2.1. Governing equations and boundary conditions

The assumption of inviscid, irrotational flow is standard for the problem of IG wave interaction with structures (e.g. Agnon et al. Reference Agnon, Choi and Mei1988). Under this assumption, let $\varPhi$ be a scalar flow potential for water waves (Mei et al. Reference Mei, Stiassnie and Yue2005); it relates to flow velocity $\boldsymbol{u}$ via

(2.1) \begin{equation} \boldsymbol{u}=\boldsymbol{\nabla }\varPhi =\dfrac {{\partial }\varPhi }{{\partial } x}\boldsymbol{i}+\dfrac {{\partial }\varPhi }{{\partial } z}\boldsymbol{k}, \end{equation}

where $y$ -independency is assumed; $x$ and $z$ are spatial variables, $\boldsymbol{i}$ and $\boldsymbol{k}$ are unit vectors along $x$ - and $z$ -axes, respectively, and $\varPhi$ is governed by the Laplace equation in the fluid domain

(2.2) \begin{equation} \dfrac {{\partial }^2\varPhi }{{\partial } x^2}+ \dfrac {{\partial }^2\varPhi }{{\partial } z^2}=0. \end{equation}

For the water wave problem, a free surface exists. Let $\eta$ denote the free-surface elevation; to $O(\eta ^2)$ , the two free-surface boundary conditions on $z=0$ are (Mei et al. Reference Mei, Stiassnie and Yue2005, § 13.2.1)

(2.3) \begin{align}&\qquad \left . -{\text{g}} \eta = \left (\dfrac {{\partial }\varPhi }{{\partial } t}+\eta \dfrac {{\partial }^2\varPhi }{{\partial } t{\partial } z}+\dfrac {\eta ^2}{2}\dfrac {{\partial }}{{\partial } t}\dfrac {{\partial }^2\varPhi }{{\partial } z^2}\right ) + \dfrac {1}{2} \left ( \boldsymbol{u}^2+ \eta \dfrac {{\partial } \boldsymbol{u}^2}{{\partial } z} \right ) \right |_{z=0}, \end{align}

(2.4) \begin{align}& \left (\dfrac {{\partial }^2\varPhi }{{\partial } t^2}+ {\text{g}}\dfrac {{\partial }\varPhi }{{\partial } z}\right )+ \eta \left [\dfrac {{\partial }}{{\partial } z}\left (\dfrac {{\partial }^2\varPhi }{{\partial } t^2}+ {\text{g}}\dfrac {{\partial }\varPhi }{{\partial } z}\right )\right ]+ \dfrac {\eta ^2}{2} \left [\dfrac {{\partial }^2}{{\partial } z^2}\left (\dfrac {{\partial }^2\varPhi }{{\partial } t^2}+ {\text{g}}\dfrac {{\partial }\varPhi }{{\partial } z}\right )\right ] \nonumber\\ &\qquad \qquad\,\,\,\qquad \left . +\,\dfrac {{\partial } \boldsymbol{u}^2}{{\partial } t}+\eta \dfrac {{\partial }^2\boldsymbol{u}^2}{{\partial } t{\partial } z}+ \dfrac {1}{2}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}^2 = 0\right |_{z=0}, \\[9pt] \nonumber \end{align}

where $\text{g}\approx 9.81\,\text{m}/\text{s}^2$ is the gravitational acceleration, $t$ is a time variable. For a sea of depth $h$ with a flat bottom, the bottom boundary condition is the no-penetration condition

(2.5) \begin{equation} \left . \dfrac {{\partial }\varPhi }{{\partial } z}=0\right |_{z=-h}. \end{equation}

Define a small book-keeping parameter related to wave steepness

(2.6) \begin{equation} \varepsilon \equiv k_{0,1} A_1 \ll 1 ,\end{equation}

where $k_{0,1}$ is the progressive wavenumber of a carrier wave, $A_1$ being its amplitude. The unknowns $\varPhi$ and $\eta$ are expanded in terms of $\varepsilon$

(2.7) \begin{align}& \varPhi =\varepsilon \varPhi _1+\varepsilon ^2\varPhi _2+\varepsilon ^3\varPhi _3+\ldots ,\end{align}

(2.8) \begin{align}&\,\, \eta =\varepsilon \eta _1+\varepsilon ^2\eta _2+\varepsilon ^3\eta _3+\ldots . \end{align}

A multiple-scale method for the water wave problem presumes that $x$ can be separated into a fast variable $\mathfrak{x}$ and a slow variable $X\equiv \varepsilon \mathfrak{x}$ , and similarly, $t$ can be separated into a fast time $\mathfrak{t}$ and a slow time $T\equiv \varepsilon \mathfrak{t}$ . Hence, in (2.7, 2.8), $\varPhi _n (n=1,2,3)$ and $\eta _n, (n=1,2,3)$ are functions of $(\mathfrak{x}, X, \mathfrak{t}, T, z)$ . It should be noted that, in principle, the slow variables $X$ and $T$ should be characterised by a different small parameter other than $\varepsilon$ (see e.g. Li et al. Reference Li, Zheng, Lin, Adcock and van den Bremer2021). However, the use of a single small parameter $\varepsilon$ aids clarity, and as explained in Appendix A, it is reasonable for some problems, such as in Benney & Roskes (Reference Benney and Roskes1969) and the current problem.

Substituting (2.7, 2.8) into the Laplace equation and the boundary conditions, at the leading order $O(\varepsilon )$ , one finds the following linear equations for $\varPhi _1$ and $\eta _1$ :

(2.9) \begin{equation} \begin{aligned} \dfrac {{\partial }^2\varPhi _1}{{\partial } \mathfrak{x}^2}+\dfrac {{\partial }^2 \varPhi _1}{{\partial } z^2}=0; \, \left .\dfrac {{\partial }^2\varPhi _1}{{\partial } \mathfrak{t}^2}+{\text{g}}\dfrac {{\partial }\varPhi _1}{{\partial } z}=0\right |_{z=0}; \ \left . -{\text{g}}\eta _1=\dfrac {{\partial }\varPhi _1}{{\partial } \mathfrak{t}}\right |_{z=0};\, \left . \dfrac {{\partial }\varPhi _1}{{\partial } z} =0\right |_{z=-h}. \end{aligned} \end{equation}

Two types of solutions exist for (2.9). One, a periodic solution that involves only fast variables, e.g. $\varPhi _{1}=\phi _{1}(z){\text{e}}^{{\text{i}}(k\mathfrak{x}-\omega \mathfrak{t})}$ , the other, a solution in the form of $\varPhi _{1}(X,T)$ , containing only slow variables (e.g. Mei et al. Reference Mei, Stiassnie and Yue2005, (13.2.21)). The former is the classical Airy solution for water waves (e.g. Manasseh Reference Manasseh2021), named carrier wave in this paper, while the latter corresponds to a slow-varying solution related to the bound IG wave. The function $\varPhi _{1}(X,T)$ is so far an arbitrary function of slow variables; its details cannot be specified at $O(\varepsilon )$ . To determine its analytical form, analyses must be carried to $O(\varepsilon ^2)$ and $O(\varepsilon ^3)$ , leading to (4.1), from which $\varPhi _{1}(X,T)$ can be solved.

Figure 2. Cross-section of a 2-D rectangular heaving block on a water layer of depth $h$ . The block’s resting draught is $d$ , with length $2L$ . Regions I, II and III are named to facilitate analysis. Here, $s_{1,2,3}$ are the names of the three immersed surfaces of the 2-D block. Second-order IG wave potentials (black fonts) in different regions are also shown. These wave potentials are specified from § 4 to § 6.

2.2. Wave modification due to the heaving body: an overview

Consider in figure 2, a block of draught $d$ and length $2L$ on a water depth $h$ , allowed to heave. It interacts with a nonlinear wave group containing, among others, two linear carrier waves and a nonlinearly generated bound IG wave. The block’s linear interaction with two carrier waves $\eta ^{\text{I},{A_1}}$ and $\eta ^{\text{I},{A_2}}$ leads to the following surface elevations in region I of figure 2 ( $\mathfrak{x}\lt -L$ ):

(2.10) \begin{equation} \eta ^{\text{I},{A_1}}+\eta ^{\text{I},{A_2}}+\eta ^{\text{I},{D_1}}+\eta ^{\text{I},{D_2}}+\eta ^{\text{I},{R_1}}+\eta ^{\text{I},{R_2}}, \end{equation}

and the following waves in region III ( $\mathfrak{x}\gt L$ ):

(2.11) \begin{equation} { \eta ^{\text{III},{D_1}}+\eta ^{\text{III},{D_2}}+\eta ^{\text{III},{R_1}}+\eta ^{\text{III},{R_2}},} \end{equation}

where superscripts I and III denote region I and region III. The superscripts ${A}_n,{D}_n,{R}_n, (n=1,2)$ denote the first and second approaching (i.e. incident), diffracted, and radiated waves, respectively. These linear waves are determined in § 3.

Nonlinear interactions between carrier waves spawn a total bound IG wave potential denoted by $\varPhi ^{\text{I},{\textit{tot}}}_{\textit{bIG}}$ in region I (the superscript tot means ‘total’)

(2.12) \begin{equation} \varPhi ^{\text{I},{\textit{tot}}}_{\textit{bIG}}= \varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}+ \varPhi ^{\text{I},{D_1D_2}}_{\textit{bIG}}+ \varPhi ^{\text{I},{R_1R_2}}_{\textit{bIG}}+ \varPhi ^{\text{I},{D_1R_2}}_{\textit{bIG}}+ \varPhi ^{\text{I},{D_2R_1}}_{\textit{bIG}}, \end{equation}

and a total bound IG wave potential in region III, $\varPhi ^{\text{III},{\textit{tot}}}_{\textit{bIG}}$

(2.13) \begin{equation} \varPhi ^{\text{III},{\textit{tot}}}_{\textit{bIG}}= \varPhi ^{\text{III},{D_1D_2}}_{\textit{bIG}}+ \varPhi ^{\text{III},{R_1R_2}}_{\textit{bIG}}+ \varPhi ^{\text{III},{D_1R_2}}_{\textit{bIG}}+ \varPhi ^{\text{III},{D_2R_1}}_{\textit{bIG}} ,\end{equation}

where the subscript bIG is for ‘bound IG wave’, while the superscript A $_1$ A $_2$ etc. indicate which two carrier waves are interacting. Note that, in region I, no bound IG wave potential is generated due to the cross-interactions between the counter-propagating waves (i.e. between incident waves and the radiated/diffracted waves), see e.g. Agnon & Mei (Reference Agnon and Mei1985, § 4). These potentials are shown in figure 2; they will be discussed in detail in § 4. Corresponding to these potentials, the total bound IG wave surface elevations in regions I and III are

(2.14a) \begin{align}& \eta _{\textit{bIG}}^{\text{I},{\textit{tot}}}= \eta _{\textit{bIG}}^{\text{I},{A_1A_2}}+ \eta _{\textit{bIG}}^{\text{I},{D_1D_2}}+ \eta _{\textit{bIG}}^{\text{I},{R_1R_2}}+ \eta _{\textit{bIG}}^{\text{I},{D_1R_2}}+ \eta _{\textit{bIG}}^{\text{I},{D_2R_1}}, \\[-6pt] \nonumber \end{align}

(2.14b) \begin{align}&\,\,\, \eta _{\textit{bIG}}^{\text{III},{\textit{tot}}}= \eta _{\textit{bIG}}^{\text{III},{D_1D_2}}+ \eta _{\textit{bIG}}^{\text{III},{R_1R_2}}+ \eta _{\textit{bIG}}^{\text{III},{D_1R_2}}+ \eta _{\textit{bIG}}^{\text{III},{D_2R_1}}. \\[9pt] \nonumber \end{align}

Figure 2 also shows free IG (subscript ‘fIG ’) wave potentials, viz. $\varPhi ^{\text{I},{M}}_{\textit{fIG}}, \varPhi ^{\text{I},{R}}_{\textit{fIG}}$ , $\varPhi ^{\text{III},{M}}_{\textit{fIG}}, \varPhi ^{\text{III},{R}}_{\textit{fIG}}$ , where the superscripts R and M represent radiated free IG waves and free IG waves required to satisfy the matching conditions. These, and the associated surface elevations, $\eta ^{\text{I},{M}}_{\textit{fIG}}, \eta ^{\text{I},{R}}_{\textit{fIG}}$ , $\eta ^{\text{III},{M}}_{\textit{fIG}}, \eta ^{\text{III},{R}}_{\textit{fIG}}$ , as well as the IG potentials in region II, $\varPhi ^{\text{II},{M}}_{\textit{IG}}$ and $\varPhi ^{\text{II},{R}}_{\textit{IG}}$ , will be discussed in §§ 5 and 6.

The sum of free and bound IG surface elevations gives the total IG waves. Denote the total IG wave potential in regions I and III by $\varPhi _{\textit{IG}}^{\text{I},{\textit{tot}}}$ and $\varPhi _{\textit{IG}}^{\text{III},{\textit{tot}}}$ , they are

(2.15a,b) \begin{align} \varPhi _{\textit{IG}}^{\text{I},{\textit{tot}}}=\varPhi ^{\text{I},{M}}_{\textit{fIG}}+\varPhi _{\textit{fIG}}^{\text{I},{R}}+\varPhi _{\textit{bIG}}^{\text{I},{\textit{tot}}}; \quad \varPhi _{\textit{IG}}^{\text{III},{\textit{tot}}}=\varPhi ^{\text{III},{M}}_{\textit{fIG}}+\varPhi _{\textit{fIG}}^{\text{III},{R}}+ \varPhi _{\textit{bIG}}^{\text{III},{\textit{tot}}}, \end{align}

which corresponds to the surface elevation: $\eta _{\textit{IG}}^{\text{I},{\textit{tot}}}$ and $\eta _{\textit{IG}}^{\text{III},{\textit{tot}}}$ , their details are

(2.16a,b) \begin{align} \eta _{\textit{IG}}^{\text{I},{\textit{tot}}}=\eta _{\textit{fIG}}^{\text{I},{M}}+\eta _{\textit{fIG}}^{\text{I},{R}}+\eta _{\textit{bIG}}^{\text{I},{\textit{tot}}}; \quad \eta _{\textit{IG}}^{\text{III},{\textit{tot}}}=\eta _{\textit{fIG}}^{\text{III},{M}}+\eta _{\textit{fIG}}^{\text{III},{R}}+ \eta _{\textit{bIG}}^{\text{III},{\textit{tot}}}. \end{align}

Our primary interest here is controlling the amplitude of $\eta _{\textit{IG}}^{\text{III},{\textit{tot}}}$ by suitable 2-D heaving block.

3.1. Governing equations

The governing equations and boundary conditions for the diffraction problem are (2.9) plus the following boundary condition on the floating body:

(3.1) \begin{equation} \left .\dfrac {{\partial }\varPhi _1}{{\partial } n}=0 \right |_{\text{on} \ s_{1,2,3}}, \end{equation}

where $n$ represents normal vectors of surfaces $s_{1,2,3}$ in figure 2. For the heaving radiation problem, the block’s vertical velocity is

(3.2) \begin{equation} \dot {\xi }(\mathfrak{t})=-{\text{i}}\omega A_{{h}} {\text{e}}^{-{\text{i}}\omega \mathfrak{t}} ,\end{equation}

where $A_{{h}}$ is an unknown complex heaving amplitude, and $\omega$ is the angular frequency of the incident wave. To determine it, the following boundary condition on the bottom of the block is required, stating that the $z$ -velocity of the flow equals the heaving velocity:

(3.3) \begin{equation} \begin{aligned} \left .\dfrac {{\partial }\varPhi _1}{{\partial } z}=-{\text{i}}\omega A_{{h}} {\text{e}}^{-{\text{i}}\omega \mathfrak{t}} \right |_{\text{on the bottom of the block}, \ s_2}. \end{aligned} \end{equation}

3.2. Incident waves

Incident waves in figure 2 approach from the negative $x$ -axis, satisfying (2.9) in region I. Sinusoidal surface profiles are well-known solutions (see Appendix B)

(3.4) \begin{equation} \eta ^{\text{I},{A}_1} =A_1\cos (k_{0,1}\mathfrak{x}-\omega _1 \mathfrak{t}), \quad \eta ^{\text{I},{A}_2} =A_2\cos (k_{0,2}\mathfrak{x}-\omega _2 \mathfrak{t}) ,\end{equation}

where the superscripts ‘I, A $_n$ ’ $(n=1,2)$ denote incident waves 1 and 2 in region I (see figure 2 for region division). The letter ‘I’ denotes region I, and the letter ‘A’ denotes ‘approaching’ waves, a term used as an alternative for ‘incident’ waves, to avoid using the letter ‘I’ again in the superscripts. The real quantities $A_n, k_{0,n}, \omega _n, (n=1,2)$ are wave amplitudes, wavenumber and wave frequencies, respectively. Note that the subscript 0 in $k_{0,n}$ arises to ensure consistency with the notations in Appendix B. The values of $k_{0,n}$ satisfy the dispersion relations

(3.5) \begin{equation} \omega _1^2={\text{g}} k_{0,1} \tanh k_{0,1} h, \quad \omega _2^2={\text{g}} k_{0,2} \tanh k_{0,2} h. \end{equation}

The majority of examples in this paper assume that $\omega _2=1.1\omega _1$ (cf. Hossain et al. Reference Hossain, Kioka and Kitano2001; Sergiienko et al. Reference Sergiienko, Cannard, Cui, Leontini, Manasseh and Cazzolato2024); other frequencie ratios are discussed briefly in § 9.2.

3.3. Diffracted and radiated waves

When waves given by (3.4) interact with the 2-D heaving block, diffracted and radiated wave fields are generated. The governing equations are (2.9), (3.1) and (3.3). Generally, the diffracted and radiated waves can be described by infinite sums of eigenfunctions involving progressive and evanescent modes. Evanescent modes are neglected because they decay rapidly away from the block and do not spawn IG waves (Agnon et al. Reference Agnon, Choi and Mei1988; Liu & Iskandarani Reference Liu and Iskandarani1991; Hermans Reference Hermans2010). Using the superscripts D $_1$ , D $_2$ to denote diffracted waves related to incident carrier waves 1 and 2, using the superscripts R $_1$ , R $_2$ to denote radiated waves 1 and 2 and using ‘I’ for waves in region I and ‘III’ for waves in region III, the progressive modes of the eight diffracted and radiated waves are

(3.6a) \begin{align} \eta ^{\text{I},{D_n}} &= \big|a_{0,1}^- \big|\cos \big[-k_{0,n}(\mathfrak{x}-(-L))-\omega _n \mathfrak{t}+\text{arg}\big(a_{0,n}^-\big)\big], \quad n=1,2, \\[-9pt] \nonumber \end{align}

(3.6b) \begin{align} \eta ^{\text{I},{R_n}}&= \big|\mathfrak{a}_{0,n}^-\big|\cos \big[-k_{0,n}(\mathfrak{x}-(-L))-\omega _n \mathfrak{t}+\text{arg}\big(\mathfrak{a}_{0,n}^-\big)\big], \quad n=1,2, \\[-9pt] \nonumber \end{align}

(3.6c) \begin{align} \eta ^{\text{III},{D_n}}&= \big|b_{0,n}^+ \big| \cos \big[k_{0,n}(\mathfrak{x}-L)-\omega _n \mathfrak{t}+\text{arg}\big(b_{0,n}^+\big)\big], \quad n=1,2, \\[-9pt] \nonumber \end{align}

(3.6d) \begin{align} \eta ^{\text{III},{R_n}}&= \big|\mathfrak{b}_{0,n}^+ \big|\cos \big[k_{0,n}(\mathfrak{x}-L)-\omega _n \mathfrak{t}+\text{arg}\big(\mathfrak{b}_{0,n}^+\big)\big], \quad n=1,2, \end{align}

where $a_{0,1}^-, a_{0,2}^-, \mathfrak{a}_{0,1}^-, \mathfrak{a}_{0,2}^-$ are unknown coefficients for reflected waves in region I, and $b_{0,1}^+, b_{0,2}^+, \mathfrak{b}_{0,1}^+, \mathfrak{b}_{0,2}^+$ for transmitted waves in region III. These coefficients are solved in Appendices B and C. To this end, (2.10) and (2.11) are fully specified. All wave surface elevations in this paper are presented in real form, like (3.6). This is because surface elevations are inherently real quantities, and unlike flow potentials, they are final results that no longer participate in important calculations. Hence, a real presentation is suitable.

4.1. Bound IG wave potentials due to interaction of incident carrier waves

Use $\varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}$ to denote the flow potential of the bound IG wave generated by the two incident carrier waves (3.4). The subscript ‘bIG ’ stands for ‘bound IG waves’, the superscript ‘I’ means that the bound IG wave is in region I and A $_1$ A $_2$ indicates that it is generated by the interaction of two incident carrier waves A $_1$ and A $_2$ where, as before, A stands for ‘approaching waves’. The potential is governed by the following equation (cf. (13.2.36) of Mei & Benmoussa Reference Mei and Benmoussa1984; Mei et al. Reference Mei, Stiassnie and Yue2005; and Liu & Iskandarani Reference Liu and Iskandarani1991):

(4.1) \begin{align} \dfrac {{\partial }^2 \varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}}{{\partial } T^2} - {\text{g}} h \dfrac {{\partial }^2\varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}}{{\partial } X^2}&={\text{i}}\left (\dfrac {{\text{g}}^2 k_{{S}} k_{{L}}}{\omega _{{S}}} + \dfrac {\omega _{{S}}^2\omega _{{L}}}{2\sinh ^2 k_{{S}} h} \right )\nonumber\\&\quad \times \left [{\text{e}}^{2{\text{i}}\left (k_{{L}}X-\omega _{{L}} T\right )}-{\text{e}}^{-2{\text{i}}\left (k_{{L}}X-\omega _{{L}} T\right )}\right ] A_1A_2 ,\end{align}

where $X$ and $T$ are slow-varying spatial and temporal scales, $k_{{S}}$ , $\omega _{{S}}$ , $k_{{L}}$ and $\omega _{{L}}$ are defined as follows:

(4.2) \begin{equation} \omega _{{L}}=\left |\dfrac {\omega _1-\omega _2}{2}\right |\!, \quad \omega _{{S}}=\dfrac {\omega _1+\omega _2}{2}, \quad k_{{L}}= \left |\dfrac {k_{0,1}-k_{0,2}}{2}\right |\!, \quad k_{{S}}= \dfrac {k_{0,1}+k_{0,2}}{2}. \end{equation}

Physically, the former two are the averaged wavenumber and angular frequency of the carrier waves. The subscript S stands for ‘short waves’, and L stands for ‘long waves’. To obtain the wavenumber and frequency of the IG wave, one should multiply both $k_{{L}}$ and $\omega _{{L}}$ by 2. This is because $k_{{L}}$ and $\omega _{{L}}$ are defined in such a way in past studies (e.g. Mei et al. Reference Mei, Stiassnie and Yue2005); we follow the tradition.

Equation (4.1) contains a homogeneous part for $\varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}$ on the left-hand side and an inhomogeneous forcing term proportional to the product of carrier waves’ amplitudes, $A_1A_2$ , on the right-hand side. The particular solution to (4.1) is

(4.3) \begin{equation} { \varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}=C_{{B}}A_1A_2 \left [{\text{e}}^{2{\text{i}} (k_{{L}} X-\omega _{{L}} T)}- \text{c.c.}\right ] } ,\end{equation}

where ‘c.c.’ stands for complex conjugate, and the coefficient $C_{{B}}$ is defined as

(4.4) \begin{equation} C_{{B}}= \dfrac {{\text{i}}}{\left (-4\omega _{{L}}^2+4{\text{g}} h k_{{L}}^2\right )} \left (\dfrac {{\text{g}}^2 k_{{S}} k_{{L}}}{\omega _{{S}}} + \dfrac {\omega _{{S}}^2\omega _{{L}}}{2\sinh ^2 k_{{S}} h} \right )\!. \end{equation}

Equation (4.3) is not a function of $z$ , as (4.1) does not contain $z$ .

4.2. Bound IG wave potentials due to diffracted and radiated carrier waves

The diffracted and radiated first-order waves expressed by (3.6) also generate bound IG wave potentials. The IG wave potentials due to interaction between diffracted waves in regions I and III are denoted by $\varPhi ^{\text{I},{D_1D_2}}_{\textit{bIG}}$ and $\varPhi ^{\text{III},{D_1D_2}}_{\textit{bIG}}$ , respectively, while IG wave potentials due to interactions between radiated waves in regions I and III are denoted by $\varPhi ^{\text{I},{R_1R_2}}_{\textit{bIG}}$ and $\varPhi ^{\text{III},{R_1R_2}}_{\textit{bIG}}$ . The bound IG wave potential arising from cross-interactions between diffracted and radiated carrier wave are $\varPhi ^{\text{I},{D_1R_1}}_{\textit{bIG}}$ , $\varPhi ^{\text{I},{D_2R_1}}_{\textit{bIG}}$ , $\varPhi ^{\text{III},{D_1R_2}}_{\textit{bIG}}$ , $\varPhi ^{\text{III},{D_2R_1}}_{\textit{bIG}}$ , $\varPhi ^{\text{I},{D_1R_2}}_{\textit{bIG}}$ . These wave potentials are governed by equations similar to (4.1), and their details are given in Appendix D. It should be noted that there is no IG wave arising from interactions between waves of the same frequency; therefore, the following IG wave potentials do not exist: $\varPhi ^{\text{I},{D_1R_1}}_{\textit{bIG}}$ , $\varPhi ^{\text{III},{D_1R_1}}_{\textit{bIG}}$ , $\varPhi ^{\text{I},{D_2R_2}}_{\textit{bIG}}$ , $\varPhi ^{\text{III},{D_2R_2}}_{\textit{bIG}}$ . Moreover, as noted in § 2, no bound IG waves are generated by the interaction between incident waves and radiated/diffracted waves in region I. To this end, all elements in (2.12) and (2.13) are explained.

4.3. Bound IG wave surface elevations

The bound IG wave potentials each give rise to surface elevations. For example, the incident bound IG wave potential $\varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}$ corresponds to the following surface elevation (cf. 2.9):

(4.5) \begin{equation} \begin{aligned} -\dfrac {1}{{\text{g}}}\dfrac { \varPhi ^{\text{I},{A_1A_2}}_{\textit{bIG}}}{{\partial } T} = 2 C_1A_1A_2\cos \left [2\left (k_{{L}}X-\omega _{{L}} T\right )\right ]\!, \end{aligned} \end{equation}

in which

(4.6) \begin{equation} C_1 = -\dfrac {\omega _{{L}}}{2{\text{g}} k_{{L}}^2\left (-C_{{g}}^2+{\text{g}} h\right )} \left (\dfrac {{\text{g}}^2 k_{{S}} k_{{L}}}{\omega _{{S}}} + \dfrac {\omega _{{S}}^2\omega _{{L}}}{2\sinh ^2 k_{{S}} h} \right )\!. \end{equation}

The surface elevation expressed by (4.5) is incomplete; it needs to be amended by another comparably small IG wave, due to $O(\varepsilon ^2)$ self-interactions of carrier wave components, cf. (13.2.33) of Mei et al. (Reference Mei, Stiassnie and Yue2005), to yield the complete surface elevation for the incident bound IG wave $\eta _{\textit{bIG}}^{\text{I},{A_1A_2}}$

(4.7) \begin{equation} \eta _{\textit{bIG}}^{\text{I},{A_1A_2}} =2 \left (C_1+C_2\right ) A_1A_2 \cos \left [2\left (k_{{L}}X-\omega _{{L}} T\right )\right ]\!, \end{equation}

where $C_2$ is the amplitude of the other IG wave

(4.8) \begin{equation} C_2=-\dfrac {k_{{S}}}{2\sinh 2k_{{S}}h}. \end{equation}

Similarly, bound IG waves due to interactions between diffracted waves in regions I and III are denoted by $\eta ^{\text{I},{D_1D_2}}_{\textit{bIG}}$ and $\eta ^{\text{III},{D_1D_2}}_{\textit{bIG}}$ , respectively, while bound IG waves generated by interactions between radiated waves in regions I and III are denoted by $\eta ^{\text{I},{R_1R_2}}_{\textit{bIG}}$ and $\eta ^{\text{III},{R_1R_2}}_{\textit{bIG}}$ . Bound IG waves due to cross-interactions between diffracted and radiated carrier waves are $\eta ^{\text{I},{D_1R_1}}_{\textit{bIG}}$ , $\eta ^{\text{I},{D_2R_1}}_{\textit{bIG}}$ , $\eta ^{\text{III},{D_1R_2}}_{\textit{bIG}}$ , $\eta ^{\text{III},{D_2R_1}}_{\textit{bIG}}$ , $\eta ^{\text{I},{D_1R_2}}_{\textit{bIG}}$ . Their details are given in Appendix E. These waves contribute to (2.14).

5.1. Free IG wave potentials and surface elevations

The total bound IG wave potentials (2.12 and 2.13) are generally discontinuous across the block. To rectify this, free IG wave potentials in regions I and III, denoted by $\varPhi ^{\text{I},{M}}_{\textit{fIG}}$ and $\varPhi ^{\text{III},{M}}_{\textit{fIG}}$ (subscript fIG for free IG waves), are introduced. In region II, the second-order potential is $\varPhi ^{\text{II},{M}}_{\textit{IG}}$ (the subscript is ‘IG’ but not ‘fIG ’; it corresponds to neither free nor bound waves). Among the unknowns, $\varPhi ^{\text{I},{M}}_{\textit{fIG}}$ and $\varPhi ^{\text{III},{M}}_{\textit{fIG}}$ should satisfy the homogeneous version of (4.1), hence

(5.1) \begin{equation} \varPhi ^{\text{I},{M}}_{\textit{fIG}}=D^{\text{I}} {\text{e}}^{-2{\text{i}} \frac {\omega _{{L}}}{\sqrt {{\text{g}} h}}X}{\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.}, \quad \varPhi ^{\text{III},{M}}_{\textit{fIG}}=D^{\text{III}} {\text{e}}^{2{\text{i}} \frac {\omega _{{L}}}{\sqrt {{\text{g}} h}}X}{\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.} \end{equation}

In (5.1), there are four unknown coefficients: $D^{\text{I}}, D^{\text{III}}$ and their complex conjugates, $\overline {D^{\text{I}}}, \overline {D^{\text{III}}}$ . The remaining potential $\varPhi _{\textit{IG}}^{\text{II},{M}}$ should satisfy the Laplace equation and the boundary conditions on the block’s bottom $s_2$ (see figure 2) and on the sea bottom. The following solution can be used (cf. (B6) in Appendix B, also see Hossain et al. Reference Hossain, Kioka and Kitano2001):

(5.2) \begin{equation} \varPhi _{\textit{IG}}^{\text{II},{M}}=(\mathcal{A}_1 X+\mathcal{A}_2) {\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.} ,\end{equation}

which contains four more unknown coefficients: $\mathcal{A}_1$ , $\mathcal{A}_2$ and their complex conjugates $\overline {\mathcal{A}_1}$ and $\overline {\mathcal{A}_2}$ , increasing the number of unknowns to eight, requiring eight equations. They are determined by the following four matching conditions, which require that the total IG wave potential and flux are continuous at both the bow ( $X=-L$ ) and the stern ( $X=L$ ) of the heaving block (Agnon et al. Reference Agnon, Choi and Mei1988; Wu & Liu Reference Wu and Liu1990; Liu & Iskandarani Reference Liu and Iskandarani1991; Hossain et al. Reference Hossain, Kioka and Kitano2001; Li et al. Reference Li, Zheng, Lin, Adcock and van den Bremer2021)

(5.3) \begin{align}&\,\, \left . \varPhi ^{\text{I},{\textit{tot}}}_{\textit{bIG}}+ \varPhi ^{\text{I},{M}}_{\textit{fIG}}\right |_{X=-L}= \left . \varPhi _{\textit{IG}}^{\text{II},{M}} \right |_{X=-L} , \end{align}

(5.4) \begin{align}&\,\, \left . \varPhi _{\textit{IG}}^{\text{II},{M}} \right |_{X=L}= \left . \varPhi ^{\text{III},{\textit{tot}}}_{\textit{bIG}}+ \varPhi ^{\text{III},{M}}_{\textit{fIG}} \right |_{X=L} , \end{align}

(5.5) \begin{align}& \left . \dfrac {\varPhi _{\textit{bIG}}^{\text{I},{\textit{tot}}}}{{\partial } X}+ \dfrac {\varPhi ^{\text{I},{M}}_{\textit{fIG}}}{{\partial } X}\right |_{X=-L} = \left .\dfrac {\varPhi _{\textit{IG}}^{\text{II},{M}}}{{\partial } X} \right |_{X=-L}, \end{align}

(5.6) \begin{align}& \left . \dfrac {\varPhi _{\textit{IG}}^{\text{II},{M}}}{{\partial } X} \right |_{X=L} = \left . \dfrac {\varPhi _{\textit{bIG}}^{\text{III},{\textit{tot}}}}{{\partial } X}+ \dfrac {\varPhi ^{\text{III},{M}}_{\textit{fIG}}}{{\partial } X} \right |_{X=L}. \end{align}

Substituting the known quantities $\varPhi ^{\text{I},{\textit{tot}}}_{\textit{bIG}}$ and $\varPhi ^{\text{III},{\textit{tot}}}_{\textit{bIG}}$ into (5.3–5.6) and collocating like time harmonics yields eight equations, see Appendix F. Solving the equations determines the unknown potentials, then the surface elevations of the free IG waves $\eta _{\textit{fIG}}^{\text{I},{M}}$ and $\eta _{\textit{fIG}}^{\text{III},{M}}$ can be calculated by

(5.7a,b) \begin{align} \eta _{\textit{fIG}}^{\text{I},{M}}=-\dfrac {1}{{\text{g}}}\dfrac {{\partial }\varPhi ^{\text{I},{M}}_{\textit{fIG}}}{{\partial } T}, X\lt -L; \quad \eta _{\textit{fIG}}^{\text{III},{M}}=-\dfrac {1}{{\text{g}}}\dfrac {{\partial }\varPhi ^{\text{III},{M}}_{\textit{fIG}}}{{\partial } T}, X\gt L. \end{align}

The free IG waves, (5.1) and (5.2), as well as the matching conditions (5.3–5.6), are approximations, as they do not satisfy the governing equations and boundary conditions on the heaving block exactly; moreover, the free IG waves are assumed to be shallow water waves (see e.g. Li et al. Reference Li, Zheng, Lin, Adcock and van den Bremer2021, § 2.6.3).

5.2. A note on total IG wave amplitude modulation

The free IG wave’s phase speed $\sqrt {{\text{g}} h}$ is generally faster than the bound IG wave’s phase speed (the group velocity)

(5.8) \begin{equation} C_{\textit{bIG}}=\dfrac {\omega _{{L}}}{k_{{L}}}=C_{{g}}. \end{equation}

As such, when a localised wave packet interacts with an obstacle (e.g. Li et al. Reference Li, Zheng, Lin, Adcock and van den Bremer2021), the free IG wave will overtake the bound IG wave, so that the total amplitude of the IG wave is location-dependent. Such an amplitude variation of second-order waves was also observed by Massel (Reference Massel1983). However, these previous studies did not systematically consider IG wave amplitude modulation from a wave reduction/coastal protection point of view using heaving bodies.

6.1. Slow heaving

The slow heaving motion due to the second-order potential $ \varPhi _{\textit{IG}}^{\text{II},{M}}$ (5.2) is characterised by its amplitude denoted by $A_{{h}}^{\textit{IG}}$ and its complex conjugate $\overline {A_{{h}}^{\textit{IG}}}$ . Assuming that slow heaving is a periodic function of time, then its displacement $\xi _{\textit{IG}}(T)$ can be expressed as

(6.1) \begin{equation} \xi _{\textit{IG}}(T)=A_{{h}}^{\textit{IG}}{\text{e}}^{-2{\text{i}}\omega _{{L}}T}+\text{c.c.} \end{equation}

Equation (6.1) implies the following slow heaving velocity and acceleration:

(6.2) \begin{equation} \dot {\xi }_{\textit{IG}}(T)=-2{\text{i}}\omega _{{L}} A_{{h}}^{\textit{IG}}{\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.}, \quad \ddot {\xi }_{\textit{IG}}(T)= -4\omega _{{L}}^2 A_{{h}}^{\textit{IG}}{\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.}. \end{equation}

The unknown amplitudes $A_{{h}}^{\textit{IG}}$ and $\overline {A_{{h}}^{\textit{IG}}}$ are decided by a force balance equation. The excitation force in the equation is calculated by

(6.3) \begin{equation} \begin{aligned} F_{\textit{IG}}^{{D}}=\int _{-L}^L \left . -\rho \dfrac {{\partial }\varPhi _{\textit{IG}}^{\text{II},{M}}}{{\partial } T} \right |_{z=-d} {\text{d}} X =4{\text{i}}\rho {\text{g}} \omega _{{L}} \left (\mathcal{A}_2{\text{e}}^{-2{\text{i}}\omega _{{L}}T} - \overline {\mathcal{A}_2}{\text{e}}^{2{\text{i}}\omega _{{L}}T}\right )\!. \end{aligned} \end{equation}

For a block section with unit width, the restoring force is calculated as $F^{{S}}=\rho {\text{g}} A_{{wp}}\xi _{\textit{IG}}$ , where the waterplane area $A_{{wp}}=2L$ . The inertia force $F^{{M}}$ is calculated by $m\ddot {\xi }_{\textit{IG}}$ , where the mass $m$ equals $2L\rho d$ for unit width. Finally, the radiation force $F^{{R}}$ equals $-\ddot {\xi }m_{{a}}-\dot {\xi }R$ , where the frequency-dependent added mass $m_{{a}}$ and radiation damping $R$ are evaluated at the frequency of the IG wave, 2 $\omega _{{L}}$ . The analytical details of $m_{{a}}$ and $R$ are derived in Appendix C.2.

Balancing all forces by $F^{{M}}+F^{{S}}=F_{\textit{IG}}^{{D}}+F^{{R}}$ and collecting different harmonics leads to two equations. Solving them yields $A_{{h}}^{\textit{IG}}$ and $\overline {A_{{h}}^{\textit{IG}}}$

(6.4) \begin{equation} A_{{h}}^{\textit{IG}}=\dfrac {2{\text{i}}\rho \omega _{{L}}\mathcal{A}_2}{ -2\omega _{{L}}^2 (2L\rho d+m_{{a}}) -{\text{i}}\omega _{{L}} R +\rho {\text{g}} L}, \quad \overline {A_{{h}}^{\textit{IG}}}=\dfrac {-2{\text{i}}\rho \omega _{{L}}\overline {\mathcal{A}_2}}{ -2\omega _{{L}}^2 (2L\rho d+m_{{a}}) +{\text{i}}\omega _{{L}} R +\rho {\text{g}} L}. \end{equation}

6.2. Radiated free IG wave potentials

Once the slow heaving amplitudes $A_{{h}}^{\textit{IG}}$ and $\overline {A_{{h}}^{\textit{IG}}}$ are determined, the slow heaving flow potential can be determined. Denote the flow potential by $\varPhi _{\textit{IG}}^{\text{II},{R}}$ (the superscript R stands for radiation); similar to $\varPhi _{\textit{IG}}^{\text{II},{M}}$ discussed in the last section, it should also satisfy the Laplace equation and the boundary conditions on $s_2$ and on the sea bottom, cf. (2.9) and (3.1). In addition, it should satisfy the following boundary condition stating that the $z$ -derivative of $\varPhi _{\textit{IG}}^{\text{II},{R}}$ gives heaving velocity on the bottom of the block, $s_2$ :

(6.5) \begin{equation} \left .\dfrac {{\partial } \varPhi _{\textit{IG}}^{\text{II},{R}}}{{\partial } z}=\dot {\xi }_{\textit{IG}}(T)\right |_{s_2}. \end{equation}

A solution is (cf. C4)

(6.6) \begin{equation} \varPhi _{\textit{IG}}^{\text{II},{R}}=\left (\mathcal{B}_1x+\mathcal{B}_2\right ){\text{e}}^{-2{\text{i}}\omega _{{L}}T}+A_{{h}}^{\textit{IG}}\dfrac {(z+h)^2-x^2}{2(h-\textit{d})}{\text{e}}^{-2{\text{i}}\omega _{{L}}T}+\text{c.c.}, \end{equation}

which contains four unknowns, $\mathcal{B}_1, \mathcal{B}_2$ and their complex conjugates $\overline {\mathcal{B}_1}, \overline {\mathcal{B}_2}$ . The potential $\varPhi _{\textit{IG}}^{\text{II},{R}}$ generates radiated waves in regions I and III, and it is envisaged that these waves assume the same form as the free IG waves discussed before. Denoting these radiated free IG wave potentials by $\varPhi _{\textit{fIG}}^{\text{I},{R}}$ and $\varPhi _{\textit{fIG}}^{\text{III},{R}}$ , their details are

(6.7) \begin{equation} \varPhi ^{\text{I},{R}}_{\textit{fIG}}=\mathfrak{D}^{\text{I}} {\text{e}}^{-2{\text{i}} \frac {\omega _{{L}}}{\sqrt {{\text{g}} h}}X} {\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.} \quad \varPhi ^{\text{III},{R}}_{\textit{fIG}}=\mathfrak{D}^{\text{III}} {\text{e}}^{2{\text{i}} \frac {\omega _{{L}}}{\sqrt {{\text{g}} h}}X} {\text{e}}^{-2{\text{i}}\omega _{{L}}T} +\text{c.c.} \end{equation}

Equation (6.7) contains four more unknown coefficients, $\mathfrak{D}^{\text{I}}$ , $\mathfrak{D}^{\text{III}}$ and their complex conjugates $\overline {\mathfrak{D}^{\text{I}}}$ and $\overline {\mathfrak{D}^{\text{III}}}$ . The unknowns are determined by the following four matching conditions stating that the vertical integrals of the potentials and their fluxes in (6.6) and (6.7) are continuous at the bow $(X=-L)$ and the stern $(X=L)$ of the heaving body:

(6.8) \begin{align}&\quad \left . \int _{-h}^{0} \varPhi _{\textit{fIG}}^{\text{I},{R}} \text{d} z = \int _{-h}^{-d} \varPhi _{\textit{IG}}^{\text{II},{R}} \text{d} z \right |_{X=-L}, \end{align}

(6.9) \begin{align}&\quad \left . \int _{-h}^{-d} \varPhi _{\textit{IG}}^{\text{II},{R}} \text{d} z = \int _{-h}^{0} \varPhi _{\textit{fIG}}^{\text{III},{R}} \text{d} z \right |_{X=L}, \end{align}

(6.10) \begin{align}&\,\,\, \left . \int _{-h}^{0} \dfrac {{\partial } \varPhi _{\textit{fIG}}^{\text{I},{R}}}{{\partial } X} \text{d} z = \int _{-h}^{-d} \dfrac {{\partial } \varPhi _{\textit{IG}}^{\text{II},{R}}}{{\partial } X} \text{d} z \right |_{X-L}, \end{align}

(6.11) \begin{align}& \left . \int _{-h}^{0} \dfrac {{\partial } \varPhi _{\textit{IG}}^{\text{II},{R}}}{{\partial } X}\text{d} z = \int _{-h}^{-d} \dfrac {{\partial } \varPhi _{\textit{fIG}}^{\text{III},{R}}}{{\partial } X} \text{d} z \right |_{X=-L}. \end{align}

Substituting the contents of $\varPhi _{\textit{fIG}}^{\text{I},{R}}$ , $\varPhi _{\textit{IG}}^{\text{II},{R}}$ and $\varPhi _{\textit{fIG}}^{\text{III},{R}}$ into the above matching conditions and collecting like terms, one obtains eight equations determining all eight unknowns $\mathcal{B}_1, \mathcal{B}_2$ , $\overline {\mathcal{B}_1}, \overline {\mathcal{B}_2}$ , $\mathfrak{D}^{\text{I}}$ , $\mathfrak{D}^{\text{III}}$ , $\overline {\mathfrak{D}^{\text{I}}}$ and $\overline {\mathfrak{D}^{\text{III}}}$ .

The radiated free IG waves can now be calculated by (cf. 5.7)

(6.12a,b) \begin{align} \eta _{\textit{fIG}}^{\text{I},{R}}=-\dfrac {1}{{\text{g}}}\dfrac {{\partial }\varPhi _{\textit{fIG}}^{\text{I},{R}}}{{\partial } T}, X\lt -L; \quad \eta _{\textit{fIG}}^{\text{III},{R}}=-\dfrac {1}{{\text{g}}}\dfrac {{\partial }\varPhi _{\textit{fIG}}^{\text{III},{R}}}{{\partial } T}, X\gt L. \end{align}

To this end, all unknowns in (2.16) are determined.

7.1. An example

We start by providing in figure 3 an example ( $L=0.1h$ , $d=0.5 h$ , $k_{0,1}h=1.20$ , $A_1=A_2=0.04$ m, $h=1$ m) to illustrate carrier and IG waves around a stationary block. Panel (a) shows carrier waves; bound and free IG waves are given in panel (b). The total transmitted IG wave is given in panel (c) for $T=0$ , while in panel (d), total IG waves are evaluated at six different times, superimposed to reveal an amplitude envelope. Panel (d) also shows that the minimum transmitted total IG wave amplitude is near zero, but the maximum is almost one. The reason for such an amplitude modulation (cf. § 5.2) can be understood in view of panel (b), where the spatial wavelength of the transmitted free IG waves is almost twice the bound IG waves, so that when they superimpose, peaks and troughs in the total IG wave amplitude are created. In this case, if reducing total IG Waves is the goal, then the optimal location is approximately 50 m in the lee of the block.

Figure 3. An illustration of wave components around a 2-D stationary block (diffraction only). The case is taken from the red curves of the second column of figure 5 where $L/h=0.1, d/h=0.5$ . The value of $k_{0,1}h$ is set to 1.20. The figures show surface elevation $\eta$ for: (a) carrier waves; (b) incident and transmitted IG waves; (c) total IG waves at $t=0$ ; (d) total IG waves evaluated at various times, revealing the modulated amplitude envelope. In all plots, bIG means bound IG waves, and fIG means free IG waves. See § 7.1.

Figure 4. Validity range. For a given value of $L/h$ , when the value of $k_{0,1} h$ is located to the right of the red curve (i.e. in the light region), then the inequality (7.1) fails. This diagram is calculated with $\omega _2= 1.1 \omega _1$ and $h = 1$ m. Gridlines are added to mark three points on the red curve corresponding to $L/h=1, 1.5$ and 2. For details, see § 7.2.

7.2. Systematic calculation of more cases

Consider 2-D blocks specified by combinations of $L=0.01h, 0.1h, 1.0h, 1.5 h, 2.0h$ and $d=0.1h, 0.3h, 0.5h$ . All blocks are evaluated for wave conditions up to $k_{0,1} h=2$ . The two carrier waves satisfy $f_2=1.1f_1$ . To ensure the smallness of the blocks compared with the IG waves, it is required that

(7.1) \begin{equation} 2L\lt \dfrac {2\lambda _{{g}}}{20} ,\end{equation}

where $2L$ gives the total length of the block, $\lambda _{{g}}=2\pi /(2k_{{L}})$ is the length of the carrier wave group (so that the IG wave length is $2\lambda _{{g}}$ ). For blocks with $L=0.01h$ and $0.1h$ , the inequality (7.1) is always satisfied in the range $k_{0,1}h\in [0,2]$ . However, for blocks with $L=1.0h$ , $k_{0,1}h$ should be smaller than 1.768, for $L=1.5h$ , $k_{0,1}h$ should be smaller than 1.349 and when $L=2.0h$ , $k_{0,1}h$ should be less than 1.113. The $k_{0,1}h$ limits for other values of $L/h$ are given in figure 4.

The transmission coefficient of carrier waves is discussed first. It is defined as

(7.2) \begin{equation} \text{transmission coefficient}=\dfrac {\text{amplitude of the transmitted carrier wave}}{\text{amplitude of the incident carrier wave}}. \end{equation}

Figure 5. Wave modification due to stationary 2-D blocks. Five columns correspond to different $L$ ; see the legends at the top of each column. Light blue shades are added as visual guides to distinguish different columns. The four rows are: row 1, transmission coefficient for monochromatic waves with non-dimensional wavenumber $k_{0,1}h$ . Row 2: the total bound IG wave amplitude in region III normalised by the incident bound IG wave amplitude. Row 3: similar to row 2 but for normalised free IG wave amplitude. Row 4: for total IG wave amplitude. In all plots, black, blue and red curves correspond to $d=0.10 h, 0.30 h$ and $0.50 h$ . In row 4, dashed and solid curves are the maximum and minimum amplitudes, respectively. In the red shades of plots (n, o, s, t, x, y), the condition (7.1) is not met, and the results may be invalid. In the light grey shades, $k_{0,1}h\lt 0.15$ , and no results are presented. In all cases, the two carrier wave frequencies satisfy $f_2=1.1 f_1$ . These results are relevant to § 7.

The first row of figure 5 presents the transmission coefficient of the first carrier wave (with wavenumber $k_{0,1}$ ). The transmission of the second carrier wave (with wavenumber $k_{0,2}$ ) will follow the same trend, so it is omitted. A general observation from the first row of figure 5 is that the carrier waves are attenuated as $L$ and $d$ increase, meaning larger-sized blocks are more effective in reflecting the incident carrier waves.

The second row of figure 5 shows the normalised amplitude of total bound IG waves, as the ratio between the total bound IG wave amplitude in region III ( $|\eta _{\textit{bIG}}^{\text{III},{\textit{tot}}}|$ ) and the incident bound IG wave amplitude in region I ( $|\eta _{\textit{bIG}}^{\text{I},{A}}|$ ). Similarly to the carrier wave’s transmission coefficient, it reduces as $L$ and $d$ increase. The light red regions in figure 5 highlight the regimes where the inequality (7.1) is not respected, so caution should be exercised when interpreting the results in these red regions. In the light grey regions where $k_{0,1}h\lt 0.15$ , no IG wave results are shown, because for the examples considered here (which assume that the water depth $h=1$ m to be relevant to laboratory conditions), the IG wavelength is too long to be realistic when $k_{0,1}h\lt 0.15$ .

The third row of figure 5 shows free IG waves in the form of the ratio between the total free IG wave amplitude in region III ( $|\eta _{\textit{fIG}}^{\text{III},{\textit{tot}}}|$ ) and the incident bound IG wave amplitude. The free IG wave increases as $L$ and $d$ increase, suggesting that reducing bound IG waves incurs a cost: the release of free IG waves. Note that in panels (n) and (o) of figure 5, the free IG wave can be noticeably larger than 1. Caution should be exercised when interpreting these results, as they occur in the light red regions where the condition (7.1) is not satisfied.

Adding up the bound and free waves in region III, one obtains the last row of figure 5, showing the total IG wave amplitude (i.e. $|\eta _{\textit{IG}}^{\text{III},{\textit{tot}}}|$ ) normalised by the incident IG wave. Due to the reasons mentioned in § 5.2, $|\eta _{\textit{IG}}^{\text{III},{\textit{tot}}}|$ is modulated: it depends not only on $k_{0,1}, L, d$ , but also on the location $x$ . For this reason, an upper bound (dashed lines) and a lower bound (solid lines) exist when evaluating the total IG wave amplitude. It is clear that although the lower bound can be as small as zero (the IG wave is wholly eliminated), the upper bound is usually close to one (the IG wave amplitude is comparable to the incident IG wave).

7.3. A summary: diffraction-only results

This section reveals that when reducing the total IG waves is the goal, the best a diffraction-only block can do is the location-dependent reduction exemplified by figure 4(d): at some locations, the total IG wave amplitude is always near zero for all times, while at some other locations, the total IG wave is not reduced. This is due to the superposition of free and bound IG waves. Such a location-dependent reduction can be achieved by small-sized blocks (e.g. $L=0.1 h$ ) as well as larger blocks. The larger blocks have a stronger impact on lower values of $k_{0,1} h$ while the reverse is true for the small-sized blocks. If bound and free IG waves are shown separately, figure 5 shows that, when bound IG waves are small, free IG waves are non-small, vice versa. This aspect concurs with Liu & Iskandarani (Reference Liu and Iskandarani1991).

8.1. An example

An example is provided to demonstrate that the total IG waves can be reduced everywhere in the lee of a block. Consider a block with $d=0.5 h, L=2.0 h$ . Set $k_{0,1} h=0.95, A_1=A_2=0.04$ m. The wave elevations around the block are presented in figure 6. Incident and transmitted carrier wave groups are given in panel (a). The IG waves bound to these wave groups are given in panel (b), which also shows bound IG waves due to cross-interaction between radiated and diffracted carrier waves, as well as free IG waves. The total IG waves are given in panel (c), evaluated at a specific time $(T=0)$ . Evaluating the waves at more values of $T$ and combining the plots, one obtains figure 6(d). The spatial amplitude modulation (cf. § 5.2) of the transmitted total IG waves is again observed. Despite the modulation, at all $x\gt L$ locations, the total IG waves are reduced everywhere.

Figure 6. An illustration of wave components around a heaving 2-D block ( $L=2h, d=0.5 h$ where $h=1$ m, $k_{0,1}h=0.95$ ). These correspond to a case on the red curves in the fifth column of figure 7. Surface elevation $\eta$ for: (a) first-order carrier waves; (b) incident and transmitted bound and free IG waves; (c) the total IG waves; (d) similar to (c) but evaluated at various times, illustrating the modulated amplitude envelope in region III. In all plots, bIG means bound IG waves, and fIG means free IG waves.

8.2. Systematic calculation of more cases

Figure 7. Wave modification due to heaving 2-D blocks. Five columns correspond to different $L$ ; see the legends at the top of each column. Light green shades are added to columns 2 and 4 as visual guides to distinguish different columns. Rows 1 and 2: transmission coefficient and RAO for monochromatic waves with non-dimensional wavenumber $k_{0,1}h$ . Row 3: the total bound IG wave amplitude in region III normalised by the incident bound IG wave amplitude. Row 4: similar to row 3, but for total free IG wave amplitude. Row 5: similar to row 3, but for total IG wave amplitude. Black, blue and red curves correspond to $d=0.10 h, 0.30 h$ and $0.50 h$ . In row 5, dashed and solid curves are the maximum and minimum IG wave amplitudes, respectively. In all cases, the two carrier wave frequencies satisfy $f_2=1.1 f_1$ . In the red shades of plots (n, o, s, t, x, y), the condition (7.1) is not met, and the results may not be valid. In grey shades, $k_{0,1}h\lt 0.15$ , results for IG waves are not presented. This figure is relevant to § 8.

Figure 7 shows results for heaving blocks. The transmission of the carrier waves is presented in the first row. Troughs of the transmission curves are related to the peaks of the heaving response amplitude operator (RAO) curves in the second row: when the values of $k_{0,1}h$ are to the right of the values of RAO peaks, the transmission coefficient quickly reduces (cf. Cui et al. Reference Cui, Sergiienko, Cazzolato, Leontini, Tothova, Cannard, Spinks and Manasseh2023). The RAO peaks correspond to linear resonance; their locations are decided by both $d$ and $L$ : as $d$ and $L$ increase, they move to lower values of $k_{0,1}h$ . The transmission of total bound IG waves is given by the third row of figure 7; their shapes are similar to the first row. These curves can be above unity, e.g. panels (m), (n), (o). The reason for this is given in Appendix G. The fourth row of figure 7 presents the amplitude of total free IG waves. Unlike the stationary blocks, the total free IG waves do not increase monotonically. The curves now have peaks and troughs, and the troughs can reach near-zero values, because the total free IG waves are the sum of induced free IG wave $\eta _{\textit{fIG}}^{\text{III},{M}}$ (5.7) and the radiated free IG wave $\eta _{\textit{fIG}}^{\text{III},{R}}$ (6.12); the two waves can interact destructively. The last row of figure 7 shows total IG wave (2.16). Like the diffraction-only case, the total IG wave amplitude has upper (dashed lines) and lower bounds. Unlike the diffraction-only case, where the upper bound is usually close to or above one, now the upper bounds can be noticeably lower than one, meaning the total IG waves can be reduced everywhere in the lee of the block. For the cases studied, these cases usually occur when $d=0.5 h$ , and when $k_{0,1} h$ is greater or smaller than the RAO peaks, indicating that avoiding linear resonance leads to reduction of IG waves. The relationship between IG wave and RAO peaks is studied in more detail in the following subsection.

8.3. Relating linear resonance (RAO peaks) to IG wave modification

Figure 8. Panels (a), (b) and (c) show normalised amplitude of total free IG waves in the lee (region III) of a 2-D block. Panels (d), (e), ( f) are similar, but for total IG waves. The white solid lines are locations of the 2-D block’s RAO peaks (cf. row 2 of figure 7). In the white-coloured regions, the condition (7.1) is not met, and the results are not presented. This plot is similar to rows 4 and 5 of figure 7, except that the parameter $L/h$ is varying continuously.

The previous subsection mentioned that for deeper-draught blocks ( $d=0.5 h$ ), when $k_{0,1}h$ is located to either side of the RAO peak curve, the maximum total IG wave amplitude can be reduced, establishing a link between RAO curves (resulting from linear analysis) and IG waves (nonlinearly generated). This subsection studies this observation further, by overlapping locations of RAO peaks with IG wave results. To identify the trend clearly, $L$ is now allowed to vary continuously, and the upper limit of $L/h$ is increased from 2 to 3. The results are plotted in figure 8 in the form of heat maps, for the total free IG waves (row 1) and the maximum total IG waves (row 2). These quantities are related to row 4 of figure 7 and the dashed curves in row 5 of figure 7, respectively.

The first row of figure 8 shows that, as $d$ increases, the RAO peak curve (white curve) relocates to largely coincide with a red-coloured ‘ridge’ on the heat map where total free IG waves assume relatively large amplitudes. To both sides of the ‘ridge’, the free IG wave amplitudes decrease as the colour turns towards blue. The same conclusion applies to the second row of figure 8, where maximum total IG waves are shown. These conclusions do not apply to small- $d$ cases (panels a and d), presumably because, according to row two of figure 7, the RAOs for the small- $d$ cases are devoid of obvious peaks; hence, the RAO peak curve ceases to be a meaningful indicator.

Overall, when $d$ is very large, a relationship exists between a block’s linear RAO peak and its ability to attenuate transmitted total IG waves.

8.4. A summary of heaving cases

Compared with diffraction-only blocks, allowing heaving fundamentally changes the behaviour of IG waves around the block. From an IG wave reduction point of view, it is possible to reduce IG waves everywhere in the lee (region III) by up to 50 %, particularly for blocks with larger $L$ and $d$ , e.g. the $L=1.5 h$ and $2.0 h$ cases with $d=0.5 h$ , see the red dashed curves in figure 7(x, y), or figure 8(c, f).