3.1. Velocity potential due to a single source: the Green function

We may first define the non-dimensional variables based on the density of fluid $\rho$ together with $g$ and $H$. Typically, $\phi ^{\prime }=\phi /H\sqrt {gH}$, $x^{\prime }=x/H$, $z^{\prime }=z/H$ and $\epsilon ^{\prime }=\epsilon \sqrt {H/g}$. For convenience, the primes are omitted in the following. In such a case, (2.5) and (2.6) can be re-expressed as

(3.1)\begin{gather} \left(\epsilon + F\frac{\partial}{\partial x}\right)^2\phi + \frac{\partial \phi}{\partial z} = 0,\quad -\infty < x < 0, \ z=0, \end{gather}

(3.2)\begin{gather} \left(D\frac{\partial^4}{\partial x^4} + M\epsilon^2 +1\right)\frac{\partial \phi}{\partial z}+\left(\epsilon + F\frac{\partial }{\partial x}\right)^2\phi = 0,\quad 0 < x <{+}\infty,\ z = 0, \end{gather}

where $F=U/\sqrt {gH}$ denotes the depth-based Froude number, $D=L/\rho gH^4$ and $M=m_i/\rho H$.

The Green function $G(x,z;x_0,z_0)$ is first introduced, which is the velocity potential at a field point $(x,y)$ induced by a single source at $(x_0,z_0)$. Here $G$ satisfies the following equation

(3.3)\begin{equation} \nabla^2G=2{\rm \pi} \delta\left(x-x_0\right)\delta\left(z-z_0\right),\quad -\infty < x <{+}\infty, \ -1 \leq z \leq 0, \end{equation}

and the boundary conditions in (2.8)–(3.2), where $\delta (x)$ is the Dirac delta function. To find $G$, we may apply the Fourier transform to $G$

(3.4)\begin{equation} \hat{G}(\alpha,z)=\int_{-\infty}^{+\infty}G(x,z){\rm e}^{{\rm i}\alpha x}\,{{\rm d}\kern 0.06em x}. \end{equation}

Here $\alpha$ is extended to the complex plane. As the conditions in (3.1) and (3.2) are divided by the sign of $x$, we may use the Wiener–Hopf technique (Noble Reference Noble1958) and further introduce the following Fourier transforms

(3.5)\begin{equation} \left. \begin{aligned} \hat{G}_{-}(\alpha,z) & =\int_{-\infty}^{0}G(x,z){\rm e}^{{\rm i}\alpha x}\,{{\rm d}\kern 0.06em x},\\ \hat{G}_{+}(\alpha,z) & =\int_{0}^{+\infty}G(x,z){\rm e}^{{\rm i}\alpha x}\,{{\rm d}\kern 0.06em x}. \end{aligned} \right\} \end{equation}

Because of the presence of $\epsilon$ in (3.1) and (3.2), $G$ decays as $x\rightarrow \pm \infty$ at a rate of ${\rm e}^{-\epsilon _0|x|}$. Here $\epsilon _0>0$ and tends to zero when $\epsilon \rightarrow 0^-$. Following Noble (Reference Noble1958), $\hat {G}_+(\alpha,z)$ is analytical in the region ${\rm Im}\{\alpha \}>-\epsilon _0$ and $\hat {G}_-(\alpha,z)$ is analytical in the region ${\rm Im}\{\alpha \}<\epsilon _0$, respectively. From (3.4) and (3.5), we have

(3.6)\begin{equation} \hat{G}(\alpha,z)=\hat{G}_{+}(\alpha,z) + \hat{G}_{-}(\alpha,z). \end{equation}

Applying (3.4) to (3.3), we obtain

(3.7)\begin{equation} \frac{\partial^2\hat{G}}{\partial z^2} - \alpha^2\hat{G}=2{\rm \pi} {\rm e}^{{\rm i}\alpha x_0}\delta(z-z_0). \end{equation}

Based on the boundary condition in (2.8), (3.7) can be solved as

(3.8)\begin{equation} \hat{G}=A(\alpha)C(z,\alpha) + \frac{2{\rm \pi} {\rm e}^{{\rm i}\alpha x_0}}{\alpha} \left\{ \begin{array}{@{}ll} S(z,\alpha)C(z_0,\alpha), & z > z_0,\\ C(z,\alpha)S(z_0,\alpha), & z < z_0 , \end{array} \right. \end{equation}

where $A(\alpha )$ is an unknown coefficient and

(3.9)\begin{equation} \left. \begin{aligned} S(z,\alpha) & = \sinh\alpha(z+1),\\ C(z,\alpha) & = \cosh\alpha(z+1). \end{aligned} \right\} \end{equation}

Substituting (3.5) and (3.8) into (3.1) and (3.2) and keeping only the leading term of $\epsilon$, we have

(3.10)\begin{align} &D_{+}(\alpha)+D_{-}(\alpha)=A(\alpha)K_1^{\epsilon}(\alpha,F)\nonumber\\ &\qquad +\frac{2{\rm \pi} {\rm e}^{{\rm i}\alpha x_0}}{\alpha}\left(\alpha \cosh\alpha - F^2\alpha^2 \sinh\alpha -2{\rm i}\epsilon F\alpha \sinh\alpha \right)C(z_0,\alpha), \end{align}

(3.11)\begin{align} &F_{+}(\alpha)+F_{-}(\alpha)=A(\alpha)K_2^{\epsilon}(\alpha,F)\nonumber\\ &\qquad +\frac{2{\rm \pi} {\rm e}^{{\rm i}\alpha x_0}}{\alpha}\left[\left(D\alpha^4+1\right)\alpha \cosh\alpha - F^2\alpha^2 \sinh\alpha - 2{\rm i}\epsilon F\alpha \sinh\alpha\right]C(z_0,\alpha), \end{align}

where

(3.12)\begin{equation} K_{i}^{\epsilon}(\alpha,F)=K_{i}(\alpha,\alpha F + {\rm i}\epsilon)\approx K_{i}(\alpha, \alpha F)-{\rm i}\epsilon\text{sgn}(\alpha),\quad i = 1,2, \end{equation}

and

(3.13a)\begin{gather} K_{1}(\alpha,\alpha F)=\alpha \sinh\alpha - \left(\alpha F\right)^2\cosh\alpha, \end{gather}

(3.13b)\begin{gather} K_{2}(\alpha,\alpha F)=\left(D\alpha^4+1\right)\alpha \sinh\alpha - \left(\alpha F \right)^2\cosh\alpha. \end{gather}

In (3.10) and (3.11) $D_{\pm }(\alpha )$ and $F_{\pm }(\alpha )$ are the Fourier transforms of (3.1) and (3.2), respectively, defined in (3.5). It should be noted that the main effect of $\epsilon$ term in (3.12) is that the location of $K_i^{\epsilon }(\alpha,F)=0$ ($i=1, 2$) will be changed slightly from those corresponding to $K_i(\alpha,\alpha F)=0$. When $\alpha$ is a fully complex number, such a change may be trivial. However, when $\alpha$ is a real number, such a change becomes significant, as it will affect the path at the singularities when the inverse Fourier transform is performed, and affect the decomposition in the Wiener–Hopf method. For this reason, the sign function ${\rm sgn}(\alpha )$ is used in (3.12) as the term is significant only when $\alpha$ is real. It should be noted that $\epsilon$ will also slightly change the double root of $K_i(\alpha,\alpha F)=0$ at $\alpha =0$. However, such a change reflected in the integration path of the inverse transform at $\alpha =0$ is actually equivalent to adding a constant term to the Green function (Li et al. Reference Li, Wu and Shi2019), and will not affect the results. Thus, the effect of $\epsilon$ on the roots at $\alpha =0$ is neglected.

From the boundary conditions in (3.1) and (3.2), we have $D_-(\alpha )=F_+(\alpha )=0$. To obtain $D_+(\alpha )$ and $F_-(\alpha )$, we may eliminate $A(\alpha )$ from (3.10) and (3.11), and remove the trivial $\epsilon$ terms, which provides

(3.14)\begin{equation} F_{-}(\alpha) = D_{+}(\alpha)K^{\epsilon}(\alpha,F)-\frac{2{\rm \pi} DF^2\alpha^6}{K_1^{\epsilon}(\alpha, F)}C(z_0,\alpha){\rm e}^{{\rm i}\alpha x_0}, \end{equation}

where

(3.15)\begin{equation} K^{\epsilon}(\alpha,F)=\frac{K_2^{\epsilon}(\alpha, F)}{K_1^{\epsilon}(\alpha, F)}. \end{equation}

Following the procedure of Wiener–Hopf method, in the complex plane of $\alpha$, (3.15) may be factorised as

(3.16)\begin{equation} K^{\epsilon}(\alpha, F)=K^{\epsilon}_{-}(\alpha, F)K^{\epsilon}_{+}(\alpha, F), \end{equation}

where $K_-^{\epsilon }(\alpha,F)$ are $K_+^{\epsilon }(\alpha,F)$ are analytical in their own regions in the complex plane. To do that, we need to find the distribution of the roots of $K_i^{\epsilon }(\alpha,F)=0$ ($i=1,2$).

As shown in McCue & Stump (Reference McCue and Stump2000) for $K_1(\alpha,\alpha F)=0$ and Appendix A for $K_2(\alpha,\alpha F)=0$, $K_1(\alpha,\alpha F)=0$ and $K_2(\alpha,\alpha F)=0$ both have an infinite number of roots at $\alpha =\pm \mathcal {k}_m$ ($m=0,1,2,\ldots$) and $\alpha =\pm \kappa _m$ ($m=-1,0,1,\ldots$), respectively. The properties of the roots are related to the Froude number $F$. Typically, for $K_1(\alpha,\alpha F)=0$, $\mathcal {k}_0$ is a positive real root when $F<1$, and $\mathcal {k}_0$ is a purely positive imaginary root between $0$ and ${\rm \pi} {\rm i}/2$ when $F>1$. Here $\mathcal {k}_m$ ($m\geq 1$) are all purely positive imaginary roots, and $\mathcal {k}_m$ is between $m{\rm \pi} {\rm i}$ and $(m{\rm \pi} +{\rm \pi} /2){\rm i}$. For $K_2(\alpha, \alpha F)=0$, there is a critical Froude number $F_c$ (Li et al. Reference Li, Wu and Shi2019). When $0< F< F_c$, $\kappa _{-1}$ and $\kappa _0$ are two complex roots with positive imaginary part, which satisfy $\kappa _0=-\bar {\kappa }_{-1}$ and ${\rm Re}\{\kappa _{-1}\}>0$. When $F_c< F<1$, $\kappa _{-1}$ and $\kappa _0$ become two positive real roots with $\kappa _{-1}>\kappa _0$. When $F>1$, $\kappa _{-1}$ remains to be a positive real root, but $\kappa _{0}$ becomes a purely positive imaginary root between $0$ and ${\rm \pi} {\rm i}/2$. Similar to $\mathcal {k}_m$ ($m\geq 1$), $\kappa _m$ ($m\geq 1$) are all purely negative imaginary roots between $m{\rm \pi} {\rm i}$ and $(m{\rm \pi} +{\rm \pi} /2){\rm i}$.

The root of $K_1^{\epsilon }(\alpha,F)=0$ in (3.12) corresponding to $\pm \mathcal {k}_0$ can be obtained as $\pm \mathcal {k}_0-{\rm i}\epsilon _0^{\prime }$ ($\epsilon _0^{\prime }\rightarrow 0$). Similarly, the roots of $K_2^{\epsilon }(\alpha,F)=0$ in (3.12) corresponding to $\pm \kappa _m$ ($m=-1, 0$) can be expressed as $\pm \kappa _m-{\rm i}\varepsilon _m^{\prime }$ ($\varepsilon _m^{\prime }\rightarrow 0$) respectively. When $\mathcal {k}_0$ and $\kappa _m$ ($m=-1, 0$) are real, we have

(3.17)\begin{equation} \left. \begin{aligned} \epsilon_0^{\prime} & ={-}\epsilon\text{sgn}(\mathcal{k}_0)\left/\frac{\partial K_1(\mathcal{k}_0,\mathcal{k}_0F)}{\partial \alpha} \right. ={-}\epsilon \text{sgn}(-\mathcal{k}_0)\left/\frac{\partial K_1(-\mathcal{k}_0,-\mathcal{k}_0F)}{\partial \alpha},\right.\\ \varepsilon^{\prime}_m & ={-}\epsilon\text{sgn}(\kappa_m)\left/\frac{\partial K_2(\kappa_m,\kappa_m F)}{\partial \alpha} \right.={-}\epsilon\text{sgn}(-\kappa_m)\left/\frac{\partial K_2(-\kappa_m, -\kappa_m F)}{\partial \alpha}\right. . \end{aligned} \right\} \end{equation}

From (3.13a), it can be shown that ${\partial K_1(\mathcal {k}_0,\mathcal {k}_0 F)}/{\partial \alpha }<0$. From (3.17), $\epsilon _0^{\prime }<0$, which means $\textrm {Im}\{\pm \mathcal {k}_0-\textrm {i}\epsilon _0^{\prime }\}>0$. Similarly, from (3.13b), ${\partial K_2(\kappa _{-1},\kappa _{-1}F)}/{\partial \alpha }>0$ and ${\partial K_2(\kappa _0,\kappa _0F)}/{\partial \alpha }<0$ and, therefore, $\varepsilon _{-1}^{\prime }>0$, $\varepsilon _{0}^{\prime }<0$, which gives $\textrm {Im}\{\pm \kappa _{-1}-\textrm {i}\varepsilon _{-1}^{\prime }\}<0$ and $\textrm {Im}\{\pm \kappa _{0}-\textrm {i}\varepsilon _{0}^{\prime }\}>0$. When $\mathcal {k}_0$ and $\kappa _m$ ($m=-1, 0$) are not purely real, $\epsilon _0^{\prime }=0$ and $\varepsilon _m^{\prime }=0$. Thus, we may write

(3.18a)\begin{gather} \left. \begin{array}{ll@{}} \epsilon_0^{\prime} <0, & \text{Im}\{\mathcal{k}_0\}=0\ (0< F<1),\\ \epsilon_0^{\prime} =0, & \text{Im}\{\mathcal{k}_0\}\neq0\ (F>1), \end{array} \right\} \end{gather}

(3.18b)\begin{gather} \left. \begin{array}{ll@{}} \varepsilon_{{-}1}^{\prime} >0, & \text{Im}\{\kappa_{{-}1}\}=0\ (F>F_c),\\ \varepsilon_{{-}1}^{\prime} =0, & \text{Im}\{\kappa_{{-}1}\}\neq0\ (0< F< F_c), \end{array} \right\} \end{gather}

(3.18c)\begin{gather} \left. \begin{array}{ll@{}} \varepsilon_{0}^{\prime} >0, & \text{Im}\{\kappa_{0}\}=0\ (F_c< F<1),\\ \varepsilon_{0}^{\prime} =0, & \text{Im}\{\kappa_{0}\}\neq0\ (0< F< F_c \ \text{and} \ F>1). \end{array} \right\} \end{gather}

In such a case, based on the Weierstrass factorisation, we have

(3.19a)$$\begin{gather} K_{1}^{\epsilon}(\alpha,F)=(1-F)^2\alpha^2\left(1-\frac{\alpha}{\mathcal{k}_0 - {\rm i}\epsilon_0^{\prime}}\right)\left(1+\frac{\alpha}{\mathcal{k}_0 + {\rm i}\epsilon_0^{\prime}}\right)\prod_{m=1}^{+\infty}\left(1-\frac{\alpha^2}{\mathcal{k}_m^2}\right), \end{gather}$$

(3.19b) $$\begin{gather}K_{2}^{\epsilon}(\alpha,F)=(1-F)^2\alpha^2\Bigg(1-\frac{\alpha}{\kappa_{{-}1} - {\rm i}\varepsilon_{{-}1}^{\prime}}\Bigg)\Bigg(1+\frac{\alpha}{\kappa_{{-}1} + {\rm i}\varepsilon_{{-}1}^{\prime}}\Bigg)\nonumber\\ \qquad\qquad\qquad\qquad\ \ \times\left(1-\frac{\alpha}{\kappa_{0} - {\rm i}\varepsilon_{0}^{\prime}}\right)\left(1+\frac{\alpha}{\kappa_{0} + {\rm i}\varepsilon_{0}^{\prime}}\right)\prod_{m=1}^{+\infty}\left(1-\frac{\alpha^2}{\kappa_m^2}\right). \end{gather}$$

From (3.13), it can be shown that $\mathcal {k}_m=\textrm {i}(m+\tfrac {1}{2}){\rm \pi} +O(m^{-1})$ and $\kappa _m=\textrm {i}m{\rm \pi} +O(m^{-3})$ when $m\rightarrow +\infty$, which mean that (3.19) is convergent. We may define

(3.20)\begin{equation} \mathcal{K}_{i}^{\epsilon}(\alpha,F)=K_{i}^\epsilon(\alpha,F)/\alpha^2,\quad i = 1,2, \end{equation}

where $\mathcal {K}_i^{\epsilon }(\alpha,F)$ ($i=1, 2$) is bounded at $\alpha =0$. In (3.19), when $F\rightarrow 1$, $(1-F^2 )\rightarrow 0$, $\mathcal {k}_0\rightarrow 0$ and $\kappa _0\rightarrow 0$, and it can be found from (3.13) that ${\lim }_{F\rightarrow 1}({1-F^2})/{\mathcal {k}_0^2}={\lim }_{F\rightarrow 1}({1-F^2})/{\kappa _0^2}=\tfrac {1}{3}$. Therefore, the Weierstrass factorisations in (3.19) are still valid. Substituting (3.19) into (3.15), we have

(3.21) \begin{align} K^{\epsilon}(\alpha,F)&=\frac{\mathcal{k}_0\gamma_{{-}1}^+(\alpha,F)\gamma_{{-}1}^-(\alpha,F)\gamma_{0}^+(\alpha,F)\gamma_{0}^-(\alpha,F)\left[(\kappa_{{-}1}\!+\!{\rm i}\varepsilon_{{-}1}^{\prime})+\alpha\right]\left[(\kappa_{0}\!-\!{\rm i}\varepsilon_{0}^{\prime})-\alpha\right]}{\kappa_{{-}1}\kappa_{0}\beta_0^+(\alpha,F)\beta_{0}^-(\alpha,F)\left[(\mathcal{k}_0 \!-\! {\rm i}\epsilon_0^{\prime})\!-\!\alpha\right]}\nonumber\\ &\quad \times\prod_{m=1}^{+\infty}\frac{\mathcal{k}_m^2\left(\kappa_m^2-\alpha^2\right)}{\kappa_m^2\left(\mathcal{k}_m^2-\alpha^2\right)}, \end{align}

where

(3.22a)\begin{gather} \beta_{0}^+(\alpha,F)=\left\{ \begin{array}{@{}ll} 1, & \text{Im}\{\mathcal{k}_0\}=0,\\ \dfrac{\mathcal{k}_0+\alpha}{\mathcal{k}_0}, & \text{Im}\{\mathcal{k}_0\}>0, \end{array} \right.\quad \beta_{0}^-(\alpha,F)=\!\left\{ \begin{array}{@{}ll} \dfrac{(\mathcal{k}_0+{\rm i}\epsilon_0^{\prime})+\alpha}{\mathcal{k}_0}, & \text{Im}\{\mathcal{k}_0\}\!=\!0,\\ 1, & \text{Im}\{\mathcal{k}_0\}>0, \end{array} \right. \end{gather}

(3.22b) \begin{gather}\left. \begin{array}{l} \gamma_{{-}1}^+(\alpha,F)=\left\{ \begin{array}{@{}ll} \dfrac{(\kappa_{{-}1}-{\rm i}\varepsilon_{{-}1}^{\prime})-\alpha}{\kappa_{{-}1}}, & \text{Im}\{\kappa_{{-}1}\}=0,\\ 1, & \text{Im}\{\kappa_{{-}1}\}>0, \end{array} \right. \nonumber\\ \gamma_{{-}1}^-(\alpha,F) =\left\{ \begin{array}{@{}ll} 1, & \text{Im}\{\kappa_{{-}1}\}=0,\\ \dfrac{\kappa_{{-}1}-\alpha}{\kappa_{{-}1}}, & \text{Im}\{\kappa_{{-}1}\}>0. \end{array} \right. \end{array}\right\} \end{gather}

(3.22c)\begin{gather} \gamma_{0}^+(\alpha,F)=\left\{ \begin{array}{@{}ll} 1, & \text{Im}\{\kappa_0\}=0,\\ \dfrac{\kappa_{0}+\alpha}{\kappa_{0}}, & \text{Im}\{\kappa_{0}\}>0, \end{array} \right.\quad \gamma_{0}^-(\alpha,F)=\left\{ \begin{array}{@{}ll} \dfrac{(\kappa_{0}+{\rm i}\varepsilon_0^{\prime})+\alpha}{\kappa_0}, & \text{Im}\{\kappa_0\}=0,\\ 1, & \text{Im}\{\kappa_0\}>0. \end{array} \right. \end{gather}

From (3.12), (3.13) and (3.15), it can be shown that $K^{{\epsilon }}(\alpha,F)\sim O(|\alpha |^3)$ when $|\alpha |\rightarrow +\infty$. Using (3.16), $K_\pm ^{\epsilon }(\alpha,F)$ can be defined as

(3.23a)\begin{gather} K_+^{\epsilon}(\alpha,F)=\frac{\gamma_{{-}1}^+(\alpha,F)\gamma_{0}^+(\alpha,F)}{\beta_0^+(\alpha,F)}\frac{\left[(\kappa_{{-}1}+{\rm i}\varepsilon_{{-}1}^{\prime})+\alpha\right]}{\kappa_{{-}1}}\prod_{m=1}^{+\infty}\frac{\mathcal{k}_m(\kappa_m+\alpha)}{\kappa_m(\mathcal{k}_m+\alpha)}, \end{gather}

(3.23b)\begin{gather} K_-^{\epsilon}(\alpha,F)=\frac{\gamma_{{-}1}^-(\alpha,F)\gamma_{0}^-(\alpha,F)}{\beta_0^-(\alpha,F)}\frac{\mathcal{k}_0\left[(\kappa_{0}-{\rm i}\varepsilon_{0}^{\prime})+\alpha\right]}{\kappa_0\left[(\mathcal{k}_0-{\rm i}\epsilon_0^{\prime})-\alpha\right]}\prod_{m=1}^{+\infty}\frac{\mathcal{k}_m(\kappa_m-\alpha)}{\kappa_m(\mathcal{k}_m-\alpha)}, \end{gather}

which is to ensure that $K_+^{\epsilon }(\alpha,F)$ and $K_-^{\epsilon }(\alpha,F)$ are analytical in the upper and lower half complex planes, respectively. As the singularities may move across the real axis when $F$ changes. $\beta _{0}^{\pm }(\alpha,F)$ and $\gamma _{-1}^\pm (\alpha,F)$, $\gamma _{0}^\pm (\alpha,F)$ are introduced to ensure $K_\pm ^{\epsilon }(\alpha,F)$ remain analytical in their corresponding regions at different ranges of $F$. In particular, when $\textrm {Im}\{\mathcal {k}_0\}=0$ ($0< F<1$) and $\textrm {Im}\{\kappa _{0}\}=0$ ($F_c< F<1$), the terms $({(\mathcal {k}_0+\textrm {i}\epsilon _0^{\prime })+\alpha })/{\mathcal {k}_0}$ and $({(\kappa _0+\textrm {i}\varepsilon _0^{\prime })+\alpha })/{\kappa _0}$ will appear in $K_-^{\epsilon }(\alpha,F)$, otherwise it will appear in $K_+^{\epsilon }(\alpha,F)$. When $\textrm {Im}\{\kappa _{-1}\}=0$ ($F>F_c$), the term $({(\kappa _{-1}-\textrm {i}\varepsilon _{-1}^{\prime })-\alpha })/{\kappa _{-1}}$ will be in $K_+^{\epsilon }(\alpha,F)$, otherwise it will appear in $K_-^{\epsilon }(\alpha,F)$.

In (3.19), we may define $\tau = \min \{ |\textrm {Im}\{\mathcal {k}_0 - \textrm {i}\epsilon _0^{\prime }\}|, |\textrm {Im}\{\kappa _{-1}-\textrm {i}\varepsilon _{-1}^{\prime }\}|, |\textrm {Im}\{\kappa _{0} - \textrm {i}\varepsilon _{0}^{\prime }\}|,$ $|\textrm {Im}\{\kappa _{1}\}|,|\textrm {Im}\{\mathcal {k}_1\}|\}$, which makes $|\textrm {Im}\{\mathcal {k}_m\}|\geq \tau$ and $|\textrm {Im}\{\kappa _m\}|\geq \tau$ for all $m$. Then, $K_+^{\epsilon }(\alpha,F)$ and $K_-^{\epsilon }(\alpha,F)$ in (3.23) are analytical in the complex planes $S_+$ with $\textrm {Im}\{\alpha \}>-\tau$ and $S_-$ with $\textrm {Im}\{\alpha \}<\tau$, respectively, as well as have no zero in $S_+$ and $S_-$, respectively. As shown in (B7) of Appendix B, $K_+^{\epsilon }(\alpha,F) \sim O(|\alpha |^{5/2})$ and $K_-^{\epsilon }(\alpha,F) \sim O(|\alpha |^{1/2})$ as $|\alpha |\rightarrow +\infty$, Thus, the asymptotic behaviour of $K_+^{\epsilon }(\alpha,F)K_-^{\epsilon }(\alpha,F)$ is consistent with that of $K^{\epsilon }(\alpha,F)$. Substituting (3.16) and (3.20) into (3.14), and dividing $K_-^{\epsilon }(\alpha,F)$ on both sides, we have

(3.24)\begin{equation} \frac{F_{-}(\alpha)}{K_{-}^{\epsilon}(\alpha,F)}=D_+(\alpha)K_{+}^{\epsilon}(\alpha,F)-\frac{2{\rm \pi} D F^2\alpha^4C(z_0,\alpha){\rm e}^{{\rm i}\alpha x_0}}{\mathcal{K}_1^{\epsilon}(\alpha,F)K_{-}^\epsilon(\alpha,F)}. \end{equation}

The second term on the right-hand side needs to be further decomposed. Noting (3.15) and (3.16), we may write

(3.25)\begin{align} \frac{DF^2\alpha^2C(z_0,\alpha){\rm e}^{{\rm i}\alpha x_0}}{\mathcal{K}_1^{\epsilon}(\alpha,F)K_{-}^{\epsilon}(\alpha,F)}=\frac{DF^2\alpha^2K_+^{\epsilon}(\alpha,F)C(z_0,\alpha){\rm e}^{{\rm i}\alpha x_0}}{\mathcal{K}_2^{\epsilon}(\alpha,F)}=M_{+}(\alpha,x_0,z_0)+M_{-}(\alpha,x_0,z_0), \end{align}

$M_{\pm }(\alpha,x_0,z_0 )$ are analytical in $S_\pm$, respectively. Following the procedure in Noble (Reference Noble1958), when $x_0>0$, $M_-(\alpha,x_0,z_0)$ can be obtained by the Cauchy integral in the upper half-plane as

(3.26)\begin{align} M_{-}(\alpha,x_0,z_0)&={-}\frac{DF^2}{2{\rm \pi}{\rm i}}\int_{-\infty -{\rm i}\sigma}^{+\infty -{\rm i}\sigma}\frac{\zeta^2K_{+}^{\epsilon}(\zeta,F)C(z_0,\zeta){\rm e}^{{\rm i}\zeta x_0}}{\mathcal{K}_2^{\epsilon}(\zeta,F)(\zeta - \alpha)}\,{\rm d}\zeta\nonumber\\ &= DF^2\sum_{\zeta\in R_+}\frac{\zeta^2K_{+}^{\epsilon}(\zeta,F)C(z_0,\zeta){\rm e}^{{\rm i}\zeta x_0}}{\mathcal{K}_2^{\epsilon\prime}(\zeta,F)(\zeta - \alpha)}, \end{align}

where $0<\sigma <\tau$, $R_+$ is the set containing all the roots of $K_2^{\epsilon }(\alpha,F)=0$ on the complex plane $S_+$, which can be expressed as

(3.27)\begin{equation} R_+{=} \left\{ \begin{array}{@{}ll} \left\{\kappa_m|m={-}1,0,1,\ldots\right\}, & 0< F< F_c,\\ \left\{\kappa_m|m=0,1,2,\ldots\right\}\cup\left\{{\pm}\kappa_{0}-{\rm i}\varepsilon_0^{\prime}\right\}, & F_c< F<1,\\ \left\{\kappa_m|m=0,1,2,\ldots\right\}, & F>1. \end{array} \right. \end{equation}

Subsequently, $M_+(\alpha,x_0,z_0)$ can be obtained from

(3.28)\begin{equation} M_{+}(\alpha,x_0,z_0)=\frac{DF^2\alpha^2C(z_0,\alpha){\rm e}^{{\rm i}\alpha x_0}}{\mathcal{K}_1^{\epsilon}(\alpha,F)K_{-}^{\epsilon}(\alpha, F)} - M_{-}(\alpha,x_0,z_0). \end{equation}

When $x_0<0$, the Cauchy integral can be applied in the lower half-plane to obtain $M_{+}(\alpha,x_0,z_0)$ or $S_-$. Substituting (3.25) into (3.24), we have

(3.29)\begin{equation} \frac{F_{-}(\alpha)}{K_{-}^{\epsilon}(\alpha,F)}+2{\rm \pi}\alpha^2M_{-}(\alpha,x_0,z_0)=D_{+}(\alpha)K_{+}^{\epsilon}(\alpha,F)-2{\rm \pi} \alpha^2M_{+}(\alpha,x_0,z_0). \end{equation}

The functions on the left- and right-hand sides of (3.29) are analytical in the complex domains $\textrm {Im}\{\alpha \}<\tau$ and $\textrm {Im}\{\alpha \}>-\tau$, respectively. As these domains overlap within $-\tau <\textrm {Im}\{\alpha \}<\tau$, based on the theorem of analytical continuation, these two functions should be identical and analytical in the entire complex plane. From the Liouville's theorem, such a function should be a polynomial $2{\rm \pi} Q(\alpha )$, which provides

(3.30a)\begin{gather} \frac{F_{-}(\alpha)}{K_{-}^{\epsilon}(\alpha,F)}+2{\rm \pi}\alpha^2M_{-}(\alpha,x_0,z_0)=2{\rm \pi} Q(\alpha), \end{gather}

(3.30b)\begin{gather} D_{+}(\alpha)K_{+}^{\epsilon}(\alpha,F)-2{\rm \pi} \alpha^2M_{+}(\alpha,x_0,z_0) = 2{\rm \pi} Q(\alpha). \end{gather}

Using (3.11) and (3.30a), $A(\alpha )$ can be expressed as

(3.31)\begin{align} A(\alpha)&=\frac{2{\rm \pi} Q(\alpha)K_{-}^{\epsilon}(\alpha,F)}{K_2^{\epsilon}(\alpha F)}-\frac{2{\rm \pi} \alpha^2M_{-}(\alpha,x_0,z_0)K_{-}^{\epsilon}(\alpha,F)}{K_2^{\epsilon}(\alpha,F)}\nonumber\\ &\quad -\frac{2{\rm \pi} {\rm e}^{{\rm i}\alpha x_0}}{\alpha}\frac{\left[(D\alpha^4 + 1)\alpha \cosh\alpha - F^2\alpha^2 \sinh\alpha\right]C(z_0,\alpha)}{K_2^{\epsilon}(\alpha,F)}. \end{align}

Substituting (3.31) into (3.8), the Green function can be obtained through the inverse Fourier transform

(3.32)\begin{equation} G=\frac{1}{2{\rm \pi}}\int_{-\infty}^{+\infty}\hat{G}{\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha. \end{equation}

Letting $\epsilon \rightarrow 0^-$, it provides

(3.33)\begin{equation} G = G_1 + G_2 + G_{ice}, \end{equation}

where

(3.34a)\begin{gather} G_1 = \int_{-\infty}^{+\infty}\frac{Q(\alpha)K_{-}(\alpha,F)C(z,\alpha){\rm e}^{-{\rm i}\alpha x}}{\alpha^2\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{gather}

(3.34b)\begin{gather} G_2={-}\int_{-\infty}^{+\infty}\frac{M_{-}(\alpha,x_0,z_0)K_{-}(\alpha,F)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{gather}

(3.34c)\begin{gather} G_{ice} = \ln r + \ln r^{\prime}-\int_{-\infty}^{+\infty}\frac{{\rm e}^{-|\alpha|}\left(D\alpha^4 + 1 + F^2|\alpha|\right)C(z,\alpha)C(z_0,\alpha) {\rm e}^{-{\rm i}\alpha(x-x_0)}}{\alpha^2\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{gather}

with $r=\sqrt {(x-x_0)^2 + (z-z_0)^2}$ and $r^{\prime }=\sqrt {(x-x_0)^2 + (z+z_0+2)^2}$, $\mathcal {K}_i(\alpha,F)=K_i(\alpha,\alpha F)/\alpha ^2$ ($i=1, 2$). In (3.34c) $G_{ice}$ corresponds to the third term in (3.31) and represents the Green function of the fluid fully covered by an ice sheet, which is obtained by applying a procedure similar to that in Li et al. (Reference Li, Wu and Shi2019). It should be noted that the singularity at $\alpha =0$ in the integrands of $G$ has been treated through the Hadamard regulation and Cauchy principal value. This is equivalent to adding a constant term to $G$. It will not affect the physics, which involves only the spatial derivatives of $G$. When $\epsilon \rightarrow 0^-$, $K_2^{\epsilon }(\alpha,F)\rightarrow K_2(\alpha,\alpha F)$ and we may consider how some roots approach the real axis of $\alpha$. For $K_-(\alpha,F)$, when $F<1$, $\pm \mathcal {k}_0-\textrm {i}\epsilon _0^{\prime }$ approach the real axis of $\alpha$ from above. For $\mathcal {K}_2(\alpha,F)$, when $F>F_c$, $\pm \kappa _{-1}-\textrm {i}\varepsilon _{-1}^{\prime }$ approach from below, and when $F_c< F<1$, $\pm \kappa _{0}-\textrm {i}\varepsilon _{0}^{\prime }$ approach from above. In such a case, the integration path in $G$ from $-\infty$ to $+\infty$ should pass under the poles at $\pm \mathcal {k}_0$ when $0< F<1$, and should pass under the poles at $\pm \kappa _0$ and pass over the poles at $\pm \kappa _{-1}$ when $F_c< F<1$, and should pass over the poles at $\pm \kappa _{-1}$ only when $F>1$.

The Green function can be also obtained by using (3.10) and (3.30b), which gives

(3.35)\begin{equation} G = G_3 + G_4 + G_{water}, \end{equation}

where

(3.36a)\begin{gather} G_3 = \int_{-\infty}^{+\infty}\frac{Q(\alpha)C(z,\alpha){\rm e}^{-{\rm i}\alpha x}}{\alpha^2K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha, \end{gather}

(3.36b)\begin{gather} G_4 =\int_{-\infty}^{+\infty}\frac{M_{+}(\alpha,x_0,z_0)C(z,\alpha){\rm e}^{-{\rm i}\alpha x}}{K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha, \end{gather}

(3.36c)\begin{gather} G_{water} = \ln r + \ln r^{\prime}-\int_{-\infty}^{+\infty}\frac{{\rm e}^{-|\alpha|}\left(1 + F^2|\alpha|\right)C(z,\alpha)C(z_0,\alpha) {\rm e}^{-{\rm i}\alpha(x-x_0)}}{\alpha^2\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha, \end{gather}

where $G_{water}$ denotes the Green function for free surface flow, and it should be noted that $G_1=G_3$. The integration path is the same as that discussed previously for (3.34).

The ice sheet deflection $\xi$ at $x>0$ can be obtained by applying (3.33) and (3.34) to (2.10), which provides

(3.37)\begin{equation} \xi(x,x_0,z_0)=\xi_1(x,x_0,z_0) + \xi_2(x,x_0,z_0) +\xi_{ice}(x,x_0,z_0),\quad x>0, \end{equation}

where

(3.38a)\begin{gather} \xi_1 = \frac{{\rm i}}{F}\int_{-\infty}^{+\infty}\frac{Q(\alpha)K_{-}(\alpha,F)\left[D\alpha^3S(z,\alpha) - F^2 C(z,\alpha)\right]}{\alpha\mathcal{K}_2(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha,\quad z\rightarrow 0^-, \end{gather}

(3.38b)\begin{gather} \xi_2 =\frac{{\rm i}}{F} \int_{-\infty}^{+\infty}\frac{M_{-}(\alpha,x_0,z_0)K_{-}(\alpha,F)\sinh\alpha}{\mathcal{K}_2(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha, \end{gather}

(3.38c)\begin{gather} \xi_{ice} ={\rm i}F\int_{-\infty}^{+\infty}\frac{C(z_0,\alpha)}{\alpha \mathcal{K}_2(\alpha,F)} {\rm e}^{-{\rm i}\alpha(x-x_0)}\,{\rm d}\alpha, \end{gather}

$z\rightarrow 0^-$ is used in $\xi _1$ is to ensure the convergence of the integral. Similarly, substituting (3.35) and (3.36) into (2.10), the free surface wave elevation at $x<0$ can be written as

(3.39)\begin{equation} \xi(x,x_0,z_0)=\xi_3(x,x_0,z_0) + \xi_4(x,x_0,z_0) +\xi_{water}(x,x_0,z_0),\quad x<0, \end{equation}

where

(3.40a)\begin{gather} \xi_3 ={-}{\rm i}F\int_{-\infty}^{+\infty}\frac{Q(\alpha)C(z,\alpha)}{\alpha K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha,\quad z \rightarrow 0^-, \end{gather}

(3.40b)\begin{gather} \xi_4 ={-}\frac{{\rm i}}{F} \int_{-\infty}^{+\infty}\frac{M_{+}(\alpha,x_0,z_0)\sinh\alpha}{K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha, \end{gather}

(3.40c)\begin{gather} \xi_{water} ={\rm i}F\int_{-\infty}^{+\infty}\frac{C(z_0,\alpha)}{\alpha \mathcal{K}_1(\alpha,F)} {\rm e}^{-{\rm i}\alpha(x-x_0)}\,{\rm d}\alpha. \end{gather}

Using (3.13b) in (3.38a), and (3.13a) in (3.40a), and removing the zero integral terms with no singularity, $\xi _1$ and $\xi _3$ can be written in the following form

(3.41a)\begin{gather} \xi_1 ={-}\frac{{\rm i}}{F}\int_{-\infty}^{+\infty}\frac{Q(\alpha)K_{-}(\alpha,F)\sinh\alpha}{\alpha^2\mathcal{K}_2(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha+\frac{\rm \pi}{F}Q(0),\quad x>0, \end{gather}

(3.41b)\begin{gather} \xi_3 ={-}\frac{{\rm i}}{F}\int_{-\infty}^{+\infty}\frac{Q(\alpha)\sinh\alpha}{\alpha^2K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)}{\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha-\frac{\rm \pi}{F}Q(0),\quad x<0, \end{gather}

where the term $\pm ({{\rm \pi} }/{F})Q(0)$ results from the difference of the residues at $\alpha =0$ of the two integrands in (3.38a) and (3.40a). In (3.41a), $K_-(\alpha,F)\sim O(|\alpha |^{1/2})$ and ${\mathcal {K}_2(\alpha,F)}/{\sinh \alpha }\sim O(|\alpha |^3)$ as $|\alpha |\rightarrow +\infty$. If there are terms of $\alpha ^n$ ($n\geq 4$) in $Q(\alpha )$, the integral will be divergent. Thus, based on the discussion above, $Q(\alpha )$ can at most be a cubic polynomial as

(3.42)\begin{equation} Q(\alpha) = b + c\alpha + {\rm d}\alpha^2 + f\alpha^3, \end{equation}

where $b$, $c$, $d$ and $f$ are four unknown coefficients. However, there are only two edge conditions, which means two of them are undetermined. Here, common in this kind of problem, when the flow leaves the edge of the plate, we impose the Kutta condition, which is achieved by assuming the free surface and ice sheet have the same elevation and same slope at $x=0$, or

(3.43)\begin{equation} \left. \begin{aligned} \xi\left(0^+,x_0,z_0\right) & = \xi\left(0^-,x_0,z_0\right),\\ \frac{\partial \xi}{\partial x}\left(0^+,x_0,z_0\right) & = \frac{\partial \xi}{\partial x}\left(0^-,x_0,z_0\right). \end{aligned} \right\} \end{equation}

Substituting (3.37)–(3.41) into (3.43) and using (3.42), we have $\xi (0^+,x_0,z_0)-\xi (0^-,x_0,z_0)=2{\rm \pi} b/F=0$, which gives $b=0$. We also have $({\partial \xi }/{\partial x})(0^+,x_0,z_0)-({\partial \xi }/{\partial x})(0^-,x_0,z_0)=f[I^{\prime \prime \prime }(0^-)-I^{\prime \prime \prime }(0^+)]$, where

(3.44)\begin{equation} I(x)={-}\frac{{\rm i}}{F}\int_{-\infty}^{+\infty}\frac{K_{-}(\alpha,F)\sinh \alpha}{\alpha \mathcal{K}_2(\alpha,F)}{\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha={-}\frac{{\rm i}}{F}\int_{-\infty}^{+\infty}\frac{\sinh\alpha}{\alpha\mathcal{K}_1(\alpha,F)K_+(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha. \end{equation}

Invoking $\sinh \alpha =({\mathcal {K}_2(\alpha,F)-\mathcal {K}_1(\alpha,F)})/{D\alpha ^3}$, (3.44) can be expressed as

(3.45)\begin{equation} I(x) ={-}\frac{{\rm i}}{DF}\int_{-\infty}^{+\infty}\frac{1}{\alpha^4}\left[K_{-}(\alpha,F)-\frac{1}{K_{+}(\alpha,F)}\right] {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha. \end{equation}

Noting that $K_\pm (\alpha,F)$ are analytical in $S_\pm$, respectively, we may consider the integral in $S_-$ when $x>0$ and in $S_+$ when $x<0$. This gives

(3.46)\begin{equation} I(x) = \left\{ \begin{array}{ll} \dfrac{{\rm i}}{DF}\displaystyle\int_{-\infty}^{+\infty}\dfrac{{\rm e}^{-{\rm i}\alpha x}}{\alpha^4K_+(\alpha,F)}\,{\rm d}\alpha, & x>0,\\ \dfrac{{\rm i}}{DF}\displaystyle\int_{-\infty}^{+\infty}\dfrac{K_{-}(\alpha,F){\rm e}^{-{\rm i}\alpha x}}{\alpha^4}\,{\rm d}\alpha, & x<0, \end{array} \right. \end{equation}

where the integral paths in (3.46) for $x>0$ and $x<0$ should pass under and over the pole at $\alpha =0$, respectively. It can be shown from (3.46) that $I^{\prime \prime \prime }(0^-)\rightarrow \infty$ while $I^{\prime \prime \prime }(0^+)$ is finite, which provides $f=0$. By further substituting (3.37), (3.38b), (3.38c) and (3.41a) into the edge conditions in (2.9), and using (3.44), we have

(3.47)\begin{equation} \left[ \begin{array}{cc} I^{\prime\prime}(0^+) & {\rm i}I^{\prime\prime\prime}(0^+)\\ I^{\prime\prime\prime}(0^+) & {\rm i}I^{(4)}(0^+) \end{array} \right] \left[ \begin{array}{c} c\\ d \end{array} \right]= \left[ \begin{array}{c} \dfrac{\partial^2}{\partial x^2}\left[\xi_2(0^+,x_0,z_0)+\xi_{ice}(0^+,x_0,z_0)\right]\\ \dfrac{\partial^3}{\partial x^3}\left[\xi_2(0^+,x_0,z_0)+\xi_{ice}(0^+,x_0,z_0)\right] \end{array} \right]. \end{equation}

Differentiating (3.46) with respect to $x$, letting $x\rightarrow 0^+$ and then applying the theorem of residue in the upper half-plane of $\alpha$, we obtain

(3.48)\begin{equation} \left. \begin{aligned} I^{\prime\prime}(0^+) & =\frac{2{\rm \pi}}{DF}K_+^{\prime}(0,F),\\ I^{\prime\prime\prime}(0^+) & ={-}\frac{2{\rm \pi} {\rm i}}{DF},\\ I^{(4)}(0^+) & = 0. \end{aligned} \right\} \end{equation}

The right-hand side of (3.47) can be also treated in a similar way as in (3.44)–(3.46) and (3.48). This gives,

(3.49)\begin{equation} \left. \begin{aligned} \frac{\partial^2}{\partial x^2}\left[\xi_2(0^+,x_0,z_0)+\xi_{ice}(0^+,x_0,z_0)\right] & ={-}\frac{2{\rm \pi}}{DF}M_{-}\left(0,x_0,z_0\right),\\ \frac{\partial^3}{\partial x^3}\left[\xi_2(0^+,x_0,z_0)+\xi_{ice}(0^+,x_0,z_0)\right] & = 0. \end{aligned} \right\} \end{equation}

Substituting (3.48) and (3.49) into (3.47), we have

(3.50)\begin{equation} \left. \begin{aligned} c & = 0,\\ d & = M_{-}\left(0,x_0,z_0\right). \end{aligned} \right\} \end{equation}

Using this with (3.34) and (3.36), and noticing $M_+(0,x_0,z_0)=-M_-(0,x_0,z_0)$, the Green function may be written as

(3.51a)\begin{gather} G = G_{ice} - \int_{-\infty}^{+\infty}\frac{N_{-}\left(\alpha,x_0,z_0\right)K_{-}(\alpha,F)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{gather}

(3.51b)\begin{gather} G = G_{water} + \int_{-\infty}^{+\infty}\frac{N_{+}\left(\alpha,x_0,z_0\right)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha, \end{gather}

where $N_{\pm }(\alpha,x_0,z_0 )=M_{\pm }(\alpha,x_0,z_0 )-M_\pm (0,x_0,z_0 )$.

3.2. Multipole expansion for a submerged circular cylinder

Once the Green function has been determined, the potentials due to multipoles or higher-order singularities can be found by differentiating the Green function in (3.33) and (3.34) with respect to the position of the source ($x_0,z_0$). Define $x-x_0=r\sin \theta$ and $z-z_0=r\cos \theta$, and apply the following operator (Wu Reference Wu1998)

(3.52)\begin{equation} \left(D_{{\pm}}\right)^n={-}\frac{1}{2^{n-1}(n-1)!}\left(\frac{\partial }{\partial z_0}\pm{\rm i}\frac{\partial}{\partial x_0}\right)^n. \end{equation}

Note that the Green function here is a real function since the problem is steady. In fact, we may use $K_{-}(-\alpha,F)=\overline {K_{-}}(\alpha,F)$ and $N_{-}(-\alpha,x_0,z_0)=\overline {N_{-}}(\alpha,x_0,z_0)$, (3.51a) and (3.51b) become

(3.53a)\begin{gather} G = G_{ice}-2\text{Re}\left\{\int_{0}^{+\infty}\frac{N_{-}(\alpha,x_0,z_0)K_{-}(\alpha,F)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha\right\}, \end{gather}

(3.53b)\begin{gather} G = G_{water}+2\text{Re}\left\{\int_{0}^{+\infty}\frac{N_{+}(\alpha,x_0,z_0)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{K_{+}(\alpha,F)\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha\right\}, \end{gather}

where

(3.54a)\begin{gather} G_{ice} = \ln r + \ln r^{\prime}-2\text{Re}\left\{\int_{0}^{+\infty}\frac{{\rm e}^{-\alpha}\left(D\alpha^4 + 1 + F^2\alpha\right)C(z,\alpha)C(z_0,\alpha){\rm e}^{-{\rm i}\alpha(x-x_0)}}{\alpha^2\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha\right\}, \end{gather}

(3.54b)\begin{gather} G_{water} = \ln r + \ln r^{\prime}-2\text{Re}\left\{\int_{0}^{+\infty}\frac{{\rm e}^{-\alpha}\left(1 + F^2\alpha\right)C(z,\alpha)C(z_0,\alpha){\rm e}^{-{\rm i}\alpha(x-x_0)}}{\alpha^2\mathcal{K}_1(\alpha,F)}\,{\rm d}\alpha\right\}. \end{gather}

In such a case, $(D_+)^n$ and $(D_-)^n$ lead to a pair of conjugate functions. Thus, we may apply only $(D_+)^n$ here. Using (3.26), (3.51a) and

(3.55) \begin{equation} \left. \begin{gathered} \left(D_{+}\right)^n \ln r = \frac{{\rm e}^{{\rm i}n\theta}}{r^n},\\ \left(D_{+}\right)^n \big[C(z_0,\alpha){\rm e}^{{\pm}{\rm i}\alpha x_0}\big] ={-}\frac{({\mp} 1)^n}{(n-1)!}\alpha^n \exp({\pm {\rm i}\alpha x_0 \mp \alpha(z_0+1)}), \end{gathered} \right\} \end{equation}

we have

(3.56)\begin{equation} G_n = G_{n,1} + G_{n,ice}, \end{equation}

where

(3.57a)\begin{equation} G_{n,1} = \frac{1}{(n-1)!}\int_{-\infty}^{+\infty}\frac{\hat{N}_n(\alpha,x_0,z_0)K_{-}(\alpha,F)C(z,\alpha) {\rm e}^{-{\rm i}\alpha x}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{equation}

(3.57b) \begin{align} G_{n,ice}&=\frac{{\rm e}^{{\rm i}n\theta}}{r^n}+\frac{({-}1)^n}{(n-1)!}\int_{0}^{+\infty}\alpha^{n-1} \exp({-\alpha(z+z_0+2)-{\rm i}\alpha(x-x_0)})\,{\rm d}\alpha\nonumber\\ &\quad+\frac{({-}1)^n}{(n-1)!}\int_{-\infty}^{+\infty}\nonumber\\ &\quad\times\frac{\alpha^{n-2} {\rm e}^{-|\alpha|}\left[(D\alpha^4\!+\!1)\!+\!F^2|\alpha|\right]C(z,\alpha) \exp({-\alpha(z_0\!+\!1)\!-{\rm i}\alpha(x\!-x_0)})}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{align}

(3.58)\begin{align} \hat{N}_n(\alpha,x_0,z_0)&={-}(n-1)!\left(D_+\right)^nN_{-}(\alpha,x_0,z_0)\nonumber\\ &=DF^2({-}1)^n\alpha\sum_{\zeta\in R_+}\frac{\zeta^{n+1}K_+(\zeta,F) \exp({{\rm i}\zeta x_0 -\zeta (z_0+1)})}{\mathcal{K}_2^{\prime}(\zeta,F)(\alpha - \zeta)}. \end{align}

Then, the velocity potential due to a submerged cylinder can be expressed in a multipole expansion form as

(3.59)\begin{equation} \phi={\rm Re}\left\{\sum_{n=1}^{+\infty}a^nA_nG_n\right\}, \end{equation}

where $A_n$ are unknown coefficient. Substituting (3.56) and (3.57) into (3.59) and using (Abramowitz & Stegun Reference Abramowitz and Stegun1968)

(3.60)\begin{equation} \left. \begin{aligned} \exp({\alpha(z+1)-{\rm i}\alpha x}) & = \exp({\alpha(z_0+1)-{\rm i}\alpha x_0})\sum_{l=0}^{+\infty}\frac{\alpha^l r^l}{l!} {\rm e}^{-{\rm i}l\theta},\\ \exp({-\alpha(z+1)-{\rm i}\alpha x}) & = \exp({-\alpha(z_0+1)-{\rm i}\alpha x_0})\sum_{l=0}^{+\infty}\frac{({-}1)^l\alpha^l r^l}{l!} {\rm e}^{-{\rm i}l\theta}, \end{aligned} \right\} \end{equation}

the velocity potential can be expressed in the polar coordinate $(r,\theta )$ as

(3.61) \begin{equation} \phi = {\rm Re}\left\{\sum_{n=1}^{+\infty}\left(\frac{a}{r}\right)^n+\sum_{n=1}^{+\infty}\sum_{l=0}^{+\infty}\frac{a^nA_n}{(n-1)!}\frac{r^l}{l!}\big[J_+(n,l) {\rm e}^{{\rm i}l\theta} + J_{-}(n,l){\rm e}^{-{\rm i}l\theta}\big]\right\}, \end{equation}

where

(3.62a)\begin{align} J_{+}(n,l)&=\frac{({-}1)^l}{2}\int_{-\infty}^{+\infty}\frac{\alpha^l\hat{N}_n(\alpha,x_0,z_0)K_{-}(\alpha,F) \exp({-\alpha(z_0+1)-{\rm i}\alpha x_0})}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha\nonumber\\ &\quad+\frac{({-}1)^{n+l}(n+l-1)!}{\left[2(z_0+1)\right]^{n+l}}\nonumber\\ &\quad +\frac{({-}1)^{n+l}}{2}\int_{-\infty}^{+\infty}\frac{\alpha^{l+n-2}{\rm e}^{-|\alpha|}\left(D\alpha^4 + 1 +F^2|\alpha|\right) {\rm e}^{{-}2\alpha(z_0+1)}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha, \end{align}

(3.62b)\begin{align} J_{-}(n,l)&=\frac{1}{2}\int_{-\infty}^{+\infty}\frac{\alpha^l\hat{N}_n(\alpha,x_0,z_0)K_{-}(\alpha,F) \exp({\alpha(z_0+1)-{\rm i}\alpha x_0})}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha\nonumber\\ &\quad +\frac{({-}1)^{n}}{2}\int_{-\infty}^{+\infty}\frac{\alpha^{l+n-2} {\rm e}^{-|\alpha|}\left(D\alpha^4 + 1 +F^2|\alpha|\right)}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha. \end{align}

Substituting (3.61) into (2.7) and noting $n_x=-\sin \theta =-(({\textrm {e}^{\textrm {i}\theta } - \textrm {e}^{-\textrm {i}\theta }})/{2\textrm {i}})$, we obtain the following system of linear equations to solve $A_n$,

(3.63)\begin{align} -l\frac{A_l}{a}+\sum_{n=1}^{+\infty}\frac{a^{l+n-1}J_{+}(n,l)}{(n-1)!(l-1)!}A_n+\sum_{n=1}^{+\infty}\frac{a^{l+n-1}\bar{J}_{-}(n,l)}{(n-1)!(l-1)!}\bar{A}_n={-}{\rm i}F\delta_{l1},\quad l = 1,2,\ldots, \end{align}

where $\delta _{l1}$ denotes the Kronecker delta function. Equation (3.63) can be solved by applying the conjugate to obtain another set of equations, or by separating the real and imaginary parts, respectively.

3.3. Hydrodynamic forces on the submerged circular cylinder

The hydrodynamic forces can be determined by the integration of the hydrodynamic pressure over the surface of the cylinder, which gives

(3.64)\begin{equation} -{\rm i}F_R + F_L ={-}a\int_{-{\rm \pi}}^{\rm \pi}\left(p{\rm e}^{{\rm i}\theta}\right)_{r=a}\,{\rm d}\theta, \end{equation}

where $F_R$ and $F_L$ represent the drag and lift forces, respectively, $p$ denotes the non-dimensionalised pressure difference between the hydrodynamic pressure and atmospheric pressure. As explained in Wu (Reference Wu1991), the local disturbance of the flow field near the cylinder may not be small and the nonlinear terms in the Bernoulli equation should be retained, which gives

(3.65)\begin{equation} p={-}\tfrac{1}{2}\big[\boldsymbol{\nabla}\left(\phi -Fx\right)\boldsymbol{\cdot}\boldsymbol{\nabla}\left(\phi -Fx\right) - F^2\big]. \end{equation}

Through using a similar procedure in Li et al. (Reference Li, Wu and Shi2019), (3.64) gives

(3.66)\begin{equation} -{\rm i}F_R + F_L = \frac{2{\rm \pi}}{a}\sum_{n=1}^{+\infty}n(n+1)A_n\bar{A}_{n+1}. \end{equation}

3.4. Wave elevation and ice sheet deflection

The ice sheet deflection can be obtained by substituting (3.56), (3.57) and (3.59) into (2.10). The ice sheet deflection $\eta$ at $x>0$ can be expressed as

(3.67)\begin{equation} \eta ={-}{\rm Im}\left\{\sum_{n=1}^{+\infty}\frac{a^nA_n}{(n-1)!}\left[\eta_1(n)+\eta_{ice}(n)\right]\right\},\quad x>0, \end{equation}

where

(3.68)\begin{equation} \left. \begin{aligned} \eta_1(n) & ={-}\frac{1}{F}\int_{-\infty}^{+\infty}\frac{\hat{N}_n(\alpha, x_0, z_0)K_{-}(\alpha, F)\sinh\alpha}{\mathcal{K}_2(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha,\\ \eta_{ice}(n) & ={-}({-}1)^nF\int_{-\infty}^{+\infty}\frac{\alpha^{n-1}}{\mathcal{K}_2(\alpha,F)} \exp({-\alpha(z_0+1)-{\rm i}\alpha (x-x_0)})\,{\rm d}\alpha. \end{aligned} \right\} \end{equation}

Using a similar procedure, the free surface wave elevation at $x<0$ can be obtained as

(3.69)\begin{equation} \eta ={-}{\rm Im}\left\{\sum_{n=1}^{+\infty}\frac{a^nA_n}{(n-1)!}\left[\eta_2(n)+\eta_{water}(n)\right]\right\},\quad x<0, \end{equation}

where

(3.70)\begin{equation} \left. \begin{aligned} \eta_2(n) & = \frac{1}{F}\int_{-\infty}^{+\infty}\frac{\check{N}_n(\alpha, x_0, z_0)\sinh\alpha}{K_{+}(\alpha, F)\mathcal{K}_1(\alpha,F)} {\rm e}^{-{\rm i}\alpha x}\,{\rm d}\alpha,\\ \eta_{water}(n) & ={-}({-}1)^nF\int_{-\infty}^{+\infty}\frac{\alpha^{n-1}}{\mathcal{K}_1(\alpha,F)} \exp({-\alpha(z_0+1)-{\rm i}\alpha (x-x_0)})\,{\rm d}\alpha, \end{aligned} \right\} \end{equation}

and

(3.71)\begin{align} \check{N}_n(\alpha,x_0,z_0)&={-}(n-1)!\left(D_+\right)^nN_{+}(\alpha,x_0,z_0)\nonumber\\ &\quad =\frac{DF^2({-}1)^n\alpha^{n+2}\exp({{\rm i}\alpha x_0 - \alpha(z_0+1)})}{\mathcal{K}_1(\alpha,F)K_{-}(\alpha,F)}-\hat{N}(\alpha,x_0,z_0). \end{align}

4.1. Analysis of the distribution of the roots of $K_2(\alpha,\alpha F)=0$

The distribution of the roots corresponding to $K_2(\alpha,\alpha F)=0$ has been proved analytically in Appendix A. Here, we give a graphic description in figure 2. When $0< F< F_c$, there are four fully complex roots that are conjugate to each other and exhibit symmetrical distribution in the complex plane, as illustrated in figure 2(a,c). As $F$ approaches $F_c$, the four complex roots move towards the real axis of $\alpha$, and eventually become two real double roots at $F=F_c$ (as shown in figure 2a–c). In such a case, the corresponding dispersion equation in (3.13b) should satisfy $K_2(\pm \kappa _c,\pm \kappa _cF_c)={\partial K_2(\pm \kappa _c,\pm \kappa _c F_c)}/{\partial \alpha }=0$. The Green function at $F=F_c$ will be infinite. Direct numerical solution at this point is not practical. One way to treat this is to modify the equation as in Liu & Yue (Reference Liu and Yue1993) and Yang, Wu & Ren (Reference Yang, Wu and Ren2022). Here we perform the calculation only at $F$ sufficiently close to $F_c$. When $F>F_c$, the two real double roots become four different real roots, as illustrated in figure 2(d,e). As $F$ continues to increase and tends to $1$, two of the real roots gradually approach $0$. When $F>1$, these two roots become purely imaginary and are located between $0$ and $\pm ({{\rm \pi} }/{2})\textrm {i}$, respectively.

Figure 2. The contours of $\log [K_2(\alpha,\alpha F)]$ at different at different Froude number, the roots of $K_2(\alpha,\alpha F)=0$ are marked by red solid circles: (a) $F=0.2$; (b) $F=0.6$; (c) $F=0.78$; (d) $F=0.79$; (e) $F=0.9$; (f) $F=0.99$; (g) $F=1.01$; (h) $F=1.1$; (i) $F=1.6$. Here $h_i=1/40$, $D=1.78\times 10^{-2}$ and $F_c=0.7868$.

4.2. Verification of the Green function

The Green function derived in § 3 using the Wiener–Hopf technique is validated through a comparison with that obtained by the method of MEE, where the detailed derivation is provided in Appendix C. A comparison of the wave elevation induced by a single source at three typical Froude numbers is given in figure 3. It can be seen that there is no visible difference between the results by these two approaches, which confirms that the Green function derived by Wiener–Hopf is consistent with that by MEE.

Figure 3. A comparison of the wave elevations by the Wiener–Hopf technique and the method of MEE: (a) $F=0.3$; (b) $F=0.9$; (c) $F=1.2$. Here $h_i=1/40$, $D=1.78\times 10^{-2}$, $F_c=0.7868$, $x_0 = 1$ and $z_0 = -0.5$.

Figure 3 also shows that when $F< F_c$ (figure 3a), there is no travelling wave in the ice sheet and there is a wave of $\mathcal {k}_0$ in the free surface region. When $F_c< F<1$ (figure 3b), in the ice sheet region, there is a travelling wave of $\kappa _{-1}$ in the upstream. The downstream $\kappa _{0}$ wave is not evident because the region $0< x< x_0=1$ is small. In the free surface region, there is still the $\mathcal {k}_0$ wave. When $F>1$ (figure 3c), the only travelling wave is $\kappa _{-1}$ at upstream. All these are consistent with the previous analysis. In addition, there is a mean elevation at $x\rightarrow \pm \infty$, which is induced by the pole at $\alpha =0$ in the integrands of (3.38c) and (3.40c), and its value can be evaluated as $\pm {\rm \pi}F/(1-F^2)$ by using the theorem of residue. This phenomenon is due to the fact that $G$ is due to a source which generates a net flow into the fluid. For a dipole obtained by taking derivatives with respect to $x_0$ and $z_0$, this term becomes a constant, which no longer generates a mean free surface elevation and mean current. For higher-order multipole, this term disappears.

4.3. Hydrodynamic forces on a submerged circular cylinder

We now consider the scenarios of a circular cylinder submerged beneath the water surface. The submergence of the cylinder $z_0/a=-4$ is used here. The resistance $F_R$ and lift $F_L$ on the cylinder against $F$ at $6$ different values of $D$ ($h_i/a$) are given in figure 4, where $x_0/a=8$ and particularly $D=0$ corresponds to the results of the free surface case. For the resistance, we may apply a procedure similar to that in appendix C of Yang, Wu & Ren (Reference Yang, Wu and Ren2021) (although the symmetry conclusion of the Green function in appendix B is wrong, due to the mistake that the real part of the complex function is not taken, the procedure in appendix C is correct) to obtain the far-field formula

(4.2)\begin{equation} F_R = I_0 + I_{+\infty} + I_{-\infty}, \end{equation}

where

(4.3) \begin{align} \left. \begin{aligned} I_0 & = \frac{1}{2}[(\eta^2)_{x=0^+}-(\eta^2)_{x=0^-}]\\ I_{+\infty} & =\left\{\begin{gathered} \frac{1}{2}\int_{{-}1}^{0}\left(\frac{\partial \phi}{\partial x} \frac{\partial \phi}{\partial x}-\phi\frac{\partial^2\phi}{\partial x^2}\right)_{x={+}\infty}\,{\rm d}z -\frac{1}{2}\left(\frac{D}{F}\frac{\partial^4\phi}{\partial x^3\partial z}+ F\frac{\partial \phi}{\partial x}\right)^2_{x={+}\infty,z=0}\\ -\frac{1}{2F}\left[\frac{\partial\phi}{\partial z}\left(F\phi+\frac{D}{F}\frac{\partial^3\phi}{\partial x^2\partial z}\right)\right]_{x={+}\infty,z=0}+\frac{D}{2F^2}\left[\left(\frac{\partial^2\phi} {\partial x\partial z}\right)^2_{x={+}\infty,z=0} - \left(\frac{\partial \phi}{\partial z} \frac{\partial^3\phi}{\partial x^2\partial z}\right)\right] \end{gathered} \right\}\\ I_{-\infty}&={-}\frac{1}{2}\int_{{-}1}^{0}\left(\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial x}-\phi\frac{\partial^2\phi}{\partial x^2}\right)_{x={-}\infty}\,{\rm d}z\nonumber\\ &\quad +\frac{1}{2}F^2\left(\frac{\partial \phi}{\partial x}\right)^2_{x={-}\infty,z=0}+\frac{1}{2}\left(\phi\frac{\partial \phi}{\partial z}\right)_{x={-}\infty,z=0}. \end{aligned} \right\} \end{align}

Here $I_0=0$ because the Kutta condition is satisfied at $x=0$; and $I_{\pm \infty }$ can be determined by using the far-field expression of $\phi$. From (3.57) and (3.59), we obtain

(4.4)\begin{equation} \phi(x,z)=\left\{ \begin{array}{ll} H(1-F){\rm Re}\left\{\chi_{-}(\mathcal{k}_0){\rm e}^{-{\rm i}\mathcal{k}_0 x}+\chi_{-}(-\mathcal{k}_0) {\rm e}^{{\rm i}\mathcal{k}_0x}\right\}C(z,\mathcal{k_0}), & x\rightarrow -\infty,\\ H(F-F_c){\rm Re}\left\{\chi_{+}(\kappa_{{-}1}){\rm e}^{-{\rm i}\kappa_{{-}1} x}+\chi_{+}(-\kappa_{{-}1}) {\rm e}^{{\rm i}\kappa_{{-}1}x}\right\}C(z,\kappa_{{-}1}), & x\rightarrow +\infty, \end{array} \right. \end{equation}

where $H(x)$ denotes the Heaviside step function, and

(4.5) \begin{equation} \left. \begin{aligned} \chi_{-}(\alpha) & =2{\rm \pi}{\rm i}\sum_{n=1}^{+\infty}\frac{a^nA_n}{(n-1)!}\frac{\hat{N}_n(\alpha,x_0,z_0)}{\mathcal{K}_1^{\prime}(\alpha,F)K_+(\alpha,F)}\\ \chi_{+}(\alpha) & ={-}2{\rm \pi}{\rm i}\sum_{n=1}^{+\infty}\frac{a^nA_n}{(n-1)!}\\ &\quad \times\left\{ \begin{aligned} & \frac{\hat{N}_n(\alpha,x_0,z_0)K_{-}(\alpha,F)}{\mathcal{K}_2^{\prime}(\alpha,F)}\\ & \quad +\frac{({-}1)^n\alpha^{n-2}{\rm e}^{-|\alpha|}\left(D\alpha^4 + 1 + F^2|\alpha|\right) \exp({-\alpha(z_0+1)+{\rm i}\alpha x_0})}{\mathcal{K}_2^{\prime}(\alpha,F)} \end{aligned} \right\} \end{aligned} \right\}. \end{equation}

Substituting (4.4) into (4.3) and using (4.2), we have $F_R$ as

(4.6)\begin{equation} F_R=\left\{ \begin{aligned} & H(F-F_c)\frac{\left|\chi_{+}(\kappa_{{-}1})+\bar{\chi}_{+}(-\kappa_{{-}1})\right|^2\sinh \kappa_{{-}1}}{4F^2}\frac{\partial K_2(\kappa_{{-}1},\kappa_{{-}1}F)}{\partial \alpha}\\ & \quad - H(1-F)\frac{\left|\chi_{-}(\mathcal{k}_0)+\bar{\chi}_{-}(-\mathcal{k}_0)\right|^2\sinh \mathcal{k}_0}{4F^2}\frac{\partial K_1(\mathcal{k}_0,\mathcal{k}_0F)}{\partial \alpha} \end{aligned} \right\}. \end{equation}

As presented in § 3.1, ${\partial K_1(\mathcal {k}_0,\mathcal {k}_0 F)}/{\partial \alpha }<0$ and ${\partial K_2(\kappa _{-1},\kappa _{-1} F)}/{\partial \alpha }>0$ and, thus, in (4.6) $I_{\pm \infty } >0$. The far-field results can be used to verify the near-field ones, as given in figure 4 for $D=1.78\times 10^{-2}$ ($h_i/a=0.2$). In addition, it can be seen that with the decrease of $D$, the influence of the ice sheet gradually becomes weak, which leads the results to become consistent with that in the free surface ocean. When $D=1.78\times 10^{-5}$ ($h_i/a=0.02$), the curves are nearly identical to those of $D=0$.

Figure 4. Forces on a circular cylinder at different values of $D$ and $h_i$: (a) resistance and (b) lift. Here $H/a=8$, $z_0/a=-4$ and $x_0/a=8$.

In figure 4, it can be seen that both $F_R$ and $F_L$ are quite small but non-zero when $F$ is small. This behaviour is different from the case fully covered by an ice sheet (Li et al. Reference Li, Wu and Shi2019), where $F_R=0$ when $F< F_c$. Here $\phi$ is a wavy function as $x\rightarrow -\infty$ when $F<1$, which makes $I_{-\infty } \neq 0$. Consequently, $F_R$ is non-zero. When $F$ tends to $F_c$, the imaginary parts of $\kappa _0$ and $\kappa _{-1}$ become very small. The downstream $\kappa _0$ wave will decay very slowly and it will then affect the free surface region significantly, where there will be a travelling wave. As a result, as $F\rightarrow F_c$, a rapid rise can be found in $F_R$ as well as $F_L$. In fact, a peak can be seen when $F$ is very close to $F_c$. After that, a drop can be observed from the curves of $F_R$ and $F_L$, and $F_L$ changes from positive (attraction to the free surface) to negative (repulsion). When $F\rightarrow 1$, noticeable abrupt changes can be observed from $F_R$ and $F_L$, which is due to the fact that the downstream $\kappa _0$ wave in the ice sheet region and $\kappa _0$ wave in the free surface disappear.

The hydrodynamic forces on the circular cylinder are also affected by the longitudinal position $x_0$ of the cylinder. The resistance and lift on a cylinder at $x_0>0$ are given in figure 5. When $F< F_c$, if $x_0$ moves towards $x=+\infty$, $F_R$ gradually decreases and $F_L$ gradually increases. At $x_0/a=80$, $F_R$ and $F_L$ are nearly visually identical to those of the case fully covered by an ice sheet (Li et al. Reference Li, Wu and Shi2019). Such an asymptotic behaviour of $F_R$ and $F_L$ can be also observed when $F>1$, and is already achieved when $x_0/a=8$. By contrast, when $F_c<1< F$, the behaviour of $F_R$ and $F_L$ vs $F$ at different $x_0$ are quite different. As $x_0$ gradually increases, a highly and persistent oscillatory behaviour can be seen in the curves of $F_R$ and $F_L$. The reason behind such phenomena can be explained from the Green function. If we let $x_0\rightarrow +\infty$ in (3.26), and note from (3.27) that only $\kappa _0$ is real at $F_c< F<1$, which provides

(4.7)\begin{align} M_{-}(\alpha,x_0,z_0)&\rightarrow H\left((F-F_c)(1-F)\right)\frac{DF^2\kappa_0^2C(z_0,\kappa_0)}{\mathcal{K}_2^{\prime}(\kappa_0,F)}\nonumber\\ &\quad \times\left[\frac{K_{+}(\kappa_{0},F){\rm e}^{{\rm i}\kappa_0x_0}}{\alpha - \kappa_0}-\frac{K_+(-\kappa_0,F) {\rm e}^{-{\rm i}\kappa_0x_0}}{\alpha+\kappa_0}\right],\quad x_0\rightarrow +\infty. \end{align}

Substituting (4.7) into (3.33) and (3.34), we have the $G$ as

(4.8) \begin{align} G&\rightarrow G_{ice} - H\left((F-F_c)(1-F)\right)\frac{DF^2\kappa_0^2C(z_0,\kappa_0)}{\mathcal{K}_2^{\prime}(\kappa_0,F)}\nonumber\\ &\quad \times\int_{-\infty}^{+\infty}\left[\frac{K_{+}(\kappa_{0},F){\rm e}^{{\rm i}\kappa_0x_0}}{\alpha \!-\! \kappa_0}-\frac{K_+(-\kappa_0,F) {\rm e}^{-{\rm i}\kappa_0x_0}}{\alpha+\kappa_0}\right]\frac{\alpha K_{-}(\alpha,F)C(z, \alpha) {\rm e}^{-{\rm i}\alpha x}}{\mathcal{K}_2(\alpha,F)}\,{\rm d}\alpha,\nonumber\\ &\quad\quad\quad x_0 \rightarrow +\infty. \end{align}

From (4.8), it can be found that $G\rightarrow G_{ice}$ when $0< F< F_c$ and $F>1$, which makes $F_R$ and $F_L$ in figure 5 tend to those in fluid fully covered by an ice sheet. However, when $F_c< F<1$, an additional term with oscillatory components $\exp ({\pm \textrm {i}\kappa _0x_0})$ always exist in $G$. In fact, $\kappa _0$ is an implicit function of $F$, which can be seen from the dispersion equation in (3.13b). In such a case, oscillatory behaviours are expected in the curves of $F_R$ and $F_L$ vs $F$, as shown in figure 5. The larger $x_0$ is, the more oscillatory the result will be.

Figure 5. Forces on a circular cylinder at different values of $x_0$ ($x_0>0$): (a) resistance and (b) lift. Here $H/a=8$, $z_0/a=-4$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

Investigations are also conducted for the resistance and lift on a circular cylinder at $x_0<0$, as presented in figure 6. The properties of $F_R$ and $F_L$ are very different from those in figure 5 at $x_0>0$. In particular, $F_R$ and $F_L$ vary smoothly in the entire range of $F$ except near $F=F_c$ and $F=1$. In addition, when $x_0/a\leq -4$, there is hardly any visible difference with the results in the fully free surface. In fact, we have

(4.9)\begin{equation} M_+(\alpha,x_0,z_0)\rightarrow 0,\quad x_0 \rightarrow -\infty. \end{equation}

Substituting (4.9) into (3.35) and (3.36), it provides

(4.10)\begin{equation} G\rightarrow G_{water},\quad x_0\rightarrow -\infty. \end{equation}

Hence, $F_R$ and $F_L$ always tend to those in the free surface problem when $x_0\rightarrow -\infty$.

Figure 6. Forces on a circular cylinder at different values of $x_0$ ($x_0<0$): (a) resistance and (b) lift. Here $H/a=8$, $z_0/a=-4$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

The above result can also be understood from the physical point of view. When $x_0>0$, there will be a travelling wave $\kappa _0$ behind the cylinder when $F_c< F<1$. When this wave arrives at the ice edge, it will be reflected and come to the cylinder. It will be further reflected by the cylinder and go back to the cylinder. This forward and backward process leads to the oscillatory behaviour. When $x_0<0$, there is no free surface travelling wave far ahead of the cylinder and there will be no reflection by the ice sheet edge. Thus, when $|x_0|$ is sufficiently large, the result tends to that corresponding to the free surface.

According to Tuck (Reference Tuck1965), the free surface nonlinear effects for a circular cylinder may be ignored for the case considered in figures 4–6. Here, to acquire some more in-depth understanding of the nature of the hydrodynamic forces at different Froude numbers, we may further use some smaller submergence to highlight the features, as shown in figure 7. It can be seen that the hydrodynamic forces on the cylinder increase rapidly generally, as $|z_0|/a$ reduces.

Figure 7. Forces on a circular cylinder at different values of $z_0$: (a) resistance and (b) lift. Here $H/a=8$, $x_0/a=8$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

4.4. Wave profiles generated by a circular cylinder

The wave profiles $\eta (x)$ with $x_0>0$ at $0< F< F_c$ are plotted in figure 8. It can be seen that there is a regular wave with wavenumber $\kappa _0$ in the free surface region, whereas in the region covered by an ice sheet, the wave amplitude decays very quickly. Such a phenomenon is due to the fact that travelling wave cannot exist in the ice sheet when $F< F_c$. With the increase of $F$, although the amplitude of the entire wave gradually increases, compared with results in fluid fully covered by an ice sheet (Li et al. Reference Li, Wu and Shi2019), a very large wave trough near $x=x_0$ occurred before $F_c$ is not observed here. The wave profiles at $F_c< F<1$ is given in figure 9. When $F$ is near $F_c$, two distinct waves become evident. One with wavenumber $\kappa _0$ in the free surface region and the other with wavenumber $\kappa _{-1}$ ahead of the circular cylinder, whereby the amplitude of the wave at $x < 0$ surpasses marginally that at $x > 0$. As $F$ continues to increase, the amplitude of the wave at $x>x_0$ progressively decreases, whereas that at $x<0$ increases. When $F\rightarrow 1^-$, the wave component $\mathcal {k}_0\rightarrow 0$, its wavelength tends to infinity and its wave amplitude becomes very large. In figure 10, the wave profile at $F>1$ is presented. In such a case, the wave components $\mathcal {k}_0$ and $\kappa _0$ disappear from the downstream region, whereas the wave component $\kappa _{-1}$ remains in the upstream region. When $F\rightarrow 1^+$, a significant wave elevation is observed near $x=8a$, which corresponds to the longitudinal position of the centre of the circular cylinder. As $F$ continues to increase, this large wave elevation becomes smaller. Similar phenomenon is also observed in Li et al. (Reference Li, Wu and Shi2019).

Figure 8. Wave profile for $0< F< F_c$. Here $H/a=8$, $z_0/a=-4$, $x_0/a=8$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

Figure 9. Wave profile for $F_c< F<1$. Here $H/a=8$, $z_0/a=-4$, $x_0/a=8$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

Figure 10. Wave profile for $F>1$. Here $H/a=8$, $z_0/a=-4$, $x_0/a=8$, $h_i/a=0.2$ and $D=1.78\times 10^{-2}$.

There are many similarities between the waves here and those in figure 3 due to a source. There is, however, one distinctive difference. Here we do not have a marked mean surface elevation at infinity. The reason for this is that for a source there is net flow into the fluid. For a cylinder, the leading term will be a dipole and there will be no net flow into the fluid. Thus, there is no mean free surface elevation.