest wave load occurs when ${\bar x_0} + \bar R$ equals -1.5, and the smallest wave load happens when ${\bar x_0} + \bar R$ equals 1.5. When $\bar \sigma $ is small, the relative positions of the focusing location and the cylinder can largely affect the wave load. To be specific, if the maximum wave crest occurs somewhere in front of the cylinder, i.e., the wave envelope reaches its highest wave height before it meets the cylinder, then large load can be induced. On the contrary, a small load is caused if the focusing position happens somewhere behind the cylinder. However, the influence of the focusing position becomes weak when the focusing degree parameter is big. For instance, ${\bar x_0}$ can hardly affect the wave load if $\bar \sigma $ equals 5 or larger.

From Eqs (59)–(62), it is found that submerging depth $\bar d$ only affects terms ${\bar W_1}$ to ${\bar W_4}$. For the finite water depth, we let $\bar h = 3$ and investigates the influence of $\bar d$, which varies from 1 to 3. Figure 15 shows the variation of the horizontal force (left) and the bending moment (right) under various $\bar d$. As is shown in Fig. 15a, the horizontal force rises with the growth of $\bar d$, which accords with common sense. However, by contrast, the bending moment experiences a decrescence at first and then begins to increase. At certain values of $\bar d$, the bending moment may decline to 0, due to the balance of the contributions from the load upon the side wall and upon the cylinder bottom. To be specific, when the submerging depth is small, the contribution from the cylinder bottom dominates the entire bending moment. However, with the increasement of the submerging depth, the influence of the side wall becomes significant, and meanwhile the contribution from the cylinder bottom is weaken since fluid particles away from the free surface almost keep still.

One may be interested in the relationship between $\bar d$ and $\bar R$ when the magnitude of bending moment reaches 0, which is revealed in Fig. 16. It can be seen that $\ln \bar R$ may be approximated by a linear function of $\bar d$, if $\bar d$ is not very close to $\bar h$. And it can be easily understood that $\bar d = \bar h$ is a singularity of the curves. It should be noted Eq. (60) begins to give incorrect results when $\bar d$ is less than $0.25\bar h$, which is decided empirically. For smaller $\bar d$ and larger $\bar h$, the deep-water approximation should be used.

Similar to Fig. 15, Fig. 17 analyzes the influence of $\bar d$ on the horizontal force (left) and the bending moment (right) under the deep water, noted that $\bar R$ does not affect the value of ${\bar W_3}$. Also, similar to Fig. 15b, the bending moment under the deep water shows a piecewise linear pattern, yet the zero points of the bending moment under the deep water, which are illustrated in Fig. 18, differs from those under the finite-depth.

On the basis of the above discussions, one can find the focusing wave-induced loads exhibit many similarities with those induced by regular waves, which is due to the reason that a focusing wave is indeed dominated by a certain monochromatic regular wave. In general, the focusing wave induced loads is largely affected by the RAO of the carrier regular wave. Normally, RAO tends to be 0 when the wave frequency is either too small or too large. A small wave frequency indicates large wave lengths. Under the scale represented by the diameter of the cylinder, a large wave length leads to a mean free-surface elevation and a uniform flow field. Under such flow, both wave forces and wave moments are close to 0. On the contrary, the large wave frequency represents small waves, and a number of waves may exist within the length of the cylinder diameter. The loads induced by various waves are offset by each other, and hence a zero load is resulted. Usually a large wave force can be induced when the wave length is close to the cylinder diameter. However, this conclusion no longer applies for the wave moment, since the effects of both side walls and cylinder bottom need to be considered. Under certain circumstances, the moment induced by the side wall may neutralize the one induced by the cylinder bottom, and a zero moment is produced. The above-mentioned phenomena can be found in both regular waves and focusing waves. However, the focusing waves are also characterized by focusing positions and focusing degrees, both of which can significantly affect the wave loads. The focusing degree is the first reason to be considered. To be specific, large focusing degrees correspond to large wave loads. Besides, the influence of the focusing positions can be neglected when the focusing degree is large. The focusing position only matters when the focusing degree is small enough. Usually, a large wave load happens when the focusing position is located somewhere in front of the fore edge of the cylinder, and a small load occurs when the focusing lies somewhere behind the aft edge of the cylinder.