3.1. Horizontally translating dipole with constant speed

When a dipole instantly accelerates from zero to a constant speed $U_0$, its trajectory is determined by the formula

(3.2a,b)\begin{equation} \xi(t)=U_0 t, \quad U_1(t)=U_0. \end{equation}

Then, in the moving coordinate system $X = U_0t - x$ associated with the stationary dipole and oriented such that the flow runs on it from the left, solution (3.1) takes the form

(3.3)\begin{equation} w(X,t) = \frac{\rho M_0U_0}{\rm \pi}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}}\int_0^t \sin{[k(U_0p - X)]}\cos{[\omega(k)p]}\,\textrm{d}p\,\textrm{d}k. \end{equation}

After evaluation of the inner integral, we obtain

(3.4)\begin{equation} w(X,t) = \frac{\rho M_0U_0}{2{\rm \pi}}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}} \left[I_c(k)\cos{kX} - I_s(k)\sin{kX}\right] \textrm{d}k, \end{equation}

where

(3.5)\begin{gather} I_c(k) = \frac{1-\cos{[(U_0k+\omega(k))t]}}{U_0k + \omega(k)} + \frac{1-\cos{[(U_0k - \omega(k))t]}}{U_0k - \omega(k)}, \end{gather}

(3.6)\begin{gather}I_s(k) = \frac{\sin{[(U_0k + \omega(k))t]}}{U_0k + \omega(k)} + \frac{\sin{[(U_0k - \omega(k))t]}}{U_0k - \omega(k)}. \end{gather}

Next, we consider a horizontally oscillating dipole with zero mean velocity.

3.2. Horizontally oscillating dipole with zero mean velocity

When a dipole oscillates horizontally with constant amplitude $\gamma$ and frequency $\varOmega$, its trajectory has the form

(3.7a,b)\begin{equation} \xi(t) = \gamma \sin{\varOmega t}, \quad U_1(t) = \gamma \varOmega\cos{\varOmega t}. \end{equation}

Then, the vertical oscillations of elastic ice plate are

(3.8)\begin{align} w(x,t) &= \frac{\rho\gamma\varOmega M_0}{\rm \pi}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}}\nonumber\\ &\quad \times \int_0^t\cos{\varOmega\tau}\sin{[k(x-\gamma\sin(\varOmega \tau))]}\cos{[\omega(k)(t-\tau)]}\,\textrm{d}\tau\,\textrm{d}k. \end{align}

For small-amplitude oscillations when $\gamma \to 0$, this formula can be simplified and presented as

(3.9)\begin{equation} w(x,t) = \gamma[w_c(x,t)\cos{\varOmega t} + w_s(x,t)\sin{\varOmega t}], \end{equation}

where

(3.10)\begin{gather} w_{c,s}(x,t) = \frac{\rho\varOmega M_0}{2{\rm \pi}}\int_0^\infty\frac{k\sin{kx}\cosh{[k(H-h)]}}{B(k)\cosh{kH}}P_{c,s}(k,t)\,\textrm{d}k, \end{gather}

(3.11)\begin{gather}P_c(k,t) = \frac{\sin{[(\varOmega + \omega(k))t]}}{\varOmega + \omega(k)} + \frac{\sin{[(\varOmega-\omega(k))t]}}{\varOmega - \omega(k)}, \end{gather}

(3.12)\begin{gather}P_s(k,t) = \frac{1-\cos{[(\varOmega+\omega(k))t]}}{\varOmega + \omega(k)} + \frac{1-\cos{[(\varOmega-\omega(k))t]}}{\varOmega - \omega(k)}. \end{gather}

Now we can combine the two types of dipole motion described above using the principle of superposition for a linear system.

3.3. Horizontally moving and oscillating dipole

Assume now that a dipole moves horizontally and periodically oscillates in the direction of motion. Its trajectory is described by the formula

(3.13a,b)\begin{equation} \xi(t) = U_0 t + \gamma \sin{\varOmega t}, \quad U_1(t) = U_0 + \gamma\varOmega\cos{\varOmega t}. \end{equation}

In the moving coordinate system $X = U_0t - x$, the solution (3.1) has the form

(3.14)\begin{align} w(X,t)&=\frac{\rho M_0}{\rm \pi}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}}\int_0^t(U_0 + \gamma\varOmega\cos{\varOmega \tau}) \nonumber\\ &\quad \times \sin{\{k[U_0(t - \tau) - X - \gamma\sin{\varOmega \tau}]\}}\cos{[\omega(k)(t - \tau)]}\,\textrm{d}\tau\,\textrm{d}k. \end{align}

For small-amplitude oscillations of the dipole this formula can be further linearized with respect to $\gamma$, as above, and presented in the form

(3.15)\begin{equation} w(X,t) = w_0(X,t) + \gamma\left[W_c(X,t)\cos{\varOmega t} + W_s(X,t)\sin{\varOmega t}\right], \end{equation}

where function $w_0(X,t)$ is the same as in (3.4), and functions $W_c(X,t)$ and $W_s(X,t)$ are given by the formulae

(3.16)\begin{align} W_c(X,t) &= \frac{\rho M_0\varOmega}{\rm \pi}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}} \nonumber\\ &\quad \times \int_0^t\cos{\varOmega p}\sin{[k(U_0p-X)]}\cos{[\omega(k) p]}\,\textrm{d}p\,\textrm{d}k,\end{align}

(3.17)\begin{align} W_s(X,t) &= \frac{\rho M_0\varOmega}{\rm \pi}\int_0^\infty\frac{k\cosh{[k(H-h)]}}{B(k)\cosh{kH}} \nonumber\\ &\quad \times \int_0^t \sin{\varOmega p}\sin{[k(U_0p-X)]}\cos{[\omega(k) p]}\,\textrm{d}p\,\textrm{d}k. \end{align}

To study the behaviour of functions $W_c$ and $W_s$ in the far-field zone when $|X|,t \to \infty$ we will use the method of stationary phase to estimate asymptotically the double integral (3.16) and (3.17). The phase functions in these integrals have the following forms:

(3.18)\begin{gather} \varPsi_{1,2}(k,p) = k(U_0p - X)\pm p[\varOmega + \omega(k)], \end{gather}

(3.19)\begin{gather}\varPsi_{3,4}(k,p) = k(U_0p - X)\pm p[\varOmega - \omega(k)]. \end{gather}

The stationary points are solutions of the following set of simultaneous equations:

(3.20)\begin{gather} \frac{\partial\varPsi_i}{\partial k} = 0, \quad \rightarrow \quad \frac{\textrm{d}\omega}{\textrm{d}k} = \pm\left(U_0 - \frac{X}{p}\right), \end{gather}

(3.21)\begin{gather}\frac{\partial\varPsi_i}{\partial p} = 0, \quad \rightarrow \quad \frac{\omega}{k} = \pm\frac{\varOmega}{k} \pm U_0. \end{gather}

We will use these equations in the following sections, and now we will present the analysis of the dispersion relation (2.11).

4.1. Numerical solution of the dispersion relation

To investigate the dispersion relation quantitatively, we set the following values for the parameters:

(4.13) \begin{equation} \left.\begin{gathered} \rho_1 = 922.5\ \mbox{kg} \ \textrm{m}^{-3},\quad h_1 = 1 \ \mbox{m}, \quad E = 5\times 10^9\ \mbox{Pa}, \ \nu = 0.3,\\ \rho = 1025\ \mbox{kg} \ \textrm{m}^{-3}, \quad h = 10 \ \mbox{m}, \quad R = 5\ \mbox{m}, \quad g = 9.81 \ \mbox{m} \ \textrm{s}^{-2}. \end{gathered}\right\} \end{equation}

We remind the reader that, here, $R$ is the radius of a cylinder moving at the depth $h$ (see § 3). Three values for the total depth were chosen, $H=100$ m, $H=40$ m and infinite depth, $(H = \infty )$.

Figure 2 shows the dispersion relation (2.11) in dimensionless form for the parameters indicated above (4.13) and $H=40$ m. The dispersion curves were plotted for five values of the parameter $\tilde {Q} \equiv Q/\sqrt {g\rho D} = 0, 1.2, 1.8, 1.95, 2$. As one can see, the dispersion is normal (the corresponding curves are monotonically increasing) for the first two values of $\tilde {Q}$ and anomalous for the other three values.

Figure 2.

Dispersion relations for the FGW for the different values of the compression parameter $Q$.

In table 1 we present the values of $k_0R$ and $\tilde {Q}_0 \equiv Q_0/\sqrt {g \rho D}$ for the parameters indicated above (4.13) for three different depths.

Table 1.

Values of $k_0R$ and $\tilde {Q}_0$ for the typical parameters indicated in (4.13) for three different depths.

The influence of water depth is noticeable only for relatively small wavenumbers. This is illustrated by figure 3, where the dispersion relation is shown in the range $0 \le kR \le 0.3$ for three values of the water depth and two values of the parameter $\tilde {Q}$ characterizing the compression effect.

Figure 3.

Dispersion relation of FGW for three values of water depth $H$ and two values of the compression parameter $\tilde {Q}$. (a) $\tilde {Q} = 1.2$, (b) $\tilde {Q} = 2$. In both panels line 1 pertains to $H = 40$ m, line 2 pertains to $H = 100$ m and line 3 pertains to infinite depth $H = \infty$ m.

In figures 4 and 5 we present the dependences of phase and group speeds at $H = 40$ m (solid lines) and $H = \infty$ (dashed-dotted lines) for two values of the compression parameter $\tilde {Q} = 1.2$ and $\tilde {Q} = 1.95$, respectively. For a finite fluid depth in the long-wave limit we obtain $c_p(0) = c_g(0) = \sqrt {gH}$, whereas for infinitely deep water when $k \to 0$ we have $c_p = 2c_g \approx \sqrt {g/k}$. When $k \to \infty$, $c_g = 2c_p \approx 2k\sqrt {D/M}$ regardless of fluid depth.

Figure 4.

Phase $c_p$ and group $c_g$ speeds of FGW for $\tilde {Q} = 1.2$. Solid lines pertain to $H = 40$ m, and dashed lines to $H = \infty$ m.

Figure 5.

The same as figure 4, but for $\tilde {Q} = 1.95$.

From these figures we see again that the effect of water depth is noticeable only at relatively small wavenumbers $0 \le kR \le 0.3$. The behaviours of both phase and group velocities are non-monotonic for FGW, since both speeds have local minima. As can be seen from the graphs for the group velocity with the abnormal dispersion, figure 5, there are two values of the wavenumber $k_1$ and $k_2$ ($k_1 < k_2$) where the group velocity turns to zero. The dispersion curve has local extrema at these points, and $\omega (k_1) > \omega (k_2)$. The dependencies $k_1$, $k_2$ and $\omega (k_1)$, $\omega (k_2)$ on the parameter $\tilde {Q}$ are shown in figures 6 and 7, respectively, for two different fluid depths, $H = 40$ m (solid lines) and $H = \infty$ m (dashed lines).

Figure 6.

Dependences of $k_1R$ and $k_2R$ on the parameter $\tilde {Q}$ for $H = 40$ m (solid line) and $H = \infty$ m (dashed line).

Figure 7.

Dependences of $\omega _1\sqrt {R/g}$ and $\omega _2\sqrt {R/g}$ on the parameter $\tilde {Q}$ for $H = 40$ m (solid line) and $H = \infty$ m (dashed line).

The $\times$-symbol in figure 6 marks the value of $\tilde {Q}_0$ for a particular water depth, so that we have the dependence of $k_1R(\tilde {Q})$ to the left and $k_2R(\tilde {Q})$ to the right of this point.

5.1. Flexural–gravity waves generated by the uniform horizontal motion of a dipole

In this particular case, when $\varOmega = 0$, we see from the expressions (3.18), (3.19) that the only solution for the stationary points is possible when

(5.1a,b)\begin{equation} c_p(k_c) = U_0, \quad p_c = \frac{X}{U_0 - c_g(k_c)}. \end{equation}

The first equation in (5.1a,b) has

1. (i) no roots for $U_0 < (c_p)_{min}$, where $(c_p)_{min}$ is the minimum phase velocity of FGW;

2. (ii) two roots $k_{1,2} \ (k_1 < k_2)$ for $(c_p)_{min} < U_0 < \sqrt {gH}$; and

3. (iii) only one root $k_{2}$ for $U_0 > \sqrt {gH}$.

Therefore, for $U_0 < (c_p)_{min}$, the wave motion far from the dipole is absent. For $(c_p)_{min} < U_0 < \sqrt {gH}$ a wave with the wavenumber $k_{1}$ exists only for $X > 0$, and a wave with the wavenumber $k_{2}$ only if $X < 0$. This means that the wave motion exists both in front and behind the moving dipole; moreover, the length of the generated wave for $X > 0$ is greater than for $X < 0$. For $U_0 > \sqrt {gH}$, a wave motion exists only in front of a moving dipole at $X < 0$. From the second equation (5.1a,b) it follows that the wave fronts (i.e. the boundaries dividing the regions of water perturbed by the waves from the regions where water is unperturbed) move relative to the source to the right and to the left with speeds $U_0 - c_g(k_{1,2})$, respectively. In the motionless coordinate system the wave fronts move in the corresponding directions with the group velocities of generated waves.

5.2. Flexural–gravity waves generated by a horizontally oscillating dipole

Consider now a horizontally oscillating dipole and the flexural–gravity waves generated by it. In this case, in expressions (3.18), (3.19) we have $U_0 = 0$, and the asymptotic analysis shows that the wave pattern has a certain symmetry relative to the origin and can contain from one to three wave harmonics. Stationary points now have only the functions $\varPsi _ {3,4}$ in (3.19), and those equations reduce to

(5.2a,b)\begin{equation} \varPsi_{3}(k,p) = kx + p[\varOmega-\omega(k)], \quad \varPsi_{4}(k,p) = kx-p[\varOmega-\omega(k)]. \end{equation}

In the case of normal dispersion, i.e. for $Q < Q_0$, each function $\varPsi _3$ and $\varPsi _4$ in (5.2a,b) has a unique stationary point with the same wavenumber $k_1$, which is defined by the expression

(5.3)\begin{equation} \omega(k_1) = \varOmega. \end{equation}

The value of $p_1$ in the immovable coordinate system equals to $\pm x/c_g (k_1)$, respectively. Therefore, the wave defined by the stationary point of the function $\varPsi _3$ exists only for $x > 0$, and the wave determined by the stationary point of the functions $\varPsi _4$ exists only for $x < 0$. The speeds of front propagations of these waves are equal to $\pm c_g(k_1)$, respectively.

In the case of anomalous dispersion when $Q_0 < Q <Q_*$, as shown above, there are values of $k_1$ and $k_2$ $(k_1 < k_2)$ in which the group velocity of the FGW vanishes, and the group velocity is negative in the interval $k_1 < k < k_2$. In the frequency range $\omega (k_2) < \varOmega < \omega (k_1)$, the functions $\varPsi _{3,4}$ in (5.2a,b) have three stationary points, which we denote $k^{(1)}$, $k^{(2)}$ and $k^{(3)}$ $(k^{(1)} < k^{(2)} < k^{(3)})$. The values of $k^{(\,j)}$ with $(\,j = 1, 2, 3)$ satisfy (5.3) with the replacement of $k_1$ by $k^{(\,j)}$. In this case, the waves which are determined by the stationary points $k^{(1)}$ and $k^{(3)}$ of the function $\varPsi _3$ exist only at $x > 0$, whereas a wave determined by $k^{(2)}$ exists only at $x < 0$. The opposite situation occurs for waves determined by the stationary points of the function $\varPsi _4$. As the result, we have a symmetrical wave pattern with respect to $x$. Far from the dipole there is a system of three waves whose front propagation velocities are $\pm c_g(k^{(\,j)})$ $(\,j = 1, 2, 3)$.

5.3. Flexural–gravity waves generated by the superposition of the translating and oscillating dipole motion

In this subsection we restrict ourselves to the consideration of an infinitely deep fluid, because in this case, the definition of stationary points of the functions $\varPsi _i(k, p)$ $(i = 1, 2, \ldots , 4)$ in (3.18), (3.19) reduces to the determination of the roots of polynomials, whereas in the case of a fluid of finite depth it is necessary to solve transcendental equations.

It is convenient to transform the dimensionless variables in which the radius of a cylinder $R$ plays a role of the length scale, and the parameter $\sqrt {R/g}$ plays a role of the time scale. Then the main dimensionless parameters are as follows:

(5.4a–e)\begin{equation} \bar{D} = \frac{D}{g\rho R^4}, \quad \bar{M} = \frac{M}{\rho R}, \quad \bar{Q} = \frac{Q}{g\rho R^2}, \quad F=\frac{U_0}{\sqrt{gR}}, \quad \sigma = \varOmega\sqrt{\frac{R}{g}}. \end{equation}

The dispersion relation (4.4) in the dimensionless variables has a form

(5.5)\begin{equation} \bar{\omega}(\bar{k}) = \sqrt{\frac{\bar{k}(\bar{D}\bar{k}^4 - \bar{Q}\bar{k}^2 + 1)}{1 + \bar{k}\bar{M}}}, \end{equation}

where $\bar {\omega } = \omega \sqrt {R/g}$, $\bar {k} = kR$.

Function $\varPsi _1$ in (3.18) does not have stationary points, because the determining equation

(5.6)\begin{equation} \bar{k}F + \sigma + \bar{\omega}(\bar{k}) = 0 \end{equation}

does not have positive real roots.

Function $\varPsi _2$ in (3.18) also does not have stationary points if $F < V_1(\sigma )\equiv \bar {c}_g(k_1^*)$, where $\bar {c}_g = c_g/\sqrt {gR}$ and the wavenumber $k_1^*$ is the root of equation

(5.7)\begin{equation} \bar{k}\bar{c}_g(\bar{k}) - \bar{\omega}(\bar{k}) = \sigma. \end{equation}

If, however, $F > V_1(\sigma )$, then function $\varPsi _2$ in (3.18) has two stationary points and the determining equation

(5.8)\begin{equation} \bar{k}F - \sigma - \bar{\omega}(\bar{k}) = 0 \end{equation}

has two roots, which we denote as $k_2^{(1)}$ and $k_2^{(2)}$ $(k_2^{(1)}<k_2^{(2)})$. The values of $k_2^{(i)}$ $(i = 1,2)$ can be found as the positive roots of the fifth-degree polynomial

(5.9)\begin{equation} \bar{k}^2[\bar{D}\bar{k}^3-\bar{k}(\bar{Q}+\bar{M}F^2)-F(F-2\sigma\bar{M})]+\bar{k}(1+2\sigma F-\sigma^2\bar{M})-\sigma^2=0, \end{equation}

satisfying (5.8).

It follows from the dispersion relation (5.5) that $k_1^* \to k_p$ and $V_1 \to F_p$ when $\sigma \to 0$, where $k_p$ determines the dimensionless wavenumber that corresponds to the minimal dimensionless phase velocity of FGW, $F_p = U_p/\sqrt {gR}$, and $U_p\equiv (c_p)_{min}$. The equation which allows us to determine $k_p$ has the form

(5.10)\begin{equation} \bar{D}k_p^4(2\bar{M}k_p+3) - \bar{Q}k_p^2-2\bar{M}k_p-1 = 0. \end{equation}

Function $\varPsi _3$ in (3.19) has at most three stationary points. Equation

(5.11)\begin{equation} \bar{k}F+\sigma-\bar{\omega}(\bar{k})=0 \end{equation}

always has one positive root $k_3^{(1)}$ and two additional roots $k_3^{(2)}$ and $k_3^{(3)}$ provided that $\sigma < \sigma ^* \equiv \bar {\omega }(k_g) - k_gF_g$ and $V_3 < F < V_2$. The value of $k_g$ is such that the dimensionless group velocity of FGW $\bar {c}_g$ has a minimum $F_g$ at $\bar k = k_g$; it can be calculated as the positive root of the tenth-degree polynomial which is obtained by equating to zero the numerator in (4.7) and in dimensionless variables has the form

(5.12)\begin{align} &\bar{D}\bar{k}^5[4\bar{D}\bar{M}(2\bar{M}\bar{k} + 5)\bar{k}^4+C_1\bar{k}^3-28\bar{Q}\bar{M}\bar{k}^2+C_2\bar{k}+48\bar{M}]\nonumber\\ &\quad + \bar{k}^4C_3-4\bar{Q}\bar{M}\bar{k}^3-6\bar{Q}\bar{k}^2-4\bar{M}\bar{k}-1 = 0, \end{align}

where $C_1 = 3(5\bar {D}-4\bar {Q}\bar {M}^2)$, $C_2 = 2(12\bar {M}^2-11\bar {Q})$, $C_3 = 30\bar {D} + 3\bar {Q}^2 - 4 \bar {Q}\bar {M}^2$. The value of $F_g$ can be evaluated from the dimensionless form of equation (4.6).

Functions $V_2(\sigma )$ and $V_3(\sigma )$ are determined as follows: $V_2 = \bar {c}_g(k_2^*)$ and $V_3 = \bar {c}_g(k_3^*)$, where $k_2^* < k_g < k_3^*$ are the roots of the equation

(5.13)\begin{equation} \bar{\omega}(\bar{k}) - \bar{k}\bar{c}_g(\bar{k}) = \sigma. \end{equation}

As follows from the dispersion relation (5.5), $k_2^* \to 0$, $k_3^* \to k_p$ and $V_2 \to \infty$, $V_3 \to F_p$ when $\sigma \to 0$, but $\{k_2^*, k_3^*\} \to k_g$ and $\{V_2, V_3\} \to F_g$ when $\sigma \to \sigma ^*$. The values of $k_3^{(\,j)}$ $(\,j=1, 2, 3)$ are the positive roots of the fifth-degree polynomial

(5.14)\begin{equation} \bar{k}^2[\bar{D}\bar{k}^3-\bar{k}(\bar{Q}+\bar{M}F^2) - F(F+2\sigma\bar{M})] + \bar{k}(1 - 2\sigma F - \sigma^2\bar{M}) - \sigma^2=0, \end{equation}

which satisfy (5.11).

Function $\varPsi _4$ in (3.19) has at most three stationary points. Equation

(5.15)\begin{equation} \bar{k}F - \sigma + \bar{\omega}(\bar{k}) = 0 \end{equation}

always has one positive root $k_4^{(1)}$ and two additional roots $k_4^{(2)}$ and $k_4^{(3)}$ provided that $Q_0 < Q < Q_*$. The values $k_4^{(i)}$ $(i = 1,2,3)$ can be determined as the positive roots of (5.9) which satisfy (5.15).

The direction of wave propagation which is determined by the stationary points of functions $\varPsi _2$ and $\varPsi _3$ depends on the sign of the expression $F - \bar {c}_g(k)$, and for the function $\varPsi _4$ it depends on the sign of the expression $F + \bar {c}_g(k)$. In these expressions, the wavenumbers k must be replaced by the stationary values of the corresponding functions. Waves with a positive value of expressions $F \pm \bar {c}_g(k)$ propagate downstream $(X > 0)$, whereas waves with a negative value propagate upstream $(X < 0)$.

Figure 8 shows the dependences of $V_j$ $(\,j = 1, 2, 3)$ on the dimensionless frequency $\sigma$ for three different values of the compression parameter $\tilde {Q} = 1.2$ (a), $\tilde {Q} = 1.8$ (b) and $\tilde {Q} = 1.95$ (c). Curves $V_1$, $V_2$, $V_3$ divide the plane of parameters $(\sigma -F)$ into several domains. In the case of normal dispersion ($\tilde {Q} = 1.2$), the plane $(\sigma -F)$ is divided into four domains $G_n$ $(n = 1, \ldots , 4)$. The domain $G_1$ (marked by vertical hatching) is bounded on the right by curve $V_1$, and on the left by curve $V_2$. The domain $G_2$ (marked by the combination of horizontal and vertical hatching) is bounded on the right by curve $V_2$, below by curve $V_1$ and on the left by the vertical $F$-axis. The domain $G_3$ (marked by horizontal hatching) is bounded on the right by curve $V_2$, on the left by curve $V_3$ and above by curve $V_1$. The remaining part of the plane $(\sigma -F)$ (not shaded domain) is denoted by $G_4$.

Figure 8.

Dependences $V_j$ $(\,j = 1, 2, 3)$ on the dimensionless frequency $\sigma$ for three values of the compression parameter $\tilde {Q} = 1.2$ (a), $\tilde {Q} = 1.8$ (b) and $\tilde {Q} = 1.95$ (c).

In the case of anomalous dispersion ($\tilde {Q} =1.8, 1.95$), the domains $G_1$ and $G_2$ have the same boundaries as above, and the domain $G_3$ is now bounded by curves $V_1$, $V_2$, $V_3$ and $-V_3$. Meanwhile, two new domains appear: $G_5$ (shown by oblique hatching), bounded by curves $V_2$, $-V_2$, $-V_3$, as well as the domain $G_6$ (shown by combined horizontal and oblique hatching), bounded by the horizontal axes $\sigma$ and curves $V_2$, $-V_3$.

The vertical arrows on the horizontal axis show the values of $\sigma ^*$, and the horizontal arrows on the vertical axis show the values of $|F_g|$. The common point of curves $V_1$ and $V_3$ on the vertical axis corresponds to the value of $F_p$.

Values of parameters $k_p$, $F_p$, $k_g$, $F_g$ and $\sigma ^*$ for the used values of $\tilde {Q}$ are given in table 2.

Table 2.

Values of parameters $k_p$, $F_p$, $k_g$, $F_g$ and $\sigma ^*$ for the used values of $\tilde {Q}$.

In the case of normal dispersion (e.g. for $\tilde {Q} = 1.2$):

1. (i) For the parameters $\sigma$ and $F$ from the domain $G_1$, there are four stationary points $k_2^{(1)}$, $k_2^{(2)}$, $k_3^{(1)}$, $k_4^{(1)}$, two of which ($k_2^{(1)}$, and $k_4^{(1)}$) cause wave perturbations running downstream and the other two ($k_2^{(2)}$, $k_3^{(1)}$) running upstream.

2. (ii) For the domain $G_2$ there are six stationary points three of which ($k_2^{(1)}$, $k_3^{(2)}$, $k_4^{(1)}$) cause wave perturbations running downstream, and the other three ($k_2^{(2)}$, $k_3^{(1)}$, $k_3^{(3)}$) running upstream.

3. (iii) For the domain $G_3$ there are four stationary points two of which ($k_3^{(2)}$ and $k_4^{(1)}$) cause wave perturbations running downstream, and the other two ($k_3^{(1)}$ and $k_3^{(3)}$) running upstream.

4. (iv) For the domain $G_4$ there are only two stationary points, one of which $k_4^{(1)}$ causes wave perturbation propagating downstream, and another, $k_3^{(1)}$, running upstream.

In the case of anomalous dispersion, (e.g. for $\tilde {Q} = 1.8$ or $\tilde {Q} = 1.95$):

1. (i) For the domain $G_5$ there are four stationary points two of which ($k_4^{(1)}$ and $k_4^{(3)}$) cause wave perturbations running downstream, and the other two ($k_3^{(1)}$ and $k_4^{(2)}$) running upstream.

2. (ii) For the domain $G_6$ there are six stationary points, three of which ($k_3^{(2)}$, $k_4^{(1)}$ and $k_4^{(3)}$) cause wave perturbations running downstream, and the other three ($k_3^{(1)}$, $k_3^{(3)}$ and $k_4^{(2)}$) running upstream.

In the conclusion to this section we mention that the main properties of FGW were investigated by Bukatov (Reference Bukatov1980) by means of a different method (see also Bukatov Reference Bukatov2017). However, in our opinion the method described above is more visual. The three-dimensional case for an infinitely deep fluid with normal dispersion was described in detail by Sturova (Reference Sturova2013).

5.4. Analysis of vertical displacements of the ice plate

To get an idea of the real displacements of an ice plate caused by the motion of a dipole, we calculated the vertical displacements for the case of an infinitely deep water for the translational motion of the dipole using the integral representation (3.4), and for the superposition of the translational and oscillatory motions based on the integral representations (3.14), (3.15). The results can be characterized by the dimensionless parameter, the Froude number F = $U_0/\sqrt {gR}$. Two typical values of the Froude number were chosen for the calculations, they are $F = 0.25$ and $F = 0.5$; the dimensionless frequency of oscillations were chosen as $\sigma = 0.2$ and 0.25; and the compression parameter was set to $\tilde {Q} = 0$ (no stress in the ice cover), $\tilde {Q} = 1.2$ and $\tilde {Q} = 1.95$.

The time variation of the vertical displacements of the ice plate in the case of translational motion of the dipole source with the Froude number $F = 0.5$ is shown in the videos for the different values of the parameter $\tilde {Q}$ (see the links below).

1. (i) $\tilde {Q} = 0$, https://eportfolio.usq.edu.au/view/view.php?t=4AoZqtbj07wsy8BWTrH1;

2. (ii) $\tilde {Q} \kern-0.2pt=\kern-0.4pt 1.2$, https://eportfolio.usq.edu.au/view/view.php?t=rDFPhJGxZNs8EKBHp2uo;

3. (iii) $\tilde {Q} = 1.95$, https://eportfolio.usq.edu.au/view/view.php?t=LAR34te01uXwV2fgkcvF.

Only in the case (iii) when $\tilde {Q} = 1.95$ is the dipole motion supercritical. As shown in Savin & Savin (Reference Savin and Savin2012) and Il'ichev & Savin (Reference Il'ichev and Savin2017), in infinitely deep water after long-term motion of a dipole, a stationary wave $w_0(X)$ sets in the elastic plate $w_0(X, \infty )$ in the co-moving coordinate frame. In the subcritical regime of motion ($F < F_p$), a symmetrical-in-$X$ perturbation is formed in the elastic plate, which is a plate reaction to the dipole motion, and the minimum of the elevation is located directly above the dipole. In the dimensional variables the solution for the plate displacement $w_0(X)$ in the subcritical regime has the form

(5.16)\begin{equation} w_0(X) = -2\rho R^2 U_0^2\int_0^\infty \frac{k \,\textrm{e}^{-kh}\cos(kX)\,\textrm{d}k}{Dk^4 - \left(Q+MU_0^2\right)k^2 - \rho U_0^2k + g\rho}. \end{equation}

In the supercritical regime ($F> F_p$), the solution for the $w_0(X)$ has a more complex form

(5.17)\begin{equation} w_0(X) = 2\rho R^2 U_0^2\left[\left(\mu + 1\right)\chi(k_1)\sin(k_1X) + (\mu-1)\chi(k_2)\sin(k_2X) + K(X)\right], \end{equation}

where

(5.18)\begin{gather} K(X) = \int_0^\infty\frac{k\left[G(k)\cos(hk) + \rho U_0^2k\sin(hk)\right]}{G^2(k) + \rho^2U_0^4k^2}\,\textrm{e}^{-|X|k}\,\textrm{d}k,\end{gather}

(5.19a,b)\begin{gather} G(k) = Dk^4 + \left(MU_0^2 + Q\right)k^2 + g\rho, \quad \chi(k_j) = \frac{{\rm \pi} k_j\,\textrm{e}^{-k_jh}}{P(k_j)}, \end{gather}

(5.20)\begin{gather} P(k) = 4Dk^3 - 2(Q + MU_0^2)k - \rho U_0^2. \end{gather}

Here, $\mu = \mbox {sign}(X)$, $k_1$ and $k_2\ (k_1 < k_2)$ are roots of the equation $kU_0 = \omega (k)$. In the case of infinitely deep fluid this equation reduces to the polynomial

(5.21)\begin{equation} Dk^4 - \left(Q + MU_0^2\right)k^2 - \rho U_0^2k + g\rho = 0. \end{equation}

In the videos presented above the limiting solutions are shown in the very end by the dotted lines for the comparison with the transient solutions.

The behaviour of the elastic ice plate in superposition of the translational and oscillatory dipole motions is shown for $F = 0.25$ and $\sigma = 0.25$ in figure 9. Panel (a) shows the plate oscillation for $\tilde {Q} = 1.2$ (this corresponds to the domain G4 in figure 8), and (b) shows the oscillation for $\tilde {Q} = 1.95$ (this corresponds to the domain G6).

Figure 9.

Vertical displacements of elastic ice plate $w(X, t)$ for superpositions of translational and oscillatory motions of the dipole with $F = 0.25$, $\sigma = 0.25$. Panel (a) pertains to $\tilde {Q} = 1.2$, and (b) to $\tilde {Q} = 1.95$. The time instants for each curve are indicated on the right of each panel. Calculations were performed on the basis of the linearized solution (3.15).

Figure 10 shows plate oscillations for $F = 0.5$ and $\sigma = 0.2$. Panel (a) shows the plate oscillation for $\tilde {Q} = 1.2$ (this corresponds to the domain G4 in figure 8), and (b) shows the oscillation for $\tilde {Q} = 1.95$ (this corresponds to the domain G3).

Figure 10.

The same as in figure 9, but for $F = 0.5$ and $\sigma = 0.2$. The dark dots for $\tilde {Q} = 1.2$ and $\bar {t} = 3$ and $\bar {t} = 5$ show the values of $w(X, t)$ calculated on the basis of the complete solution (3.14).

The calculations were performed on the basis of the linearized solution (3.15) for $\gamma /R = 0.5$ (the applicability of this solution presumes that $\varOmega R \ll 2U_0$). For comparison, in figure 10(a) we show by dark dots the complete solution (3.14) for the following parameters $F = 0.5$, $\sigma = 0.2$, $\tilde {Q} = 1.2$ and $\bar {t} = 3, \ 5$. It can be seen that, for the chosen amplitude of the dipole oscillation, the complete (3.14) and approximate (3.15) solutions practically coincide, whereas the calculation of the double integral in (3.14) requires significantly more computer resources and time-consuming calculation than the calculation of the single integral in (3.15) after analytic evaluation of the inner integrals in (3.16) and (3.17).

An increase in the compression coefficient $\tilde {Q}$ leads to an increase of wave intensity both for the translational motion of the dipole and in the superposition of translational and oscillatory motions.

6.1. The problem statement

We assume that the wave motion is caused in the initially quiescent fluid by forced oscillations of a horizontal circular cylinder of a radius $R$ with the frequency $\varOmega$ and amplitudes of horizontal and vertical displacements $\eta _{1,2}$ respectively. Assuming that the fluid motion is steady state, we write down the full velocity potential in the form

(6.1)\begin{equation} \varPhi(x,y,t) = \mbox{Re} \left[\textrm{i}\varOmega\sum_{j=1}^2\eta_j\varphi_j(x,y)\exp(\textrm{i}\varOmega t)\right]. \end{equation}

Vertical displacements of the ice cover $W(x,t)$ are determined from the relation

(6.2)\begin{equation} \frac{\partial W}{\partial t} = \left.\frac{\partial \varPhi}{\partial y}\right|_{y=0} \end{equation}

and by analogy with (6.1), it is convenient to write

(6.3)\begin{equation} W(x,t) = \mbox{Re}\left[\sum_{j=1}^2\eta_jw_j(x)\exp(\textrm{i}\varOmega t)\right]. \end{equation}

In the bulk of water for the radiation potentials $\varphi _j(x,y) \ (\,j = 1,2)$ the Laplace equation is satisfied

(6.4)\begin{equation} {\rm \Delta} \varphi_j=0 \quad (|x| < \infty,\ -\infty < y\leq 0). \end{equation}

At the upper boundary of the fluid, $y = 0$, the kinematic and dynamic conditions are satisfied in accordance with conditions (2.2) and (2.3)

(6.5a,b)\begin{equation} w_j(x)=\frac{\partial \varphi_j}{\partial y}, \quad \left(D\frac{\partial^4}{\partial x^4}+Q\frac{\partial^2}{\partial x^2}-M\varOmega^2 +\rho g\right)w_j-\rho \varOmega^2\varphi_j=0 \ (|x|<\infty). \end{equation}

On the circular contour of the cylinder $\{S: \ x^2 + (y + h)^2 = R^2\}$, the non-leakage condition is assumed

(6.6)\begin{equation} \partial \varphi_j/\partial n=n_j \ (x,y \in S), \quad (\,j=1,2), \end{equation}

where ${\boldsymbol {n}} = (n_1, n_2)$ is the internal normal to the contour $S$, $h$ is the distance of the cylinder centre from the upper boundary of the fluid $(h > R)$. For the infinitely deep fluid, the following condition secures the absence of fluid motion in the depth:

(6.7)\begin{equation} \nabla \varphi_j \to 0 \quad (y\to -\infty). \end{equation}

One more condition we use in the far-field zone requires the fulfilment of the radiation condition, that means that the generated waves are outgoing.

6.2. The multipole expansion method

To solve problem (6.4)–(6.7), the multipole expansion method is used (see, for example, Linton & McIver Reference Linton and McIver2001), which is the most effective method in studying bodies of a simple geometry: in the two-dimensional case – circles, in the three-dimensional case – spheres. Even and odd in $x$ multipoles, $\cos (m\theta )/r^m$ and $\sin (m\theta )/r^m$, where $r = \sqrt {x^2 + (y + h)^2}$, $\theta = \arctan [x/(y + h)]$, are fundamental solutions of the Laplace equation singular at the point $x = 0$, $y = -h$.

The boundary condition (6.6) in terms of $r,\theta$ has the form

(6.8a,b)\begin{equation} \partial \varphi_1/\partial r = \sin{\theta}, \quad \partial \varphi_2/\partial r = \cos{\theta} \quad (r = R). \end{equation}

According to these boundary conditions, function $\varphi _1(x,y)$ is odd, and function $\varphi _2(x,y)$ is even in the variable $x$.

For the vertical oscillations of the cylinder solution $\varphi _2(x,y)$ has the form

(6.9)\begin{equation} \varphi_2(x,y) = \sum_{m=1}^\infty p_m R^m\left[\frac{\cos(m\theta)}{r^m}+f_m(x,y)\right], \end{equation}

where

(6.10)\begin{gather} f_m(x,y) = \frac{1}{(m-1)!}\int_0^\infty k^{m-1}\cos(kx)A(k)\,\textrm{e}^{k(y+h)}\,\textrm{d}k, \end{gather}

(6.11)\begin{gather}A(k) = \frac{kP(k)+\rho \varOmega^2}{Z(k)}\,\textrm{e}^{-2kh}, \end{gather}

(6.12)\begin{gather}P(k) = Dk^4-Qk^2-M\varOmega^2+\rho g, \end{gather}

(6.13)\begin{gather}Z(k) = kP(k)-\rho\varOmega^2. \end{gather}

The integrand in (6.10) has simple poles that are the roots of the equation $Z(k) = 0$ which can be written as $\varOmega ^2 = \omega ^2(k)$, where $\omega (k)$ is the dispersion relation (4.4) for flexural–gravity waves in an infinitely deep fluid.

The function $Z(k)$ is the fifth-degree polynomial which has only one real root $k = k_1 > 0$ if $Q < Q_0$, whereas for $Q_0 < Q < Q_*$ it has three real positive roots $k_n \ (n = 1,2,3)$ which we will arrange in the ascending order: $k_1 < k_2 < k_3$. The remaining roots of the polynomial are complex.

Taking into account the radiation condition in the far-field zone, we can present (6.10) in the form

(6.14)\begin{align} f_m(x,y) &= \frac{1}{(m-1)!}\left[P.V.\int_0^\infty k^{m-1}A(k)\,\textrm{e}^{k(y+h)}\cos{(kx)}\,\textrm{d}k \right.\nonumber\\ &\quad -\left. \textrm{i}{\rm \pi}\sum_{n=1}^N\chi_n k_n^{m-1}A^{(n)}\,\textrm{e}^{k_n(y+h)}\cos{(k_nx)}\right]. \end{align}

Here, $N = 1$ for $Q < Q_0$ and $N = 3$ for $Q_0 <Q < Q_*$; $P.V.$ means that the principal value of the integral should be considered; $A^{(n)}$ is the residue of function $A(k)$ in $k=k_n$; $\chi _n = 1 \ (\chi _n = -1)$ if the group velocity is positive (negative).

Taking into account the dispersion relation, we obtain $\omega (k_n) = \varOmega \ (n=1,2,3)$, and the group velocity (4.6) for $k = k_n$ is

(6.15)\begin{equation} c_g(k_n) = \frac{5Dk_n^4-3Qk_n^2 + \rho g - M\varOmega^2} {2\varOmega(\rho+k_nM)}. \end{equation}

The constructed solution satisfies (6.4) and the boundary conditions (6.5a,b) and (6.7), but does not satisfy the non-leakage condition on the cylinder surface (6.8b). To account for this boundary condition, the known relation is used:

(6.16)\begin{equation} \exp[k(\textrm{i}x+y+h)]=\sum_{l=0}^\infty\frac{(kr)^l}{l!}\exp(\textrm{i}l\theta). \end{equation}

Then, (6.10) takes the form

(6.17)\begin{equation} f_m(x,y)=\frac{1}{(m-1)!}\sum_{l=0}^\infty \frac{r^l}{l!}\cos{(l\theta)}I_{ml}, \end{equation}

where

(6.18)\begin{equation} I_{ml} = P.V.\int_0^\infty k^{m+l-1}A(k)\,\textrm{d}k -\textrm{i}{\rm \pi}\sum_{n=1}^N\chi_n k_n^{m+l-1}A^{(n)}, \quad I_{ml}=I_{lm}. \end{equation}

Using (6.17), we can present $\varphi _2(x,y)$ in (6.9) in the form

(6.19)\begin{equation} \varphi_2 = \sum_{m=1}^\infty p_m R^m\left[\frac{\cos{(m\theta)}}{r^m} + \frac{1}{(m-1)!}\sum_{l=1}^\infty\frac{r^l}{l!}\cos{(l\theta)}I_{ml}\right]. \end{equation}

In contrast to (6.17), the term with $l = 0$ is not taken into account in (6.19) when we sum over $l$, since the radiation potential is determined up to an arbitrary constant.

This problem was solved by Das & Sahu (Reference Das and Sahu2019) in the absence of compression forces in the elastic ice plate $(Q = 0)$ by performing numerical integration in (6.18). In the current paper, we use the recurrence formulae and integral exponential functions to calculate the integral in (6.18). To this end, we make the following conversion:

(6.20)\begin{equation} \int_0^\infty k^{m+l-1}\,\textrm{e}^{-2kh}\left[1 + \frac{2\rho \varOmega^2}{Z(k)}\right]\textrm{d}k = \frac{(m+l-1)!} {(2h)^{m+l}} + 2\rho\varOmega^2\int_0^\infty\frac{ k^{m+l-1}\,\textrm{e}^{-2kh}}{Z(k)}\,\textrm{d}k. \end{equation}

The denominator in the integrand, which is the fifth-degree polynomial in $k$, can be represented as

(6.21)\begin{equation} Z(k) = D\prod_{n=1}^5(k-k_n), \end{equation}

where $k_n$ are the roots of the polynomial. Then, we use the expansion

(6.22)\begin{equation} \frac{1}{Z(k)} = \frac{1}{D}\sum_{n=1}^5\frac{\alpha_n}{k-k_n}, \end{equation}

where the coefficients $\alpha _n$ can be calculated from the solution of the corresponding system of linear algebraic equations obtained from the requirement of equality of the numerator on the right- and left-hand sides of (6.22).

Further we will use the recurrent formulae to calculate the integrals in (6.20) using the substitution equation (6.22). Denoting

(6.23)\begin{equation} I_J = \int_0^\infty \frac{k^J\,\textrm{e}^{-2kh}}{k-k_n}\,\textrm{d}k. \end{equation}

For the real roots $k_n$, taking into account the radiation condition in the far-field zone, we obtain

(6.24)\begin{equation} I_J = \tilde{I}_J-i{\rm \pi}\chi_nk_n^J\,\textrm{e}^{-2k_nh}, \quad \mbox{where} \ \tilde{I}_J = P.V.\int_0^\infty\frac{k^J\,\textrm{e}^{-2kh}}{k-k_n}\,\textrm{d}k. \end{equation}

The recurrent formulae for $\tilde {I}_J$ are

(6.25a,b)\begin{equation} \tilde{I}_{J+1}=\frac{J!}{(2h)^{J+1}}+k_n\tilde{I}_J, \quad \tilde{I}_0=-\textrm{e}^{-2k_nh}{Ei}(2k_nh), \end{equation}

where ${Ei}$ is the integral exponential function of a real argument (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

For the complex roots $k_n$, we obtain in (6.23)

(6.26)\begin{equation} I_J=\frac{1}{(2h)^J}G_J(-2hk_n), \end{equation}

where

(6.27a,b)\begin{equation} G_{J+1}(z) = J! - zG_J(z) \quad (J\geq 1), \quad G_1(z) = 1 - z\textrm{e}^z{E}_1(z), \end{equation}

where ${E}_1$ is the integral exponential function of a complex argument (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

Differentiating (6.9) with respect to $r$ and taking into account the boundary conditions (6.8b) and the orthogonality of trigonometric functions, we obtain a set of linear algebraic equations for the determining of the coefficients $p_m$ in (6.9)

(6.28)\begin{equation} p_m - \sum_{l=1}^\infty \frac{p_l R^{l+m}I_{ml}} {m!(l-1)!} = -R\delta_{ml}, \end{equation}

where $\delta$ is the Kronecker symbol.

After determining the coefficients $p_m$, we can calculate all the characteristics of the fluid motion and ice plate oscillations. The radiation forces ${\boldsymbol {F}} = (F_1, F_2)$ are usually written in the matrix form (for more details see, for example, Linton & McIver Reference Linton and McIver2001)

(6.29a,b)\begin{equation} F_k = \sum_{j=1}^2\eta_j\tau_{kj} \ (k=1,2), \quad \tau_{kj} = \rho\varOmega^2\int_S\varphi_jn_k\,\textrm{d}s = \varOmega^2\mu_{kj}-\textrm{i}\varOmega\lambda_{kj}, \end{equation}

where $\mu _{kj}$ and $\lambda _{kj}$ are the coefficients of the added mass and damping, respectively. Similarly to Das & Sahu (Reference Das and Sahu2019), we have $\tau _{21} = 0$, $\tau _{22} = -{\rm \pi} \rho R\varOmega ^2 (2p_1+R)$.

For the horizontal oscillations of the cylinder, the solution for $\varphi _1(x,y)$ can be presented in a form similar to (6.9)

(6.30)\begin{equation} \varphi_1(x,y) = \sum_{m=1}^\infty q_mR^m\left[\frac{\sin(m\theta)}{r^m}+g_m(x,y)\right], \end{equation}

where

(6.31)\begin{equation} g_m(x,y)=\frac{1}{(m-1)!}\int_0^\infty k^{m-1}\sin(kx)A(k)\,\textrm{e}^{k(y+h)}\,\textrm{d}k, \end{equation}

and $A(k)$ is presented in (6.11).

The derivation of the solution presented above is largely repeated for the vertical oscillations, and for an infinitely deep liquid we ultimately obtain

(6.32a–c)\begin{equation} q_m = p_m, \quad \tau_{11} = \tau_{22}, \quad \tau_{12}=0. \end{equation}

Further, we use the following notations for non-zero components of the radiation load: $\mu = \mu _{jj}$, $\lambda = \lambda _{jj} \ (\,j=1,2)$.

6.3. Vertical displacement of the ice cover

For the horizontal oscillations of the cylinder, from (6.30), taking into account (6.5a), we obtain

(6.33)\begin{equation} w_1(x)=2\rho \varOmega^2\sum_{m=1}^\infty \frac{p_m R^m} {(m-1)!}\int_0^\infty\frac{k^m\sin(kx)}{Z(k)}\,\textrm{e}^{-kh}\,\textrm{d}k. \end{equation}

The function $w_2(x)$ for the vertical oscillations of the cylinder has the same form but with the replacement of the factor $\sin {(kx)}$ in the integrand by $\cos {(kx)}$.

The integral in (6.33) can be represented in alternative form taking into account the poles of the integrand in (6.33) (see § 6.2) and the radiation conditions in the far-field zone

(6.34)\begin{equation} \int_0^\infty\frac{k^m\sin{(kx)}}{Z(k)}\,\textrm{e}^{-kh}\,\textrm{d}k = P.V.\int_0^\infty\frac{k^m\sin{(kx)}}{Z(k)} \,\textrm{e}^{-kh}\,\textrm{d}k - \textrm{i}{\rm \pi}\sum_{n=1}^N \chi_n\frac{k_n^m\sin{(k_nx)}}{Z'(k_n)}\,\textrm{e}^{-k_nh}, \end{equation}

where $Z'(k_n) \equiv (\textrm {d}Z/\textrm {d}k)|_{k=k_n}$.

Vertical displacements of the elastic plate in the far-field zone at $|x| \to \infty$ are determined only by the second term in (6.34), since the integral term vanishes for large values of $|x|$. Therefore, displacement in the far-field zone can be written as

(6.35)\begin{equation} (w_1(x),w_2(x))\approx\sum_{n=1}^N C_n(\sin{(k_nx)},\cos{(k_nx)})\quad (|x|\to \infty), \end{equation}

where

(6.36)\begin{equation} C_n = -2\textrm{i}{\rm \pi} \rho\varOmega^2\frac{\chi_n\,\textrm{e}^{-k_n h}}{Z'(k_n)}\sum_{m=1}^\infty \frac{p_mR^mk_n^m}{(m-1)!}. \end{equation}

In the case of the normal dispersion $(N = 1)$, there are reciprocity relations connecting the damping coefficients and wave amplitudes in the far-field zone. These relations can be used for the verification of the accuracy of numerical calculations. In the absence of compression in the elastic plate $(Q = 0)$, there is a relation given in the paper by Li, Wu & Ji (Reference Li, Wu and Ji2018) for the case of a fluid of finite depth. In the presence of compression $(Q < Q_0)$, i.e. when the dispersion for flexural–gravity waves is normal, this relation for an infinitely deep fluid has the form

(6.37)\begin{equation} \lambda = 2|C_1|^2\left[\frac{k_1}{\varOmega}(2Dk_1^2-Q) + \frac{\varOmega\rho}{2k_1^2}\right]. \end{equation}

In the dipole approximation, wave amplitudes in the far-field zone for each of the generated modes $C_n^{(d)} \ (n = 1,2,3)$ can be determined using the stationary phase method or the solution of Savin & Savin (Reference Savin and Savin2013) on wave generation by a pulsating source in infinitely deep water under an ice sheet

(6.38)\begin{equation} |C_n^{(d)}| = \frac{{\rm \pi} \rho \varOmega R^2 k_n\,\textrm{e}^{-k_nh}}{(\rho+k_nM)c_g(k_n)} = \frac{2{\rm \pi} \rho \varOmega^2 R^2 k_n\,\textrm{e}^{-k_nh}}{5Dk_n^4-3Qk_n^2+\rho g - M\varOmega^2}. \end{equation}

Here, the (6.15) has been used for $c_g(k_n)$.

6.4. The numerical results

The added mass and damping coefficients for the various values of bending stiffness $D$ and submerged depth $h$ of a cylinder have been presented by Das & Sahu (Reference Das and Sahu2019) for $Q = 0$. Our results completely coincide with their values. To conduct the numerical computations, the infinite summations in (6.9) and (6.30) were truncated at a finite number $m = \mathcal {M}$. The results for the hydrodynamic load were obtained by setting $\mathcal {M} = 6$. It was found that a further increase of $\mathcal {M}$ does not affect the first four decimal places.

The added mass and damping coefficients for a submerged cylinder under a free surface ($h_1 = 0$) and under an ice cover of various thicknesses ($h_1 = 0.5$, 1, 2 m) without compressive forces ($Q = 0$) are shown in figure 11 for the parameters presented in (4.13) besides the value $h_1$. The following dimensionless quantities are used for the added mass and damping factors: $\bar {\mu } = \mu /({\rm \pi} \rho R^2)$, $\bar {\lambda } = \lambda /({\rm \pi} \rho \varOmega R^2)$.

Figure 11.

The added masses (a) and damping coefficients (b) as functions of dimensionless frequency $\sigma$ for the free surface $(h_1 = 0)$ and ice cover with $Q = 0$ and various ice thicknesses: $h_1 = 0.5$, 1, 2 m. Thick lines pertain to the coefficients for a circular cylinder, and thin lines pertain to the dipole approximation of the damping coefficients.

It is seen from this figure that the extreme hydrodynamic loads in the case of uncompressed ice are less than in the case of a free surface. It should be noted that, regardless of the upper boundary conditions (the free surface or ice cover), the solution in the limit of low-frequency oscillations of a cylinder reduces to the solution for the cylinder moving under a rigid horizontal cover. Therefore, for $\sigma \to 0$, the added mass depends only on the depth of cylinder submersion, and for $h/R = 2$, the value of $\bar {\mu }(0) \approx 1.135$ (see, for example, Eatock Taylor & Hu Reference Eatock Taylor and Hu1991). In figure 11(b), the values for the damping coefficients obtained for a circular cylinder by the multipole expansion method (thick lines) and in the dipole approximation (thin lines) are compared using (6.38) in (6.37). It can be seen that the discrepancy between the exact solution and the dipole approximation decreases with increasing ice-cover thickness.

The effect of compressive forces on the hydrodynamic loads in the ice cover at the thickness $h_1 = 1$ m is shown in figure 12. A few cases of $\tilde {Q}$ were considered: $\tilde {Q} = 1.2$, 1.4, 1.8. In the two former cases the normal dispersion of FGW takes place, whereas in the latter case it is anomalous so that FGW with the negative group velocities appear in the range of dimensionless frequencies $0.240 < \sigma < 0.319$. It can be seen that, in the presence of ice-cover compression, the extreme value of the hydrodynamic load significantly increases in the vicinity of frequencies where the group velocity of FGW becomes very small or negative. The damping coefficient becomes very small at the frequency $\sigma \approx \sigma _2$, which corresponds to the local maximum of the dispersion curve in the case of anomalous dispersion.

Figure 12.

The added masses (a) and damping coefficients (b) as functions of dimensionless frequency $\sigma$ for the ice cover of thickness $h_1 = 1$ m and $\tilde {Q} = 1.2$, 1.4, 1.8.

The displacement amplitudes of the ice cover in the far-field zone are presented in figure 13. The solutions obtained by the multipole expansion method (6.36) are compared with the solutions obtained in the dipole approximation (6.38) for $h_1 = 1$ m and various values of the normalized longitudinal stress parameter: $\tilde {Q} = 0$, 1.2, 1.4, 1.8. Panels (a–c) in figure 13 pertain to the case of normal dispersion when only one wave propagates in the far-field zone. In the case of anomalous dispersion, there are three waves with different wavenumbers propagating in the far-field zone; the amplitudes of these three generated waves are shown separately in figure 13(d). Arrows on the horizontal axis of figure 13(d) show the values $\sigma _1 = 0.240$ and $\sigma _2 = 0.319$ for the considerate case. It is seen that the maxima of wave amplitudes in the far-field zone increase with increasing compression of the ice cover. The solution obtained by the multipole expansion method and within the dipole approximation are in a good agreement. In the presented calculations, the reciprocity relation (6.37) is fulfilled with high accuracy in the case of the normal dispersion of the FGW.

Figure 13.

Amplitudes of ice-sheet displacements in the far-field zone determined by the multipole expansion method (solid lines) and within the dipole approximation (dots) as functions of the dimensionless frequency of the oscillating cylinder at $h_1 = 1$ m and various compression parameters: (a) $\tilde {Q} = 0$, (b) $\tilde {Q} = 1.2$, (c) $\tilde {Q} = 1.4$, (d) $\tilde {Q} = 1.8$.