3.1. Green function for a channel covered fully by an ice sheet

To solve the boundary value problem, the Green function $G(x,y,z;x_0,y_0,z_0)$ is first derived, which is the velocity potential at point $(x,y,z)$ due to a single source at point $(x_0,y_0,z_0)$. This $G$ should satisfy the following equation in the entire fluid domain:

(3.1)\begin{equation} \nabla^{2}G + \frac{\partial^{2} G}{\partial z^{2}}=2{\rm \pi}\,\delta(x-x_0)\,\delta(y-y_0)\,\delta(z-z_0), \end{equation}

where $\delta (\cdot )$ denotes the Dirac delta function. The same boundary conditions in (2.4), (2.6)–(2.10) and at far field also need to be satisfied by $G$. To obtain the solution, we may apply the Fourier transform along the $x$-direction,

(3.2)\begin{equation} \hat{G}=\frac{1}{2{\rm \pi}}\int_{-\infty}^{+\infty}G\, {\rm e}^{-{\rm i}kx}\,{{\rm d}\kern0.7pt x} \end{equation}

to (3.1). The governing equation becomes

(3.3)\begin{equation} -k^{2}\hat{G}+\frac{\partial^{2}\hat{G}}{\partial y^{2}}+ \frac{\partial^{2}\hat{G}}{\partial z^{2}}={\rm e}^{-{\rm i}k x_0}\,\delta(y-y_0)\,\delta(z-z_0). \end{equation}

To derive the solution of (3.3), $\hat {G}$ can be expressed as

(3.4)\begin{equation} \hat{G} = \hat{G}_p + \hat{G}_g, \end{equation}

where $\hat {G}_p$ is a particular solution of (3.3) satisfying conditions in (2.4) and (2.7) – or the solution corresponding to the problem in unbounded ocean with an ice cover – while $\hat {G}_g$ is a general solution of (3.3) with zero right-hand side, which is introduced to satisfy the remaining boundary conditions. Based on the procedure of Wehausen & Laitone (Reference Wehausen and Laitone1960), $\hat {G}_p$ can be derived through the Fourier transform method as

(3.5)\begin{equation} \hat{G}_p={-}\frac{{\rm e}^{{-{\rm i}}kx_0}}{2{\rm \pi}}\int_{-\infty}^{+\infty} \frac{{\rm e}^{-{\rm i}\sigma|y-y_0|}\,f(\alpha,z_>,z_<)}{\alpha\, K(\alpha,\omega)}\, {{\rm d}} \sigma, \end{equation}

where

(3.6)\begin{gather} f(\alpha,z_>,z_<)=[(L\alpha^{4}+\rho g -m_i\omega^{2})\alpha\cosh(\alpha z_>)+\rho \omega^{2}\sinh(\alpha z_>)]\cosh\alpha(z_<{+}H), \end{gather}

(3.7)\begin{gather}K(\alpha,\omega) = (L\alpha^{4} + \rho g -m_i\omega^{2})\alpha \sinh\alpha H - \rho\omega^{2}\cosh \alpha H, \end{gather}

with $\alpha =(\sigma ^{2}+k^{2})^{1/2}$, and $z_>$ and $z_<$ are defined as $z_>=\max \{z,z_0\}$ and $z_<=\min \{z,z_0\}$. Here, we may denote the roots of $K(\alpha,\omega )=0$ as $\alpha =\pm \kappa _m$ ($m=-2,-1,0,\ldots$), where $\kappa _0$ is the purely positive real root, $\kappa _{-2}$ and $\kappa _{-1}$ are two complex roots with positive imaginary part, and $\kappa _m$ ($m=1,2,3,\ldots$) are an infinite number of purely positive imaginary roots. When $\kappa _0^{2}>k^{2}$, there will be singularities in the integrand of (3.5) at $\sigma =\pm (\kappa _0^{2}-k^{2})^{1/2}$. To satisfy the outgoing wave radiation condition at far field, the integration path should pass under (over) the poles at $\sigma =-(\kappa _0^{2}-k^{2})^{1/2}$ ($\sigma =+(\kappa _0^{2}-k^{2})^{1/2}$). In fact, if we deform the integration path in (3.5) downwards into the lower half of the complex plane and use the residue theorem, then $\hat {G}_p$ can be expressed further in a form of eigenfunction series as

(3.8)\begin{equation} \hat{G}_p={\rm i}{\rm e}^{-{\rm i}kx_0}\sum_{m={-}2}^{+\infty}\frac{{\rm e}^{-{\rm i}\sigma_m|y-y_0|}\,\psi_m(z)\,\psi_m(z_0)}{2\sigma_mQ_m}, \end{equation}

where

(3.9)\begin{gather} \psi_m(z) = \frac{\cosh\kappa_m(z+H)}{\cosh\kappa_mH}, \end{gather}

(3.10)\begin{gather}Q_m = \frac{2\kappa_mH+\sinh 2\kappa_mH}{4\kappa_m\cosh^{2}\kappa_mH} +\frac{2L\kappa_m^{4}}{\rho\omega^{2}}\tanh^{2}\kappa_mH, \end{gather}

and $\sigma _m=-\textrm {i}(k^{2}-\kappa _m^{2})^{1/2}$. The details of the derivation of (3.8) can be found in Appendix A.

The general solution $\hat {G}_g$ can be determined through a variable separation procedure (Li et al. Reference Li, Shi and Wu2020) as

(3.11)\begin{equation} \hat{G}_g(y,z)={\rm e}^{-{\rm i}kx_0}\sum_{m={-}2}^{+\infty}\varphi_m(y)\,\psi_m(z), \end{equation}

where $\varphi _m(y)$ is governed by

(3.12)\begin{equation} \frac{{{\rm d}}^{2}\varphi_m}{{{\rm d} y}^{2}} + \sigma_m^{2}\varphi_m = 0,\quad -b\leq y \leq b. \end{equation}

To establish the boundary conditions of $\varphi _m(y)$, an orthogonal inner product proposed by Sahoo, Yip & Chwang (Reference Sahoo, Yip and Chwang2001) gives

(3.13)\begin{equation} \langle \psi_m,\psi_{\tilde{m}}\rangle =\int_{{-}H}^{0}\psi_m\psi_{\tilde{m}}\, {{\rm d}}z +\left.\frac{L}{\rho \omega^{2}}\left(\frac{{{\rm d}}\psi_m}{{{\rm d}}z}\, \frac{{{\rm d}}^{3}\psi_{\tilde{m}}}{{{\rm d}}z^{3}}+\frac{{{\rm d}}^{3}\psi_m}{{{\rm d}}z^{3}}\, \frac{{{\rm d}}\psi_{\tilde{m}}}{{{\rm d}}z}\right)\right|_{z=0}=\delta_{m\tilde{m}}Q_m, \end{equation}

which is used here, where $\delta _{ij}$ denotes the Kronecker delta function. Therefore,

(3.14)\begin{align} \left.\left\langle \frac{\partial \hat{G}}{\partial y},\psi_{\tilde{m}}\right\rangle \right|_{y={\pm} b}&=\int_{{-}H}^{0}\left.\frac{\partial \hat{G}}{\partial y}\right|_{y={\pm} b}\psi_{\tilde{m}}\, {{\rm d}}z+\left.\frac{L}{\rho \omega^{2}}\left(\frac{\partial^{2}\hat{G}}{\partial y\,\partial z}\,\frac{{{\rm d}}^{3}\psi_{\tilde{m}}}{{{\rm d}}z^{3}}+\frac{\partial^{4} \hat{G}}{\partial y\, \partial z^{3}}\,\frac{{{\rm d}} \psi_{\tilde{m}}}{{{\rm d}}z}\right)\right|_{y={\pm} b,z=0}\nonumber\\ &={\rm e}^{-{\rm i}kx_0}\,Q_{\tilde{m}}\left[\left.\frac{{{\rm d}} \varphi_{\tilde{m}}}{{\rm d} y}\right|_{y={\pm} b}\pm \frac{{\rm e}^{-{\rm i}\sigma_{\tilde{m}}(b\mp y_0)}\,\psi_{\tilde{m}}(z_0)}{2Q_{\tilde{m}}}\right]. \end{align}

Applying the impermeable condition in (2.6) to (3.14), and letting

(3.15a,b)\begin{equation} \left.\frac{\partial^{2}\hat{G}}{\partial y\,\partial z}\right|_{y={\pm} b, z=0} =\frac{{\rm e}^{-{\rm i}kx_0}(\beta_3\pm\beta_1)}{2} \quad\text{and}\quad \left.\frac{\partial^{4}\hat{G}}{\partial y\,\partial z^{3}} \right|_{y={\pm} b,z=0}=\frac{{\rm e}^{-{\rm i}kx_0}(\beta_4\pm\beta_2)}{2}, \end{equation}

where $\beta _j$ ($j=1,2,3,4$) are four unknown coefficients to be determined from the edge conditions on channel walls, we have

(3.16)\begin{equation} \left.\frac{{{\rm d}} \varphi_m}{{\rm d} y}\right|_{y={\pm} b}=\frac{L\kappa_m\tanh\kappa_mH}{2\rho\omega^{2}Q_m}\times [\kappa_m^{2}(\beta_3\pm\beta_1)+(\beta_4\pm\beta_2)]\mp\frac{{\rm e}^{-{\rm i}\sigma_m(b\mp y_0)}\,\psi_m(z_0)}{2Q_m}. \end{equation}

Based on (3.12) and (3.16), $\varphi _m$ can be found as

(3.17)\begin{equation} \varphi_m(y)=\varphi_m^{(1)}(y)+\varphi_m^{(2)}(y), \end{equation}

where

(3.18a)\begin{gather} \varphi_m^{(1)}(y) = \frac{\psi_m(z_0)}{Q_m}\times\left[\frac{\cos\sigma_m(y+y_0)+{\rm e}^{{-}2{\rm i}\sigma_mb}\cos\sigma_m(y-y_0)}{2\sigma_m\sin 2\sigma_mb}\right], \end{gather}

(3.18b)\begin{gather}\varphi_m^{(2)}(y)=C_m\cos\sigma_my+D_m\sin\sigma_my, \end{gather}

with

(3.19a)\begin{gather} C_m ={-}\frac{L}{\rho \omega^{2}}\times\frac{\tanh\kappa_mH}{Q_m\sigma_m\sin\sigma_mb}\times (\kappa_m^{3}\beta_1+\kappa_m\beta_2), \end{gather}

(3.19b)\begin{gather}D_m = \frac{L}{\rho \omega^{2}}\times\frac{\tanh\kappa_mH}{Q_m\sigma_m\cos\sigma_mb}\times (\kappa_m^{3}\beta_3+\kappa_m\beta_4). \end{gather}

In fact, it can be seen from (3.18) and (3.19) that $\varphi _m^{(1)}$ is introduced to satisfy the impermeable condition on the channel walls, while $\varphi _m^{(2)}$ is introduced for edge conditions at $y=\pm b$ & $z=0$. To obtain $\beta _j$ ($j=1,2,3,4$), we may apply the edge conditions given in (2.8) to $\hat {G}$, and use (3.4), (3.8), (3.11) and (3.17)–(3.19). A system of linear equations of the following form can be established:

(3.20)\begin{equation} \left[ \begin{array}{cccc} A_{11} & A_{12} & A_{13} & A_{14}\\ A_{21} & A_{22} & A_{23} & A_{24}\\ A_{31} & A_{32} & A_{33} & A_{34}\\ A_{41} & A_{42} & A_{43} & A_{44}\\ \end{array} \right]\left[ \begin{array}{c} \beta_1\\ \beta_2\\ \beta_3\\ \beta_4\\ \end{array} \right]=\left[ \begin{array}{c} B_1\\ B_2\\ B_3\\ B_4\\ \end{array} \right], \end{equation}

where the expressions for elements $A_{ij}$, $B_j$ and the solution $\beta _j$ ($i,j=1,2,3,4$) are given in Appendix B. Substituting $\beta _j$ in (B8) and (B9) into (3.18b), and using (B1) and (B3), $\varphi _m^{(2)}$ can be written as

(3.21)\begin{align} \varphi_m^{(2)}(y)&=\sum_{m^{\prime}={-}2}^{+\infty}\frac{I_mI_{m^{\prime}}\,\psi_{m^{\prime}}(z_0)}{Q_m Q_{m^{\prime}}}\left[\frac{\cos\sigma_my \cos\sigma_{m^{\prime}}y_0}{\mathcal{F}_S (k,\omega)\sin\sigma_mb\sin\sigma_{m^{\prime}}b}\right.\nonumber\\ &\quad +\left.\frac{\sin\sigma_my \sin\sigma_{m^{\prime}}y_0}{\mathcal{F}_A(k,\omega)\cos\sigma_mb \cos\sigma_{m^{\prime}}b}\right], \end{align}

where

(3.22)\begin{gather} I_m(k)=\zeta_m(k)\times\frac{\kappa_m\tanh\kappa_mH}{\sigma_m},\end{gather}

(3.23a)\begin{gather} \mathcal{F}_S(k,\omega)={-}2\sum_{m={-}2}^{+\infty}\frac{\kappa_m^{2} \tanh^{2}\kappa_mH}{Q_m\sigma_m}\times\frac{\zeta_m^{2}(k)}{\tan\sigma_mb}, \end{gather}

(3.23b)\begin{gather}\mathcal{F}_A(k,\omega)=2\sum_{m={-}2}^{+\infty}\frac{\kappa_m^{2} \tanh^{2}\kappa_mH}{Q_m\sigma_m}\times\frac{\zeta_m^{2}(k)}{\cot\sigma_mb} \end{gather}

and

(3.24)\begin{equation} \zeta_m(k)=\left\{ \begin{array}{@{}ll} 1, & \text{clamped-clamped},\\[3pt] \sigma_m^{2}(k)+\nu k^{2}, & \text{free-free}.\\ \end{array}\right. \end{equation}

Once $\hat {G}_p$ and $\hat {G}_g$ are found, the Green function $G$ can be obtained by performing the inverse Fourier transform. Using (e.g. Linton et al. Reference Linton, Evans and Smith1992)

(3.25)\begin{equation} \int_{-\infty}^{+\infty}\frac{{\rm e}^{{\rm i}k(x-x_0)-{\rm i} \sigma_m|y-y_0|}}{\sigma_m}\, {{\rm d}}k={-}{\rm \pi}\mathscr{H}_0^{(1)}(\kappa_mR), \end{equation}

where $R = [(x-x_0)^{2}+(y-y_0)^{2}]^{1/2}$ and $\mathscr {H}_n^{(1)}$ denotes the $n$th-order Hankel function of the first kind, we have the Green function

(3.26)\begin{align} &G(x,y,z;x_0,y_0,z_0)={-}\frac{{\rm i}{\rm \pi}}{2}\sum_{m={-}2}^{+\infty} \frac{\psi_m(z)\,\psi_m(z_0)}{Q_m}\,\mathscr{H}_0^{(1)}(\kappa_mR)\nonumber\\ &\quad +\sum_{m={-}2}^{+\infty}\frac{\psi_m(z)\,\psi_m(z_0)}{Q_m} \int_{0}^{+\infty}\left[\frac{\cos\sigma_m(y+y_0)+{\rm e}^{{-}2{\rm i} \sigma_mb}\cos\sigma_m(y-y_0)}{\sigma_m\sin 2\sigma_mb}\right] \cos k(x-x_0)\, {{\rm d}}k\nonumber\\ &\quad +\sum_{m={-}2}^{+\infty}\sum_{m^{\prime}={-}2}^{+\infty} \frac{2\psi_m(z)\,\psi_{m^{\prime}}(z_0)}{Q_mQ_{m^{\prime}}} \int_{\mathscr{L}}I_mI_{m^{\prime}}\left[ \begin{aligned} & \frac{\cos\sigma_my \cos\sigma_{m^{\prime}}y_0}{\mathcal{F}_S(k, \omega)\sin\sigma_mb\sin\sigma_{m^{\prime}}b}\\ & \quad +\frac{\sin\sigma_my \sin\sigma_{m^{\prime}}y_0}{\mathcal{F}_A (k,\omega)\cos\sigma_mb\cos\sigma_{m^{\prime}}b} \end{aligned} \right]\cos k(x-x_0)\, {{\rm d}}k. \end{align}

In (3.26), there will be singularities in the integrand when $\mathcal {F}_S (k_j,\omega )=0$ or $\mathcal {F}_A (k_j,\omega )=0$ ($j=1,2,\ldots,N_s$), where $k_j$ denotes all the corresponding purely positive real roots with $k_1< k_2< \cdots < k_{N_s}$, and $N_s$ is the number of roots. To satisfy the radiation condition at far field, which requires the disturbed waves to propagate away from the source, the integration path $\mathscr {L}$ in (3.26) from $0$ to $+\infty$ should pass over all the poles at $k_j$. In fact, $\mathcal {F}_S (k,\omega )\times \mathcal {F}_A (k,\omega )=0$ corresponds to the dispersion equation (Ren et al. Reference Ren, Wu and Li2020), or relationship between wavenumber and frequency for the propagating wave in the channel. Furthermore, it can be observed (3.26) that $\mathcal {F}_S(k,\omega )$ is combined with $\cos \sigma _my$, which means that $\mathcal {F}_S(k_j,\omega )$ corresponds to a symmetric progressing wave about $y=0$ with wavenumber $k_j$, while $\mathcal {F}_A(k,\omega )$ with $\sin \sigma _my$ corresponds to antisymmetric waves.

3.2. The velocity potential of the incident wave

For the problems in the free surface channel, one form of the incident wave could be assumed as two-dimensional along the channel length and has no transverse variation. However, in the ice-covered channel, such a form is not possible due to the physical constraints at the ice sheet edges. The propagating wave will be always three-dimensional (Ren et al. Reference Ren, Wu and Li2020), and there is always variation in the transverse direction. In fact, there are an infinite number of modes in the $y$-direction, and all these modes are coupled. Here, when the edge conditions on channel walls are the same, we may consider an incident wave symmetric about $y=0$. Following a similar procedure for solving the Green function shown above, $\phi _I$ can be obtained by finding the non-trivial solution of the homogeneous problem, which provides

(3.27)\begin{equation} \phi_I={-}{\rm i}\,\frac{Ag}{\omega\,\chi(\lambda)}\times {\rm e}^{{\rm i}\lambda (x-x_c)}\times\sum_{m={-}2}^{+\infty}\frac{I_m(\lambda)\, \psi_m(z)}{Q_m}\,\frac{\cos[\sigma_m(\lambda)\,y]}{\sin[\sigma_m(\lambda)\,b]}, \end{equation}

where $A$ is a parameter related to the amplitude of the incident wave, $\sigma _m(\lambda )=-\textrm {i}(\lambda ^{2}-\kappa _m^{2})^{1/2}$, $\lambda$ is the wavenumber along the $x$-direction, or the solution of the dispersion equation which also requires $\mathcal {F}_S(\lambda,\omega )=0$. Similar to the problem in the free surface channel, $\lambda$ is taken as the largest positive real root here, or $\lambda =k_{N_s}$. Now, $\chi (\lambda )$ in (3.27) can be expressed as

(3.28)\begin{equation} \chi(\lambda)=\frac{1}{\kappa_0\tanh\kappa_0 H}\sum_{m={-}2}^{+\infty} \frac{I_m(\lambda)\,\kappa_m\tanh\kappa_mH}{Q_m\sin[\sigma_m(\lambda)\,b]}. \end{equation}

The ice sheet deflection due to the incident wave can be obtained from $\eta _I=-({\textrm {i}}/{\omega })({\partial \phi _I}/{\partial z})|_{z=0}$. Then, on $y=0$, we have

(3.29)\begin{equation} \eta_I(x,0)={-}\frac{Ag}{\omega}\times\kappa_0\tanh\kappa_0H \times {\rm e}^{{\rm i}\lambda(x-x_c)}. \end{equation}

It is interesting to see that along the centre line of the tank, the expression for the incident wave is similar to that in unbounded ocean given in Ren et al. (Reference Ren, Wu and Ji2018b).

3.3. Solution through the source distribution method

Once the Green function is derived, the velocity potential can be determined from a boundary integral equation. For the problem of wave diffraction by a vertical cylinder in a free surface channel, the boundary integral equation can be established directly through distributing sources over the body surface (e.g. Linton et al. Reference Linton, Evans and Smith1992). However, when there is an ice sheet, the integral equation has to be re-derived, and the edge conditions must be imposed. The detailed derivation is given in Appendix C. In the result, there is an extra line integral along the edge $\mathcal {L}$ between the body surface and the ice sheet (see (C3)), which contains terms ${\partial ^{4}\phi _D}/{\partial n\,\partial z^{3}}$ and ${\partial ^{2}\phi _D}/{\partial n\,\partial z}$. This is similar to that in Ren et al. (Reference Ren, Wu and Ji2018a); however, their procedure becomes difficult here due to the presence of the channel walls, therefore a different one is introduced here. From the derivation given in Appendix C, using (C9) and the symmetric property of the Green function, or $G(x,y,z;x_0,y_0,z_0 )=G(x_0,y_0,z_0;x,y,z)$, we have

(3.30)\begin{equation} \phi_D(x,y,z)=a\oint_{\mathcal{L}}\langle G(x,y,z;x_0,y_0,z_0), \varPsi(x_0,y_0,z_0)\rangle \, {{\rm d}} \theta_0. \end{equation}

where $x_0-x_c=a\sin \theta _0$ and $y_0-y_c=a\cos \theta _0$, the operator $\langle \ \rangle$ is defined in (3.13), and $\varPsi$ is the strength of the source distributed on the body surface. To obtain $\phi _D$, we may expand $\varPsi$ into a double series as

(3.31)\begin{equation} \varPsi(a,\theta_0,z_0)=\frac{1}{2{\rm \pi} a}\sum_{n={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \frac{b_{n,m}}{Q_m\,\mathcal{J}_n(\kappa_ma)}\times\psi_m(z_0)\, {\rm e}^{-{\rm i}n\theta_0}, \end{equation}

where $b_{n,m}$ are unknown coefficients, and $\mathcal {J}_n$ denotes the $n$th-order Bessel function of the first kind. The Green function can also be expressed in the cylindrical coordinate system. Similar to Wu (Reference Wu1998), we may define

(3.32a,b)\begin{equation} k = \kappa_m\cos\gamma_m\quad \text{and}\quad \sigma_m=\kappa_m\sin\gamma_m. \end{equation}

Using the two identities (Abramowitz & Stegun Reference Abramowitz and Stegun1970)

(3.33a)\begin{gather} \mathscr{H}_0^{(1)}(\kappa_mR)=\sum_{n={-}\infty}^{+\infty} \mathscr{H}_n^{(1)}(\kappa_mr)\,\mathcal{J}_n(\kappa_ma)\, {\rm e}^{{\rm i}n(\theta_0-\theta)}, \end{gather}

(3.33b)\begin{gather}{\rm e}^{{\rm i}[k(x-x_c)\pm\sigma_m(y-y_c)]}=\sum_{n={-}\infty}^{+\infty}\mathcal{J}_n(\kappa_mr)\, {\rm e}^{{\rm i}n(\theta\pm\gamma_m)}, \end{gather}

(3.26) can be transferred to coordinates $(r,\theta,z)$ and $(a,\theta _0,z_0 )$ as

(3.34)\begin{align} &G(r,\theta,z;a,\theta_0,z_0)={-}\frac{{\rm i}{\rm \pi}}{2}\sum_{n={-}\infty}^{n={+}\infty}\sum_{m={-}2}^{+\infty} \frac{\psi_m(z)\,\psi_m(z_0)}{Q_m}\,\mathscr{H}_n^{(1)}(\kappa_mr)\, \mathcal{J}_n(\kappa_ma)\, {\rm e}^{{\rm i}n(\theta_0-\theta)}\nonumber\\ &\quad +\sum_{n={-}\infty}^{+\infty}\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \mathcal{C}_{n,n^{\prime},m}\,\psi_m(z)\,\psi_m(z_0)\,\mathcal{J}_{n^{\prime}}(\kappa_mr)\, \mathcal{J}_n(\kappa_ma)\, {\rm e}^{{\rm i}(n^{\prime}\theta+n\theta_0)}\nonumber\\ &\quad +\sum_{n={-}\infty}^{+\infty}\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \sum_{m^{\prime}={-}2}^{+\infty}\mathcal{D}_{n,n^{\prime},m,m^{\prime}}\,\psi_{m^{\prime}}(z)\, \psi_m(z_0)\,\mathcal{J}_{n^{\prime}}(\kappa_{m^{\prime}}r)\,\mathcal{J}_n(\kappa_ma)\, {\rm e}^{{\rm i}(n^{\prime}\theta+n\theta_0)}, \end{align}

where

(3.35a)\begin{gather} \mathcal{C}_{n,n^{\prime},m}=\frac{1}{Q_m}\int_{0}^{+\infty} \frac{G_{n,n^{\prime},m}+G_{n^{\prime},n,m}}{2\sigma_m\sin 2\sigma_mb}\, {{\rm d}}k, \end{gather}

(3.35b)\begin{gather}\mathcal{D}_{n,n^{\prime},m,m^{\prime}}=\frac{1}{Q_mQ_{m^{\prime}}} \int_{\mathscr{L}}I_mI_{m^{\prime}}\left[ \begin{aligned} & \frac{({-}1)^{n^{\prime}}E_{n,m}E_{{-}n^{\prime},m^{\prime}}+({-}1)^{n} E_{{-}n,m}E_{n^{\prime},m^{\prime}}}{\mathcal{F}_S(k,\omega)\sin \sigma_mb\sin\sigma_{m^{\prime}}b}\\ & \quad +\frac{({-}1)^{n^{\prime}}F_{n,m}F_{{-}n^{\prime},m^{\prime}} +({-}1)^{n}F_{{-}n,m}F_{n^{\prime},m^{\prime}}}{\mathcal{F}_A(k,\omega) \cos\sigma_mb\cos\sigma_{m^{\prime}}b} \end{aligned} \right] {{\rm d}}k, \end{gather}

with

(3.36a)\begin{gather} G_{n,n^{\prime},m}(k)=({-}1)^{n^{\prime}}\cos [2\sigma_my_c+(n-n^{\prime})\gamma_m]+{\rm e}^{{-}2{\rm i}\sigma_mb}({-}1)^{n^{\prime}}\cos(n+n^{\prime})\gamma_m, \end{gather}

(3.36b)\begin{gather}E_{n,m}(k) = \cos(\sigma_my_c+n\gamma_m), \end{gather}

(3.36c)\begin{gather}F_{n,m}(k)=\sin(\sigma_my_c+n\gamma_m). \end{gather}

Substituting (3.31) and (3.34) into (3.30), we obtain

(3.37)\begin{align} \phi_D(r,\theta,z)&={-}\frac{{\rm i}{\rm \pi}}{2}\sum_{n={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\frac{b_{n,m}}{Q_m}\,\mathscr{H}_n^{(1)}(\kappa_mr)\, \psi_m(z)\, {\rm e}^{-{\rm i}n\theta}\nonumber\\ &\quad +\sum_{n={-}\infty}^{+\infty}\sum_{n^{\prime}={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}b_{n,m}\,\mathcal{C}_{n,n^{\prime},m}\, \mathcal{J}_{n^{\prime}}(\kappa_mr)\,\psi_m(z)\, {\rm e}^{{\rm i}n^{\prime}\theta}\nonumber\\ &\quad +\sum_{n={-}\infty}^{+\infty}\sum_{n^{\prime}={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\sum_{m^{\prime}={-}2}^{+\infty}b_{n,m}\, \mathcal{D}_{n,n^{\prime},m,m^{\prime}}\,\mathcal{J}_{n^{\prime}} (\kappa_{m^{\prime}}r)\,\psi_{m^{\prime}}(z)\, {\rm e}^{{\rm i}n^{\prime}\theta}. \end{align}

Similarly, $\phi _I$ can be expressed in the cylindrical coordinate system by applying (3.33b)–(3.27). This gives

(3.38)\begin{equation} \phi_I(r,\theta,z)={-}\frac{{\rm i}Ag}{\omega\,\chi(\lambda)} \sum_{n={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty}\frac{I_m(\lambda)\, E_{n,m}(\lambda)}{Q_m\sin[\sigma_m(\lambda)\,b]}\,\mathcal{J}_n(\kappa_mr)\, \psi_m(z)\, {\rm e}^{{\rm i}n\theta}. \end{equation}

To obtain $b_{n,m}$, applying the inner product in (3.13) to $\partial \phi /\partial r$ and $\psi _{\tilde {m}}$ on $r=a$, we have

(3.39)\begin{equation} \left.\left\langle \frac{\partial \phi}{\partial r},\psi_{\tilde{m}}\right\rangle \right|_{r=a}=\int_{{-}H}^{0}\left.\left\langle \frac{\partial \phi}{\partial r} \psi_{\tilde{m}}\right\rangle \right|_{r=a}\, {{\rm d}}z+\frac{L}{\rho \omega^{2}} \left.\left(\frac{\partial^{2}\phi}{\partial r\,\partial z}\, \frac{{{\rm d}}^{3}\psi_{\tilde{m}}}{{{\rm d}}z^{3}}+\frac{\partial^{4}\phi}{\partial r\,\partial z^{3}}\,\frac{{{\rm d}} \psi_{\tilde{m}}}{{{\rm d}}z}\right)\right|_{r=a,z=0}. \end{equation}

Substituting the impermeable condition on $r=a$ (2.5) into (3.39) and letting

(3.40a,b)\begin{equation} \left.\frac{\partial^{2}\phi}{\partial r\,\partial z}\right|_{r=a,z=0} ={-}\sum_{n={-}\infty}^{+\infty}c_n\,{\rm e}^{{\rm i}n\theta} \quad\text{and}\quad\left.\frac{\partial^{4}\phi}{\partial r\, \partial z^{3}}\right|_{r=a,z=0}={-}\sum_{n={-}\infty}^{+\infty}d_n\,{\rm e}^{{\rm i} n\theta}, \end{equation}

a system of linear equations of the following form can be obtained:

(3.41)\begin{align} &\frac{{\rm i}{\rm \pi}}{2}\,\frac{({-}1)^{n+1}\mathscr{H}_n^{(1){\prime}} (\kappa_ma)}{Q_m\,\mathcal{J}_n^{\prime}(\kappa_ma)}\, b_{{-}n,m}+\sum_{n^{\prime}={-}\infty}^{+\infty}\mathcal{C}_{n^{\prime}, n,m}\,b_{n^{\prime},m}+\sum_{n^{\prime}={-}\infty}^{+\infty} \sum_{m^{\prime}={-}2}^{+\infty}\mathcal{D}_{n^{\prime},n,m^{\prime},m}\, b_{n^{\prime},m^{\prime}}\nonumber\\ &\quad +\frac{L\tanh\kappa_mH}{\rho\omega^{2}Q_m\,\mathcal{J}_n^{\prime} (\kappa_ma)}\,(\kappa_m^{2}c_n+d_n)=\frac{{\rm i}Ag}{\omega\,\chi(\lambda)}\, \frac{I_m(\lambda)\,E_{n,m}(\lambda)}{Q_m\sin[\sigma_m(\lambda)\,b]}, \end{align}

where $-\infty < n<+\infty$ and $-2\leq m <+\infty$, and $\mathscr {H}_n^{(1)\prime }(z)$ and $\mathcal {J}_n^{\prime }(z)$ denote the derivatives of $\mathscr {H}_n^{(1)}(z)$ and $\mathcal {J}_n(z)$, respectively. In addition to the imposed impermeable condition on the body surface, the edge conditions also need to be applied to $\phi$. Here, we may give an example of the clamped edge at the intersection line of the ice sheet and the body surface $\mathcal {L}$, and other conditions can be treated in a similar way. Substituting (3.37) and (3.38) into (2.9), the condition of zero deflection provides

(3.42)\begin{align} &\frac{{\rm i}{\rm \pi}}{2}\sum_{m={-}2}^{+\infty}\frac{({-}1)^{n+1} \mathscr{H}_n^{(1)}(\kappa_ma)\,\kappa_m\tanh\kappa_mH}{Q_m}\,b_{{-}n,m}\nonumber\\ &\qquad +\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \mathcal{C}_{n^{\prime},n,m}\,\mathcal{J}_n(\kappa_ma)\,b_{n^{\prime},m} \kappa_m\tanh\kappa_mH\nonumber\\ &\qquad +\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \sum_{m^{\prime}={-}2}^{+\infty}\mathcal{D}_{n^{\prime},n,m^{\prime},m}\, \mathcal{J}_n(\kappa_ma)\,b_{n^{\prime},m^{\prime}}\kappa_m\tanh\kappa_mH\nonumber\\ &\quad=\frac{{\rm i}Ag}{\omega\,\chi(\lambda)}\sum_{m={-}2}^{+\infty} \frac{I_m(\lambda)\,E_{n,m}(\lambda)\,\mathcal{J}_n(\kappa_ma)\, \kappa_m\tanh\kappa_mH}{Q_m\sin[\sigma_m(\lambda)\,b]},\quad -\infty< n<{+}\infty. \end{align}

The condition of zero slope gives

(3.43)\begin{equation} c_n = 0,\quad -\infty< n<{+}\infty. \end{equation}

In the numerical computation, the infinite series in (3.41) and (3.42) are truncated at $n=\pm N$ and $m=M$, respectively. We have $(2N+1)(M+5)$ unknowns in total, $(2N+1)(M+3)$ of which are $b_{n,m}$, and $(2N+1)$ are $c_n$ and $d_n$. From (3.41), we obtain $(2N+1)(M+3)$ equations, while (3.42) and (3.43) provide an additional $2\times (2N+1)$ equations. Thus there is a total of $(2N+1)(M+5)$ equations, which is the same as the number of unknowns.

After the coefficients $b_{n,m}$, $c_n$ and $d_n$ are found, substituting (3.41) into (3.37) and (3.38), the total velocity potential $\phi$ can be further expressed as

(3.44)\begin{align} \phi(r,\theta,z)&={-}\frac{{\rm i}{\rm \pi}}{2}\sum_{n={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\frac{b_{n,m}}{Q_m}\left[\frac{\mathscr{H}_n^{(1)} (\kappa_mr)}{\mathcal{J}_n(\kappa_mr)}-\frac{\mathscr{H}_n^{(1)\prime} (\kappa_ma)}{\mathcal{J}_n^{\prime}(\kappa_ma)}\right]\mathcal{J}_n(\kappa_mr)\, \psi_m(z)\, {\rm e}^{-{\rm i}n\theta}\nonumber\\ &\quad -\frac{L}{\rho \omega^{2}}\sum_{n={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\frac{(\kappa_m^{2}c_n +d_n)\tanh\kappa_mH}{Q_m}\, \frac{\mathcal{J}_n(\kappa_mr)}{\mathcal{J}_n^{\prime}(\kappa_ma)}\, \psi_m(z)\, {\rm e}^{{\rm i}n\theta}. \end{align}

3.4. Hydrodynamic forces and vertical shear forces on the vertical cylinder

Once the velocity potential is found, the hydrodynamic forces on the vertical cylinder can be obtained through the integration of hydrodynamic pressure over the body surface, which can be expressed as

(3.45)\begin{equation} F_j={\rm i}\omega\rho\iint_{S_B}\phi n_j\, {{\rm d}}S,\quad j=1,2,3,4, \end{equation}

where $j=1,2$ correspond to the forces $F_x$ and $F_y$, and $j=3,4$ correspond to the moments $M_x$ and $M_y$ about the bottom of the channel on $z=-H$; also, $(n_1,n_2,n_3,n_4 )=(n_x,n_y,-(z+H)n_y,(z+H)n_x )$. For the present case of a vertical circular cylinder, we have $n_x=-\sin \theta =-(\textrm {e}^{\textrm {i}\theta }-\textrm {e}^{-\textrm {i}\theta })/2\textrm {i}$ and ${n_y=-\cos \theta =-(\textrm {e}^{\textrm {i}\theta }+\textrm {e}^{-\textrm {i}\theta })/2}$. Substituting these and (3.44) into (3.45), we obtain

(3.46a)\begin{align} &\left[ \begin{array}{c} F_x \\ F_y \\ \end{array} \right]={\rm \pi}\omega\rho a\times\left[ \begin{array}{cc} 1 & -1\\ {\rm i} & {\rm i}\\ \end{array} \right]\nonumber\\ &\quad \times\left\{ \begin{aligned} & \frac{1}{a}\sum_{m={-}2}^{+\infty}\frac{\tanh\kappa_mH}{\kappa_m^{2} Q_m\,\mathcal{J}_{1}^{\prime}(\kappa_ma)}\times\left[ \begin{array}{c} b_{1,m} \\ -b_{{-}1,m}\\ \end{array} \right]\\ & \quad +\frac{L}{\rho\omega^{2}}\sum_{m={-}2}^{+\infty} \frac{\mathcal{J}_1(\kappa_ma)\tanh^{2}\kappa_mH}{\kappa_mQ_m\,\mathcal{J}_1^{\prime} (\kappa_ma)}\times\left[ \begin{array}{c} \kappa_m^{2}c_{{-}1} + d_{{-}1}\\ \kappa_m^{2}c_1 + d_1\\ \end{array} \right] \end{aligned} \right\}, \end{align}

(3.46b)\begin{align} &\left[ \begin{array}{c} M_x \\ M_y \\ \end{array} \right]={-}{\rm \pi}\omega\rho a\times\left[ \begin{array}{cc} {\rm i} & {\rm i}\\ -1 & 1\\ \end{array} \right]\nonumber\\ &\quad \times\left\{ \begin{aligned} & \frac{1}{a}\sum_{m={-}2}^{+\infty}\frac{\kappa_mH\sinh\kappa_mH -\cosh\kappa_mH+1}{\mathcal{J}_{1}^{\prime}(\kappa_ma)\, \kappa_m^{3}Q_m\cosh\kappa_mH}\times\left[ \begin{array}{c} b_{1,m} \\ -b_{{-}1,m}\\ \end{array} \right]\\ & \quad +\frac{L}{\rho\omega^{2}}\sum_{m={-}2}^{+\infty} \frac{\mathcal{J}_1(\kappa_ma)\,(\kappa_mH\sinh\kappa_mH- \cosh\kappa_mH\!+\!1)}{\mathcal{J}_1^{\prime}(\kappa_ma)\, \kappa_m^{2}Q_m\cosh\kappa_mH\coth\kappa_mH}\times\left[ \begin{array}{@{}c@{}} \kappa_m^{2}c_{{-}1} \!+\! d_{{-}1}\\ \kappa_m^{2}c_1 \!+\! d_1\\ \end{array} \right] \end{aligned} \right\}. \end{align}

When the ice sheet is clamped to the surface of the cylinder, there will be a vertical shear force on the body. The total vertical shear force $V$ can be obtained from

(3.47)\begin{equation} V = \int_{0}^{2{\rm \pi}}\tau(\theta)\,a\, {{\rm d}} \theta, \end{equation}

where $\tau (\theta )$ is the shear stress distribution along the intersection line, which can be expressed as (Ugural Reference Ugural1999)

(3.48)\begin{equation} \tau(\theta)={-}{\rm i}\,\frac{L}{\omega}\,\frac{\partial }{\partial r} \left.\left(\nabla^{2}\frac{\partial \phi}{\partial z}\right) \right|_{r=a,z=0}={\rm i}\,\frac{L}{\omega}\left.\frac{\partial^{4}\phi}{\partial r\,\partial z^{3}}\right|_{r=a,z=0}={-}{\rm i}\,\frac{L}{\omega}\sum_{n={-}\infty}^{+\infty}d_n\,{\rm e}^{{\rm i}n\theta}. \end{equation}

Substituting (3.48) into (3.47), we have

(3.49)\begin{equation} V ={-}{\rm i}\,\frac{2{\rm \pi} aL}{\omega}\,d_0. \end{equation}

3.5. Behaviour of the solution at the natural frequencies

For a given $\omega$, the residual in (3.26) at a singularity $\mathcal {F}_S(k,\omega )=0$ ($\mathcal {F}_A(k,\omega )=0$) can be obtained from the standard method in complex analysis. The result contains $\mathcal {F}_S^{\prime }(k,\omega )=0$ ($\mathcal {F}_A^{\prime }(k,\omega )=0$) in the denominator, where the prime represents the derivative with respect to $k$. At some $\omega$, $\mathcal {F}_S^{\prime }(k,\omega )$ ($\mathcal {F}_A^{\prime }(k,\omega$)) is also equal to zero when $\mathcal {F}_S(k,\omega )=0$ ($\mathcal {F}_A(k,\omega )=0$), and then the Green function $G$ will be infinite. Physically, $\omega$ in this case is the natural frequency of the ice-covered channel. In fact, from (3.23), $k=0$ is always the solution of $\mathcal {F}_S^{\prime }(k,\omega )=0$ ($\mathcal {F}_A^{\prime }(k,\omega )=0$). At a given $\omega$, if we further have $\mathcal {F}_S^{\prime }(k,\omega )=0$ ($\mathcal {F}_A^{\prime }(k,\omega )=0$), then this $\omega$ will be a natural frequency. This is similar to $\kappa _0=\textrm {i}{\rm \pi} /2b$ and $\omega =[({\textrm {i}{\rm \pi} }/{2b})\tanh ({\textrm {i}{\rm \pi} H}/{2b})]^{1/2}$ ($i=1,2,\ldots$) in the free surface channel (Linton et al. Reference Linton, Evans and Smith1992; Wu Reference Wu1998). The ice-covered channel also has an infinite number of natural frequencies, which are denoted as $\omega _c^{(i)}$ ($i=1,2,\ldots$) here, with $\omega _c^{(1)}<\omega _c^{(2)}<\omega _c^{(3)}<\cdots$ In particular, even $i$ corresponds to $\mathcal {F}_S(0,\omega _c^{(i)})=0$, while odd $i$ corresponds to $\mathcal {F}_A(0,\omega _c^{(i)})=0$. The results near the natural frequencies can change rapidly. Here, we will show that even though the Green function is infinite at one of the natural frequencies, the velocity potential $\phi$ and hydrodynamic force may remain finite. We may consider the even modes $2i$ as an example, and the odd modes $2i-1$ can be done in a similar way. When $\omega \to \omega _c^{(2i)}$, $\mathcal {F}_S(k,\omega )$ at $k\to 0$ can be expressed asymptotically as

(3.50)\begin{equation} \mathcal{F}_S(k,\omega)\to\mathcal{F}_{S,asy}(k,\omega) =\mathcal{F}_S(0,\omega)+\tfrac{1}{2}\mathcal{F}_S^{\prime \prime}(0,\omega){\rm i}k^{2},\quad k\to 0, \end{equation}

where $\mathcal {F}_S(0,\omega )\to 0^{\pm }$ when $\omega \to \omega _c^{(2i)}+0^{\pm }$, and $\mathcal {F}_S^{\prime \prime }(0,\omega )<0$, which can be confirmed from (3.23a). For the integrand as $\mathcal {G}(k)/\mathcal {F}_S(k,\omega )$ in (3.35b), we may re-express it as $[\mathcal {G}(k)/\mathcal {F}_S(k,\omega )- \mathcal {G}(0)/\mathcal {F}_{S,asy}(k,\omega )]+\mathcal {G}(0)/\mathcal {F}_{S,asy}(k,\omega )$. Then the first term is non-singular, and the second term can be integrated explicitly. Also, $\mathcal {D}_{n,n^{\prime },m,m^{\prime }}$ in $G$ can be written as

(3.51)\begin{equation} \mathcal{D}_{n,n^{\prime},m,m^{\prime}}=\frac{2{\rm \pi}}{\varDelta}\, \frac{I_m(0)\,I_{m^{\prime}}(0)\,E_{n,m}(0)\,E_{n^{\prime},m^{\prime}} (0)}{Q_mQ_{m^{\prime}}\sin\kappa_mb\sin\kappa_{m^{\prime}}b} +\tilde{\mathcal{D}}_{n,n^{\prime},m,m^{\prime}}+O(\varDelta), \quad\varDelta\to 0, \end{equation}

where $\varDelta =\mu \times |2\mathcal {F}_S^{\prime \prime }(0,\omega )\,\mathcal {F}_S(0,\omega )|$, and $\mu$ is a constant depending on whether $\omega _c^{(2i)}$ is approached from the left- or right-hand side. When $\omega \to \omega _c^{(2i)}+0^{-}$, $\mu =1$ and the singular term in $\mathcal {D}_{n,n^{\prime },m,m^{\prime }}$ is from the principal value integration. When $\omega \to \omega _c^{(2i)}+0^{+}$, $\mu =-\textrm {i}$ and the singular term is from the residue term. Also, $\tilde {\mathcal {D}}_{n,n^{\prime },m,m^{\prime }}\sim O(1)$ in (3.51) is the leading term of the remaining regular part of the Green function. To analyse the behaviour of the velocity potential $\varphi$ at natural frequencies, we may employ a procedure similar to that used by Liu & Yue (Reference Liu and Yue1993) for the forward speed problem in free surface flow. Substituting (3.51) into (3.41) and rearranging the equation, we obtain

(3.52)\begin{equation} b_{{-}n,m}+\frac{4{\rm i}}{\varDelta}\times\varLambda \times\frac{({-}1)^{n}\mathcal{J}_n^{\prime}(\kappa_ma)\,I_m(0)\, E_{n,m}(0)}{\mathscr{H}_n^{(1)\prime}(\kappa_ma)\sin\kappa_mb}=\xi_{n,m}+O(\varDelta), \end{equation}

where

(3.53)\begin{gather} \varLambda=\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m^{\prime} ={-}2}^{+\infty}\frac{I_{m^{\prime}}(0)\,E_{n^{\prime},m^{\prime}} (0)}{Q_{m^{\prime}}\sin\kappa_{m^{\prime}}b}\,b_{n^{\prime},m^{\prime}}, \end{gather}

(3.54)\begin{gather}\xi_{n,m}=\frac{2{\rm i}({-}1)^{n}\mathcal{J}_n^{\prime}(\kappa_ma)}{{\rm \pi}\, \mathscr{H}_n^{(1)\prime}(\kappa_ma)}\times\!\left[ \begin{aligned} & \frac{{\rm i}Ag}{\omega\,\chi(\lambda)}\,\frac{I_m(\lambda)\,E_{n,m}(\lambda)}{Q_m \sin[\sigma_m(\lambda)\,b]}-\sum_{n^{\prime}={-}\infty}^{+\infty} \mathcal{C}_{n^{\prime},n,m}\,b_{n^{\prime},m}\\ & \quad -\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m^{\prime}={-}2}^{+\infty} \tilde{\mathcal{D}}_{n,n^{\prime},m,m^{\prime}}\,b_{n^{\prime},m^{\prime}} +\!\frac{L\tanh\kappa_mH}{\rho\omega^{2}Q_m\,\mathcal{J}_n^{\prime}(\kappa_ma)}\,d_n \end{aligned} \right]. \end{gather}

In (3.52), $c_n=0$ in (3.43). Multiplying (3.41) by $\kappa _m\tanh \kappa _mH$, taking summation with respect to $m$ from $-2$ to $+\infty$, and subtracting (3.42) from the result, $d_n$ can be further expressed using $b_{n,m}$ as

(3.55)\begin{equation} d_n={-}\frac{\rho\omega^{2}}{La}\times\frac{\displaystyle\sum_{m={-}2}^{+\infty}\frac{\tanh\kappa_mH}{Q_m} \times\frac{b_{{-}n,m}}{\mathcal{J}_{{-}n}^{\prime}(\kappa_ma)}}{\displaystyle\sum_{m={-}2}^{+\infty} \frac{\kappa_m\tanh^{2}\kappa_mH}{Q_m}\times\frac{\mathcal{J}_n (\kappa_ma)}{\mathcal{J}_{n^{\prime}}(\kappa_ma)}}, \end{equation}

which indicates that $d_n$ has the same magnitude as $b_{n,m}$. Substituting (3.52) into (3.53), $\varLambda$ can be represented as

(3.56)\begin{equation} \varLambda=\frac{\varDelta}{\varDelta+4\varGamma {\rm i}} \times\sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m^{\prime}={-}2}^{+\infty} \frac{({-}1)^{n^{\prime}}I_{m^{\prime}}(0)\,E_{n^{\prime},m^{\prime}} (0)}{Q_{m^{\prime}}\sin\kappa_{m^{\prime}}b}\,\xi_{n^{\prime},m^{\prime}}+O(\varDelta^{2}), \end{equation}

where

(3.57)\begin{equation} \varGamma=\sum_{n={-}\infty}^{+\infty}\sum_{m={-}2}^{+\infty} \frac{I_m^{2}(0)\,E_{n,m}^{2}(0)}{Q_m\sin^{2}\kappa_mb}\times \frac{\mathcal{J}_n^{\prime}(\kappa_ma)}{\mathscr{H}_n^{(1)\prime}(\kappa_ma)}. \end{equation}

At the natural frequency, we will first check whether $\varGamma =0$. When $\varGamma \neq 0$, we may substitute (3.57) back into (3.52) and let $\varDelta =0$, then a modified matrix equation can be obtained as

(3.58)\begin{equation} b_{{-}n,m}\!+\!\frac{1}{\varGamma}\,\frac{({-}1)^{n}\mathcal{J}_n^{\prime}(\kappa_ma)\, I_m(0)\,E_{n,m}(0)}{\mathscr{H}_n^{(1)\prime}(\kappa_ma)\sin\kappa_mb} \sum_{n^{\prime}={-}\infty}^{+\infty}\sum_{m^{\prime}={-}2}^{+\infty} \frac{({-}1)^{n^{\prime}}I_{m^{\prime}}(0)\,E_{n^{\prime},m^{\prime}} (0)}{Q_{m^{\prime}}\sin\kappa_{m^{\prime}}b}\,\xi_{n^{\prime},m^{\prime}}\!=\!\xi_{n,m}. \end{equation}

This equation is not singular and can be used at natural frequencies. It can be seen from (3.43), (3.55) and (3.58) that the solutions $b_{n,m}$, $c_n$ and $d_n$ are bounded at the natural frequencies. Furthermore, using (3.44), (3.46) and (3.49), the velocity potential $\phi$ and forces are also non-singular at natural frequencies. However, whether $\varGamma$ could be $0$ in some cases, and the corresponding solution could be singular, needs further investigation.

4.1. Verification of the dispersion equation

As discussed in § 3.1, $\mathcal {F}_S(k,\omega )\times \mathcal {F}_A(k,\omega )=0$ corresponds to the dispersion relation for propagating waves in an ice-covered channel. To verify this, a comparison with the dispersion relation obtained by Ren et al. (Reference Ren, Wu and Li2020) through a different approach is presented in figure 2, and very good agreement can be found. For a given $k$, Ren et al. (Reference Ren, Wu and Li2020) pointed out that there is an infinite number of solutions $\omega$. Here, we may denote each root $\omega$ as $\omega _i(k)$ ($i=0,1,2,\ldots$), with $\omega _0(k)<\omega _1(k)<\omega _2(k)<\cdots$, where the points on curves $\omega _{2i}(k)$ ($i=0,1,2,\ldots$) are solutions of $\mathcal {F}_S(k,\omega )=0$ and correspond to waves symmetric about $y=0$, while those on $\omega _{2i+1}(k)$ are the roots of $\mathcal {F}_A(k,\omega )=0$ and correspond to waves antisymmetric about $y=0$. As discussed in § 3.5, the points on the vertical axis or $\omega _i(0)=\omega _c^{(i)}$ ($i=1,2,3,\ldots$) are the natural frequencies of the channel.

Figure 2. Dispersion relations of the ice-covered channel: (a) clamped–clamped edges; (b) free–free edges. Here, $b/H=2$ and $h_i/H=1/50$.

Figure 2 verifies the present method and equation numerically. In fact, we can further show that although the present expression for the dispersion relation is different from that in Ren et al. (Reference Ren, Wu and Li2020), they are identical mathematically. To do that, we may first construct a function of the form

(4.2)\begin{equation} h(\alpha) = \frac{\alpha^{2}\sinh\alpha H}{K(\alpha,\omega)}\times\frac{\zeta^{2}(\alpha)}{\sigma\tan\sigma b} \end{equation}

in the complex plane $\alpha$, where $\sigma =-\textrm {i}(k^{2}-\alpha ^{2})^{1/2}$ and

(4.3) \begin{equation} \zeta(\alpha)=\left\{ \begin{array}{@{}ll} 1, & \text{clamped\unicode{x2013}clamped},\\ \sigma^{2}+\nu k^{2}, & \text{free\unicode{x2013}free}. \end{array} \right. \end{equation}

Consider the integral of $h(\alpha )$ along a circle $C_R$ of radius $R \to +\infty$ and centred at the origin in the complex plane. Applying the residue theorem at singularities of $K(\alpha,\omega )$ and $\sigma \tanh \sigma b$, we have

(4.4)\begin{equation} \frac{1}{2{\rm \pi} {\rm i}}\oint_{C_R}h(\alpha)\, {{\rm d}} \alpha=2\left[ \sum_{m={-}2}^{+\infty}\frac{\kappa_m^{2}\sinh\kappa_mH}{K^{\prime}(\kappa_m, \omega)}\,\frac{\zeta_m^{2}(k)}{\sigma_m\tan\sigma_mb}+\sum_{n=0}^{+\infty} \frac{\alpha_{2n}\sinh\alpha_{2n}H}{K(\alpha_{2n},\omega)}\, \frac{\zeta^{2}(\alpha_{2n})}{b(1+\delta_{n0})}\right], \end{equation}

where $\alpha _n=(k^{2}+n^{2}{\rm \pi} ^{2}/4b^{2})^{1/2}$. When $R\to +\infty$, $|h(\alpha )|\sim O(1/R^{4})$ for clamped–clamped edges and $|h(\alpha )|\sim O(1/R^{2})$ for free–free edges. Thus the integral in (4.4) tends to zero. Then, using (A4) and (3.23a), we further obtain

(4.5)\begin{equation} \mathcal{F}_S(K,\omega)={-}2\sum_{m={-}2}^{+\infty}\frac{\kappa_m^{2} \tanh^{2}\kappa_mH}{Q_m\sigma_m}\,\frac{\zeta_m^{2}(k)}{\tan\sigma_mb}= \frac{4\rho \omega^{2}}{b}\sum_{n=0}^{+\infty}\frac{\alpha_{2n} \sinh\alpha_{2n}H}{K(\alpha_{2n},\omega)}\,\frac{\zeta^{2}(\alpha_{2n})}{(1+\delta_{n0})}. \end{equation}

Similarly, $\mathcal {F}_A(K,\omega )$ in (3.23b) can be expressed as

(4.6)\begin{equation} \mathcal{F}_A(K,\omega)=2\sum_{m={-}2}^{+\infty}\frac{\kappa_m^{2}\tanh^{2} \kappa_mH}{Q_m\sigma_m}\,\frac{\zeta_m^{2}(k)}{\cot\sigma_mb}= \frac{4\rho \omega^{2}}{b}\sum_{n=0}^{+\infty}\frac{\alpha_{2n+1} \sinh\alpha_{2n+1}H}{K(\alpha_{2n+1},\omega)}\,\zeta^{2}(\alpha_{2n+1}). \end{equation}

The system of linear equations in (2.24) in Ren et al. (Reference Ren, Wu and Li2020) can be split into those for symmetric and antisymmetric modes, respectively. Noticing that coefficient $\varDelta _n$ in (2.20$a$) in their work can be linked to $K(\alpha,\omega )$ in (3.7) here as $\varDelta _n=-16b^{4}\,K(\alpha _n,\omega )/n^{4}{\rm \pi} ^{4}$ ($n>0$), through some algebra, it can be shown that $\textrm {det}(\boldsymbol {A})=0$ in (2.25) of Ren et al. (Reference Ren, Wu and Li2020) gives the same series as those on the right-hand sides of (4.5) and (4.6). The above analysis shows that the two different methods give the same dispersion relation, while the present formulations in (4.5) and (4.6) are in a much neater and direct forms. It should also be noted that the other edge conditions can be dealt with in a similar way.

4.2. The natural frequencies at different channel widths and ice sheet thickness

It may also be interesting to investigate how the natural frequencies $\omega _c^{(i)}$ vary with the channel width $b$ and ice sheet thickness $h_i$. As an example, $\omega _c^{(i)}$ ($i=1,2,3,4$) under free–free edges are given in figure 3. It can be seen from figure 3(a) that all the $\omega _c^{(i)}$ decrease as $b$ increases. At sufficiently large values of $b/H$, all the $\omega _c^{(i)}$ ($i=1,2,3,4$) will tend to zero. The natural frequencies at different $h_i$ are given in figure 3(b), and values at $h_i/H=0$ correspond to those of the free surface channel. It can be observed that $\omega _c^{(1)}$ is hardly affected by $h_i$. The effect of ice sheet thickness on $\omega _c^{(i)}$ becomes more apparent when $i$ increases.

Figure 3. Natural frequencies of the ice-covered channel under free–free edges: (a) variation with $b$ at $h_i/H=1/50$; (b) variation with $h_i$ at $b/H=2$.

4.3. Forces on a cylinder standing at the centre of the channel

Hydroelastic wave diffraction by a vertical circular cylinder standing at the centre of the channel is considered in this subsection. Since the problem is symmetric about $y=0$, we have $b_{n,m}=b_{-n,m}$ in (3.37). In such a case, the infinite series in (3.41)–(3.43) about $n$ and $n^{\prime }$ need to be considered only from $0$ to $+\infty$. Also, the coefficients $\mathcal {C}_{n,n^{\prime },m}$ and $\mathcal {D}_{n,n^{\prime },m,m^{\prime }}$ given in (3.35) can be further simplified as

(4.7a)\begin{gather} \mathcal{C}_{n,n^{\prime},m}=\frac{({-}1)^{n}+({-}1)^{n^{\prime}}}{Q_m(1+ \delta_{n0})(1+\delta_{n^{\prime}0})}\int_{0}^{+\infty} \frac{{\rm e}^{-{\rm i}\sigma_mb}\cos n\gamma_m\cos n^{\prime} \gamma_m}{\sigma_m\sin\sigma_mb}\, {{\rm d}}k, \end{gather}

(4.7b)\begin{gather}\mathcal{D}_{n,n^{\prime},m,m^{\prime}}=\frac{2[({-}1)^{n} +({-}1)^{n^{\prime}}]}{Q_mQ_{m^{\prime}}(1+\delta_{n0})(1+ \delta_{n^{\prime}0})}\int_{\mathscr{L}}\frac{I_mI_{m^{\prime}}\cos n\gamma_m\cos n^{\prime}\gamma_{m^{\prime}}}{\mathcal{F}_S(k, \omega)\sin\sigma_mb\sin\sigma_{m^{\prime}}b}\, {{\rm d}}k. \end{gather}

As expected, $\mathcal {F}_A(k,\omega )$ does not appear here. Its singularities have no effect or there will be no wave antisymmetric about $y=0$. Furthermore, noticing that $\mathcal {C}_{2n,2n^{\prime }+1,m}=\mathcal {C}_{2n+1,2n^{\prime },m}=\mathcal {D}_{2n,2n^{\prime }+1,m,m^{\prime }}=\mathcal {D}_{2n+1,2n^{\prime },m,m^{\prime }}$ ($n,n^{\prime }=0,1,2,\ldots$), the unknown coefficients $b_{2n,m}$, $c_{2n}$ and $d_{2n}$ in (3.41)–(3.43) are completely independent of $b_{2n+1,m}$, $c_{2n+1}$ and $d_{2n+1}$.

We first investigate the hydrodynamic forces and vertical shear force at different channel widths when the ice sheet is clamped to the surface of the vertical cylinder. The numerical results for wave forces are shown in figure 4, where the black solid lines correspond to forces on a single vertical circular cylinder standing in the unbounded ocean with an ice cover, which is calculated using the method in Ren et al. (Reference Ren, Wu and Li2020) (as below). Here, $F_x^{*}$ is defined as $F_x^{*}=F_x/\rho g a^{2}A$; similarly, $M_y^{*}=M_y/\rho ga^{3}A$ and $V^{*}=V/\rho gaA$. When $b/a=5$, it can be seen that $F_x^{*}$ values in two different sets of edge conditions along the tank walls are both significantly different from that in the unbounded ice-covered ocean. In particular, as shown in figure 4(b) for channels with clamped–clamped edges, a couple of peaks can be observed in the curve of $F_x^{*}$ versus $\kappa _0a$. By contrast, there is only one obvious peak in the curve given in figure 4(a) for free–free edges. As $b$ increases, the peaks decrease and gradually become less visible. The curve of $F_x^{*}$ versus $\kappa _0a$ generally shows a variation trend similar to that in the unbounded ocean but with a continuous fluctuation, where the amplitude of fluctuation becomes smaller as $b$ increases. When $b$ is sufficiently large, the results in both cases tend to that in the unbounded ice-covered ocean, which shows that the effects from two side walls and edge conditions at $y=\pm b$ on the hydrodynamic forces become very insignificant. In addition to $F_x^{*}$, similar phenomena can also be observed in the curve of $M_y^{*}$ shown in figure 5.

Figure 4. Wave forces in the $x$-direction on the cylinder at the centre of the channel with different widths, when the ice sheet is clamped to the cylinder: (a) channel with free–free edges; (b) channel with clamped–clamped edges. Here, $H/a=5$ and $h_i/a=1/10$.

Figure 5. Moments in the $y$-direction on the cylinder at the centre of the channel with different widths, when the ice sheet is clamped to the cylinder: (a) channel with free–free edges; (b) channel with clamped–clamped edges. Here, $H/a=5$ and $h_i/a=1/10$.

The results for the vertical shear forces on the cylinder are shown in figure 6. It may seem to be a surprise that the variation trend of $V^{*}$ versus $\kappa _0a$ is quite different from that of $F_x^{*}$ and $M_y^{*}$ given in figures 4 and 5. As $\kappa _0a$ increases, $V^{*}$ in the unbounded ice-covered ocean varies smoothly. However, the results in the ice-covered channel oscillate persistently. In particular, rapid changes can be observed when $\kappa _0a$ is close to one of natural frequencies of the channel. This rapid change always exists even when $b$ is sufficiently large, which means that the results for $V^{*}$ will always be different from those in the unbounded ice-covered ocean. The difference between two neighbouring natural frequencies becomes smaller when $b$ is larger. Correspondingly, more oscillatory behaviour of the curves at larger $b$ can be observed in figure 6.

Figure 6. Vertical shear forces on the cylinder at the centre of the channel with different widths, when the ice sheet is clamped to the cylinder: (a) channel with free–free edges; (b) channel with clamped–clamped edges. Here, $H/a=5$ and $h_i/a=1/10$.

To explain the differences between the behaviours of $F_x^{*} (M_y^{*})$ and $V^{*}$ when $\kappa _0a$ is near a natural frequency, we may have a closer look at the behaviour of coefficient $\mathcal {D}_{n,n^{\prime },m,m^{\prime }}$ in (4.7b), which is from the Green function in (3.34). It can be seen from (3.46) and (3.49) that $F_x^{*} (M_y^{*})$ is related to $b_{\pm 1,m}$, $c_{\pm 1}$ and $d_{\pm 1}$, while $V^{*}$ is related to $d_0$. As mentioned above, the system of linear equations for $b_{2n,m}$, $c_{2n}$ and $d_{2n}$ is independent of that of $b_{2n+1,m}$, $c_{2n+1}$ and $d_{2n+1}$. Thus $F_x^{*} (M_y^{*})$ is in fact related only to $\mathcal {D}_{2n+1,2n^{\prime }+1,m,m^{\prime }}$ ($n,n^{\prime }=0,1,2\ldots$), while $V^{*}$ is related only to $\mathcal {D}_{2n,2n^{\prime },m,m^{\prime }}$. From (4.7b), when $\omega =\omega _c^{(2i)}$ ($i=1,2,3\ldots$), the residue term of $\mathcal {D}_{2n+1,2n^{\prime }+1,m,m^{\prime }}$ corresponding to wave component $k=0$ gives

(4.8)\begin{equation} \lim_{k\to 0}\frac{I_m(k)\,I_{m^{\prime}}(k)\cos(2n+1)\gamma_m \cos(2n^{\prime}+1)\gamma_{m^{\prime}}}{\mathcal{F}_S^{\prime}(k, \omega_c^{(2i)})\sin\sigma_mb\sin\sigma_{m^{\prime}}b}=0, \end{equation}

where $\gamma _m\to {\rm \pi}/2$ when $k\to 0$, which can be seen from (3.32a,b), and $\mathcal {F}_S^{\prime }(k,\omega _c^{(2i)})\sim O(k)$ Thus $\mathcal {D}_{2n+1,2n^{\prime }+1,m,m^{\prime }}$ is bounded at natural frequencies. However, for the residue term of $\mathcal {D}_{2n,2n^{\prime },m,m^{\prime }}$, we obtain

(4.9)\begin{equation} \lim_{k\to 0}\frac{I_m(k)\,I_{m^{\prime}}(k)\cos 2n\gamma_m \cos 2n^{\prime}\gamma_{m^{\prime}}}{\mathcal{F}_S^{\prime}(k, \omega_c^{(2i)})\sin\sigma_mb\sin\sigma_{m^{\prime}}b}= \frac{({-}1)^{n+n^{\prime}}I_m(0)\,I_{m^{\prime}}(0)}{\sin\kappa_mb \sin\kappa_{m^{\prime}}b}\,\frac{1}{\mathcal{F}_S^{\prime}(0, \omega_c^{(2i)})}\to\infty. \end{equation}

This indicates that $\mathcal {D}_{2n,2n^{\prime },m,m^{\prime }}$ are singular at natural frequencies. Since the behaviours of $\mathcal {D}_{2n,2n^{\prime },m,m^{\prime }}$ and $\mathcal {D}_{2n+1,2n^{\prime }+1,m,m^{\prime }}$ at natural frequencies are different, one is regular and the other singular, the behaviours of $F_x^{*}(M_y^{*})$ and $V^{*}$ are also not expected to be the same. On the other hand, as shown in § 3.5, although singular terms existing in the original boundary integral equation, it can be modified into a regular equation and the solution $\phi$ is still finite when $\varGamma \neq 0$. For this case, it is found that $\varGamma$ is indeed non-zero over the full range of $\kappa _0a$ in figure 6. Thus at the natural frequencies, we may use the modified equation given in (3.58) to find the vertical shear force $V^{*}$. The results at several natural frequencies for $b/a=10$ and $20$ are presented in table 1. It is also interesting to see the effect of the edge conditions at the channel walls on $V^{*}$. When the channel is relatively narrow or $b/a =5$ and $10$, significant differences in the curves of $V^{*}$ can be observed in figures 6(a) and 6(b), which indicates that the effect of edge conditions at the channel walls on $V^{*}$ is very strong. As $b$ increases, this effect gradually becomes weaker, and the curves in these two cases show a closer trend.

Table 1. Vertical shear forces at natural frequencies when the ice sheet is clamped to the surface of cylinder but free–free on two side walls. (Here, $H/a=5$ and $h_i/a=1/10$).

We next consider the cases with different combinations of ice edge conditions on the channel walls and on the cylinder surface. The results for $F_x^{*}$ and $M_y^{*}$ are given in figure 7 as an example. It can be observed that the curves of wave force and moment vary relatively smoothly when the ice edge is free both on the cylinder surface and on the two channel walls, while there are obvious peaks in the curves of the other three cases. When $\kappa _0a$ is small, the influence of edge conditions on $F_x^{*}$ and $M_y^{*}$ is relatively weak, while it is very strong when $\kappa _0a$ is relatively large.

Figure 7. Wave forces and moments on the cylinder under different types of edge conditions: (a) wave forces; (b) moments. Here, X and Y in X-Y refer the edge conditions on the channel walls and cylinder, respectively, where F means free edge, and C means clamped edge. Here, $b/a=5$, $H/a=5$ and $h_i/a=1/10$.

We then consider the hydrodynamic forces at different ice sheet thicknesses. A comparison with the hydrodynamic force in the free surface case is given in figure 8, where the corresponding results are calculated through the procedures given in Linton et al. (Reference Linton, Evans and Smith1992). In figure 8(a), the ice edge is free at all boundaries. When $h_i/a=1/10$, some difference from the result of the free surface case can be observed. As $h_i$ decreases, the difference is very much reduced. When $h_i/a=1/1000$, the difference between the two curves becomes hardly visible. By contrast, the results in figure 8(b) for the ice edges clamped into all boundaries are quite different. Here, $F_x^{*}$ is significantly influenced by $h_i$. There are obvious local peaks in the curve of $h_i/a=1/10$. These peaks decrease or become hardly visible when $h_i/a=1/100$ and $1/1000$. Furthermore, even when the ice sheet becomes very thin at $h_i/a=1/1000$, there are still some visible differences between the results of this case and the free surface case. This indicates that the cases with free edges may resemble the free surface case better when the ice sheet thickness decreases. A similar phenomenon is also reported by Ren et al. (Reference Ren, Wu and Li2020) for the wave diffraction problem of multiple circular cylinders in the unbounded ocean with ice cover.

Figure 8. Wave forces in the $x$-direction on the cylinder at the centre of the channel with different thicknesses of the ice sheet: (a) edge conditions FF-F; (b) edge conditions CC-C. Here, $b/a=5$ and $H/a=5$.

4.4. Forces on a cylinder standing in off-centre positions of the channel

Computations are also carried out to investigate the forces on a vertical cylinder standing in off-centre positions of the channel. In such a case, both the poles caused by the symmetric modes or $\mathcal {F}_S (k,\omega )=0$, and the antisymmetric modes or $\mathcal {F}_A (k,\omega )=0$, will exist in the coefficients $\mathcal {D}_{n,n^{\prime },m,m^{\prime }}$ given in (3.35b). Here, we may give an example of the case when the ice sheet is free along two side walls and is clamped into the surface of the cylinder. The numerical results for hydrodynamic forces are presented in figure 9. In figure 9(a), the curves of $F_x^{*}$ versus $\kappa _0a$ show small differences only at $y_c/a=0$, $2$ and $4$. However, if the cylinder moves to the channel wall further, or at $y_c=6$ and $8$, $F_x^{*}$ at some $\kappa _0a$ is significantly increased. Similar to the case at $y_c=0$, $F_x^{*}$ still varies relatively smoothly when $y_c\neq 0$. By contrast, obvious local peaks and rapid changes can be observed in the curves of $F_y^{*}$ shown in figure 9(b) when $\kappa _0a$ approaches the values corresponding to the natural frequencies of the channel. A similar phenomenon also occurs in the vertical shear force provided in figure 10. Compared with the results at $y_c=0$, since both $\omega _c^{(2i)}$ and $\omega _c^{(2i-1)}$ ($i=1,2,3,\ldots$) will affect $V^{*}$, more peaks in $V^{*}$ can be seen. In fact, if we define $b_{n,m}^{\pm }=b_{n,m}\pm (-1)^{n}b_{-n,m}$, $c_n^{\pm }=c_n\mp (-1)^{n}c_{-n}$ and $d_n^{\pm }=d_n\mp (-1)^{n}d_{-n}$ ($n=0,1,2,\ldots$), the matrix equation in (3.41)–(3.43) for $b_{n,m}$, $c_n$ and $d_n$ can be further converted into two independent submatrix equations. The one for $b_{n,m}^{+}$, $c_n^{+}$ and $d_n^{+}$ has singular terms at natural frequencies, and it is related to $F_y^{*}$ and $V^{*}$. The other, for $b_{n,m}^{-}$, $c_n^{-}$ and $d_n^{-}$, is regular and related to $F_x^{*}$. Thus different behaviours are observed from $F_x^{*}$ and $F_y^{*}$ & $V^{*}$ in figures 9 and 10 when $\omega$ is near a natural frequency.

Figure 9. Wave forces on the cylinder at various off-centre positions of the channel, for edge conditions FF-C: (a) force in the $x$-direction; (b) force in the $y$-direction. Here, $b/a=10$, $H/a=5$ and $h_i/a=1/10$.

Figure 10. Vertical shear forces on the cylinder at various off-centre positions of the channel, for edge conditions FF-C. Here, $b/a=10$, $H/a=5$ and $h_i/a=1/10$.

4.5. Wave patterns and principal strain distributions in the ice-covered channel

The wave elevation or ice sheet deflection can be obtained from $\eta =-({i}/{\omega })({\partial \phi }/{\partial z})|_{z=0}$, together with (3.44). We have

(4.10)\begin{align} \eta(r,\theta)\!&=\!{-}\frac{\rm \pi}{2\omega}\sum_{n={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\frac{b_{n,m}\kappa_m\tanh\kappa_mH}{Q_m} \left[\frac{\mathscr{H}_n^{(1)}(\kappa_mr)}{\mathcal{J}_n(\kappa_mr)}- \frac{\mathscr{H}_n^{(1)\prime}(\kappa_ma)}{\mathcal{J}_n^{\prime} (\kappa_ma)}\right]\mathcal{J}_n(\kappa_mr)\, {\rm e}^{-{\rm i}n\theta}\nonumber\\ &\quad +\frac{{\rm i}L}{\rho \omega^{3}}\sum_{n={-}\infty}^{+\infty} \sum_{m={-}2}^{+\infty}\frac{(\kappa_m^{2}c_n+d_n)\kappa_m\tanh^{2}\kappa_mH}{Q_m}\, \frac{\mathcal{J}_n(\kappa_mr)}{\mathcal{J}_n^{\prime}(\kappa_ma)}\, {\rm e}^{{\rm i}n\theta}. \end{align}

An example of $|\eta |/A$ at $\kappa _0a=0.8$ under four different combinations of edge conditions is provided in figure 11, where the vertical cylinder is located at the centre of the channel. It can be seen that the wave pattern is affected significantly by the edge conditions. In figure 11(a), when the ice edges are free at all boundaries, the maximum amplitude may occur at the front surface of the vertical cylinder or at two side walls. In figure 11(b), if the ice sheet is clamped to the cylinder, then the wave amplitude on its surface will be zero, and then the maximum amplitude may occur at the region in front of the cylinder or at two side walls. By contrast, when the ice edges are clamped to two channel walls, as given in figures 11(c) and 11(d), the maximum wave amplitude may appear only on the front surface or front region of the cylinder.

Figure 11. Wave amplitude $|\eta |/A$ in the ice-covered channel at $\kappa _0a=0.8$ under different edge conditions: (a) FF-F; (b) FF-C; (c) CC-F; (d) CC-C. Here, $b/a=10$, $H/a=5$ and $h_i/a=1/10$.

The strain of the ice sheet is also a very important physical parameter related to the fracture and breakup of the ice. The principal strain $\boldsymbol {\epsilon }$ can be calculated by determining the eigenvalues of the strain tensor matrix (Fung Reference Fung1977)

(4.11)\begin{equation} \boldsymbol{\epsilon} = \frac{h_i}{2}\left[ \begin{array}{cc} \epsilon_{rr} & \epsilon_{r\theta}\\ \epsilon_{r\theta} & \epsilon_{\theta\theta}\\ \end{array} \right]=\frac{h_i}{2}\left[ \begin{array}{cc} \dfrac{\partial^{2} W}{\partial r^{2}} & \dfrac{\partial^{2} W}{r\,\partial r\,\partial \theta}-\dfrac{\partial W}{r^{2}\,\partial \theta}\\[6pt] \dfrac{\partial^{2} W}{r\,\partial r\,\partial \theta}-\dfrac{\partial W}{r^{2}\,\partial \theta} & \dfrac{\partial W}{r\,\partial r}+\dfrac{\partial^{2}W}{r^{2}\,\partial\theta^{2}}\\ \end{array} \right], \end{equation}

where the ice sheet deflection $W(r,\theta,t)$ can be expressed as

(4.12)\begin{equation} W(r,\theta,t)={\rm Re}\{\eta(r,\theta)\, {\rm e}^{{\rm i}\omega t}\} ={\rm Re}\{\eta(r,\theta)\}\cos\omega t - {\rm Im} \{\eta(r,\theta)\}\sin\omega t. \end{equation}

The eigenvalues $\varsigma _{1,2}$ of the strain tensor matrix $\boldsymbol {\epsilon }$ can be obtained as

(4.13)\begin{align} \varsigma_{1,2}&=\frac{h_i}{4}\left\{\left(\frac{\partial^{2} W}{\partial r^{2}}+ \frac{\partial W}{r\,\partial r}+\frac{\partial^{2}W}{r^{2}\,\partial\theta^{2}}\right) \pm\left[\left(\frac{\partial^{2} W}{\partial r^{2}}-\frac{\partial W}{r\,\partial r}- \frac{\partial^{2}W}{r^{2}\,\partial\theta^{2}}\right)^{2}\right.\right.\nonumber\\ &\quad \left.\left.{}+4 \left(\frac{\partial^{2} W}{r\,\partial r\,\partial \theta}- \frac{\partial W}{r^{2}\,\partial \theta}\right)^{2}\right]^{1/2}\right\}. \end{align}

Substituting (4.12) and (4.10) into (4.13), the maximum principal strain $\epsilon _{max}$ at a given location can be found as the maximum value of $|\varsigma _{1,2}|$ as $t$ varies from $0$ to $2{\rm \pi} /\omega$. The distributions of $\epsilon _{max}$ at $\kappa _0a=0.8$ under four different combinations of edge conditions are given in figure 12. Compared with figure 11, the position of the largest value of $\epsilon _{max}$ is different from that of $|\eta |/A$. In figures 12(b) and 12(d), when the ice sheet is clamped to the surface of the cylinder, the largest $\epsilon _{max}$ is at the front surface of the vertical cylinder. However, when the ice sheet is free on the cylinder surface, the largest $\epsilon _{max}$ is on the left and right sides of the cylinder, as shown in figures 12(a) and 12(c).

Figure 12. Distribution of the maximum principal strain $\epsilon _{max}$ in the ice-covered channel at $\kappa _0a=0.8$ under different edge conditions: (a) FF-F; (b) FF-C; (c) CC-F; (d) CC-C. Here, $b/a=10$, $H/a=5$ and $h_i/a=1/10$.