4.1. Uniform current in the upper and lower layers

The dispersion relation (3.10) for the currents with equal velocities $U$ in each layer provides within the deep-water approximations, $A_1 = \rho _1 (1 + s)$ , $A_2 = \rho _1 g |p|(1-s) + (1+s) |p| \mathcal{A}$ , $A_3 = (1-s) g p^2 \mathcal{A}$ , $\mathcal{A} = EI p^4 + \rho _1 g$ and $\Omega _U = \Omega - pU$ . The limiting dispersion relation can be decomposed as

(4.1) \begin{align} A_1 \big(\Omega _U^2 - \Omega _{US}^2\big)\big(\Omega _{U}^2 - \Omega _{UI}^2\big) = 0, \end{align}

where

(4.2) \begin{align} \Omega _{US}^2 &= \frac {|p|}{\rho _1}\big(EI p^{4} + \rho _1 g\big),\\[-10pt]\nonumber \end{align}

(4.3) \begin{align} \Omega _{UI}^2 &= r g |p|, \end{align}

and $r = (1-s)/(1+s)$ . Equations (4.2) and (4.3) are the dispersion relations of flexural-gravity and internal gravity waves with current within the deep-water approximations, see equation (10) in Mohanty, Mondal & Sahoo (Reference Mohanty, Mondal and Sahoo2014), which leads to (4.2) without accounting for both compression in the floating plate and the plate mass.

Equation (4.2) describes flexural-gravity waves for infinitely deep fluid with a current. In the deep-water limit, these waves are independent of the presence of the interface at $y = -h$ . Equation (4.2) provides two wave systems

(4.4) \begin{align} \Omega _{S}^{\pm } = pU \pm \sqrt {\frac {|p|}{\rho _1}\big(EI p^{4} + \rho _1 g\big)}. \end{align}

Without loss of generality, we take the current velocity $U$ to be negative, which implies that the current goes from right to left, see figure 1. Then $\Omega _{S}^+\gt 0$ for $p\lt 0$ and the phase velocity $\Omega _{S}^+(p)/p$ is negative, which means that the corresponding waves propagate in the same direction as the current, from right to left, see (3.1). For positive $p$ , we have $pU\lt 0$ . Therefore, it is possible that $\Omega _{S}^+(p) = 0$ at a certain positive $p$ for some values of the parameters of these waves. The equation $\Omega _{S}^+(p) = 0$ yields the following equation with respect to $p$ :

(4.5) \begin{align} \rho _1 p U^{2} = EI p^{4} + \rho _1 g . \end{align}

This equation has no roots for $|U|\lt U_c$ , two roots for $|U|\gt U_c$ and one double roots for $|U| = U_c$ , where $U_c$ is such that the derivatives in $p$ of the left-hand and right-hand sides are equal

(4.6) \begin{align} \rho _1 U^{2} = 4 EI p^{3}. \end{align}

Substituting (4.6) in (4.5), we find $p_c^4 = (\rho _1 g/3 EI)$ , and then (4.6) provides

(4.7) \begin{align} U_c = -\left (\frac {4g}{3}\right )^{1/2}\left (\frac {3EI}{\rho _1 g}\right )^{1/8}. \end{align}

The parameters of the problem, which will be used below, are $g = 9.81$ m s⁻ $^{2}$ , $E = 5\times 10^8$ Pa, $\rho _1 = 1003.5$ kg m⁻ $^{3}$ , $\rho _2 = 1024$ kg m⁻ $^{3}$ , $\nu = 0.3$ and $d = 1$ cm providing $I = 45.78$ m $^{3}$ and $U_c = -2.127$ m s⁻¹. Therefore, the waves of the system $\Omega _{S}^+(p)$ propagate in the direction of the current for negative $p$ and in the opposite direction for positive $p$ if $|U|\lt U_c$ . For positive $p$ and $|U|\gt U_c$ , waves with $0\lt p\lt p_1$ and $p\gt p_2$ , where $p_2\gt p_1\gt 0$ are the roots of (4.5), propagate opposite to the current, and waves with $p_1\lt p\lt p_2$ , propagate in the same direction as the current.

Figure 1.

Schematic diagram of the physical domain representing a two-layer fluid with a plate-covered surface and interface.

Equation (4.4) gives $\Omega _{S}^+(-p) = -\Omega _{S}^-(p)$ , which means that the wave system with $\Omega _{S}^-(p)$ does not require special analysis. The waves with $p = \pm p_1$ and $p = \pm p_2$ have zero phase speeds and zero frequencies for the wave systems $\Omega _{S}^+(p)$ and $\Omega _{S}^-(p)$ , respectively, see figure 2. Note also that waves with $\Omega ,\, p$ and $-\Omega ,\, -p$ are the same wave, see (3.1), propagating in the same direction and having the same frequency and wavelength. The waves corresponding to $\Omega _{S}^+(p)$ with positive $\Omega$ are shown in figure 2 by the solid line, and their ‘mirror’ images corresponding to $\Omega _{S}^-(p)$ are shown by dashed lines. For a moving oscillatory load on a floating elastic plate with frequency $\Omega$ , a single wave propagates in both the following and opposing directions of the load velocity for $U \lt -1.4$ m s⁻¹ and $\Omega \gt 0$ . When $U \gt -1.4$ m s⁻¹, within the blocking points, three waves are present: two with positive group velocity and one with negative group velocity, all propagating in the opposite direction to the load. Nevertheless, a single wave propagates in the direction of the load.

The group velocity of the flexural-gravity waves on the current with $U\lt 0$ is given by

(4.8) \begin{align} c_g^{\pm }(p ) = \frac {{\rm d} \Omega _{S}^\pm }{d p} = -|U| \pm \frac {1}{2 \sqrt {\rho _1|p|}} \frac {5 EI p^4 + \rho _1 g}{\sqrt {(EI p^4 + \rho _1 g)}} sgn(p), \end{align}

where $sgn(p) = -1$ for $p\lt 0$ and $sgn(p) = +1$ for $p\gt 0$ .

It is seen that $c_g^+(p) \lt 0$ for $p\lt 0$ and $c_{g}^+(p)$ may change its sign for $p\gt 0$ . Correspondingly, $c_{g}^-(p)\lt 0$ for $p\gt 0$ . Let us find the roots of the equation $c_{g}^+(p) = 0$ . The roots are positive. It is convenient to introduce a new unknown $\alpha$ as $EI p^4 = \rho _1 g \alpha$ . Then the equation $c_{g}^+(p) = 0$ reads

(4.9) \begin{align} \frac {5 \alpha + 1}{\alpha ^{1/8} (\alpha + 1)^{1/2}} = \frac {4}{3^{3/8}}\frac {|U|}{|U_c|}. \end{align}

The left-hand side of (4.9) is a smooth and positive function of $\alpha$ , which tends to $+\infty$ as $\alpha \to 0$ and as $\alpha \to \infty$ . The left-hand side of (4.9) is minimum at $\alpha = \sqrt {1 + 1/15} - 1$ , where it is equal to $1.7557$ . Therefore, (4.9) has no roots for $|U|\lt 0.6627|U_c|$ . For larger speeds of the current, there are two roots of (4.9), which provide two wavenumbers of the waves with zero group speed and frequency. Note that both the phase and group velocities vanish for current speed $U_c$ at a certain wavenumber, as given in (4.7). It is recalled that wave blocking occurs when the group velocity vanishes. We conclude that, for the selected parameters of the problem, there are no waves with zero phase speed if the speed of the current is smaller than $2.127$ m s⁻¹, and there are no wave blocking for current speed smaller than $1.4$ m s⁻¹. Abnormal waves occur only at relatively high speeds of the current. Note that the group velocity of the flexural-gravity wave remains continuous here, regardless of the current speed. This contrasts with the jump discontinuity observed at extreme values of plate compression, as discussed in Das et al. (Reference Das, Sahoo and Meylan2018c ). For example, Lin (Reference Lin2021) reported that the speed of the under-ice current can be larger than $1.2$ m s⁻¹ flowing south-eastward from the coastal regions of southern Newfoundland towards deep waters.

Figure 2.

The dispersion curves $\Omega _{S}^+$ (solid lines) and $\Omega _{S}^-$ (dashed lines) for different values of current velocity.

In figure 2, the roots of (4.8) are shown by brown circles for $U = -1.9$ m s⁻¹. The primary blocking point is associated with the lower values of the wavenumber, whereas the other is known as the secondary blocking point, (see Boral et al. Reference Boral, Meylan, Sahoo and Ni2023b ). Figure 2 reveals that, for any frequency within the primary and secondary blocking points, the dispersion relation (4.2) possesses three positive real wavenumbers, $p$ , of the waves. By analysing the slopes of the three roots, it is confirmed that the first and the last waves are associated with positive group velocity, whilst the middle wave corresponds with negative group velocity. This implies that the middle wave propagates in the opposing direction to the current, whilst the wave energy, known as the group velocity, propagates in the direction of the current. The three roots of the dispersion relation merge at the saddle point, marked with a black square, for $U = -1.4$ m s⁻¹. Moreover, figure 2 demonstrates that wave blocking occurs for negative frequency $\Omega _{S}^+$ and positive wavenumber $p_3$ lying in the range $p_1\lt p_3\lt p_2$ for $|U|\gt U_c$ . This can be interpreted as waves in the range $p_3\lt p\lt p_2$ propagating in the direction of the current, whereas wave energy propagates in the opposite direction. On the contrary, both the wave energy and wave in the range $p_1\lt p\lt p_3$ propagate in the direction of the current.

Figure 3.

The dispersion curves for positive $\Omega _I^+$ (solid lines) and $\Omega _I^-$ (dashed lines) with different values of current speed $U$ .

The dispersion relation (4.3), which is for waves localised at the interface between two infinitely deep fluids, gives two wave systems

(4.10) \begin{align} \Omega _{I}^{\pm } = p U \pm \sqrt {rg|p|}. \end{align}

For $U\lt 0$ , the frequency $\Omega _{I}^+\gt 0$ for $p\lt 0$ and the phase speed $\Omega _{I}^+/p$ is negative, which implies that the short interfacial wave propagates in the direction of current, see (3.1). For positive $p$ , the expression $pU\lt 0$ . Thus, the frequency $\Omega _{I}^+$ may be zero for certain values of $p$ , which is given by

(4.11) \begin{align} p_c = rg/U^{2}. \end{align}

Equation (4.11) provides that $p_c$ approaches zero for large values of current speed $U$ , see figure 3. Figure 3 is plotted by taking the same parameters used in figure 2. Equation (4.10) gives $\Omega _{I}^+\lt 0$ for $p\gt p_c$ , which implies that the phase speed is negative. Equation (4.10) provides $\Omega _{I}^+(p) = \Omega _{I}^-(-p)$ , which means that $\Omega _{I}^-\gt 0$ for $p\lt -p_c$ in the case of $p\lt 0$ , as presented by dashed lines in figure 3. Note that, in figure 3, only the positive values of $\Omega _I^+(p)$ and $\Omega _I^-(p)$ are exhibited. This is due to the fact that $\Omega _I^+(p) = \Omega _I^-(-p)$ , which means that no additional analysis is needed for $\Omega _I \lt 0$ .

The following group velocity for the short interfacial waves is obtained:

(4.12) \begin{align} c_{g}^{\pm } = \frac {{\rm d} \Omega _{I}^\pm }{d p} = -|U| \pm \frac {1}{2}\sqrt {\frac {rg}{|p|}} sgn(p). \end{align}

Equation (4.12) gives $c_{g}^-\lt 0$ for $p\lt 0$ . It is obvious that the group velocity becomes zero, $c_{g}^+ = 0$ , for positive wavenumber $p = p_{cg} = p_c/4$ . Equation (4.11) implies that no wave blocking occurs for infinitely large values of current $U$ , as demonstrated in figure 3. Furthermore, (4.12) gives that group velocity is positive for waves in the range $0\lt p\lt p_{cg}$ , and negative for $p_{cg}\lt p\lt p_c$ . Equations (4.9) and (4.12) imply that flexural-gravity wave blocking occurs beyond a certain value of uniform current; however, wave blocking occurs in the short interfacial wave in the presence of a uniform current in a two-layer fluid. In figure 3, the wavenumbers $p_c$ and $p_{cg}$ are shown. Figure 3 demonstrates that interfacial waves do not propagate opposite to the current for frequencies beyond the blocking point. For any positive frequency, before the blocking point, two interfacial waves with positive and negative group velocities propagate in the opposite direction to the current. Moreover, two interfacial waves with negative group velocity propagate in the direction of the current for any frequency $\Omega _{I}$ .

4.2. Moving upper layer over a stationary lower layer

The dispersion relation (3.15) for a moving upper layer with speed $U$ over a stationary lower layer provides, within the deep-water approximations,

(4.13) \begin{align} \left (\rho _1 \Omega _{r1} + \rho _2 \Omega \right ) \rho _1 \Omega _{r1}^2 \Omega &+ \left \{\rho _1\Delta U\Omega _{r1} - (\rho _2 - \rho _1) g \right \} \rho _1 p \Omega _{r1}^2 + \mathcal{A} \mathcal{D}_{r_1} = 0, \end{align}

where $\mathcal{A} = {EIp}^4 + \rho _1 g$ and ${\mathcal{D}}_{r1} = -\rho _2 \left \{sp \Omega \Omega _{r1} + p \Omega ^{2} - (1-s) g p^{2}\right \} - \rho _1 \Delta U p^{2} \Omega _{r1}$ with $\Omega _{r1} = \Omega - p U_1$ . This limiting dispersion relation can be decomposed as

(4.14) \begin{align} (\Omega _{r1}^2 - \Omega _{r1+}^2)\big \{\rho _2 \Omega _I^{2} + \rho _1 \Omega _{r1} \Omega _I + \rho _1 \Delta U |p| \Omega _{r1} - (\rho _2 - \rho _1) g |p|\big\} = 0, \end{align}

where

(4.15) \begin{align} \Omega _{r1+}^2 & = \frac {|p|}{\rho _1}(EI p^4 + \rho _1 g), \end{align}

(4.16) \begin{align} \Omega _I^{\pm } &= \frac { U p s \pm \sqrt {g |p| (1-s^2) -sp^2 U^2}}{(1+s)}. \end{align}

Equations (4.15) and (4.16) are associated with flexural-gravity waves where $\eta _{20}/\eta _{10} \ll 1$ and the interfacial wave with $\eta _{10}/\eta _{20} \ll 1$ , respectively. It is worth mentioning that (4.15) is the same as in (4.4), which are both independent of the presence of the interface and the flow under the interface within the short wave approximations. Here, $\Omega _I^+$ and $\Omega _I^-$ represent the upper and lower branches of the dispersion curve, respectively. It is recalled that the Kelvin–Helmholtz (KH) instability of short internal gravity waves for given values of the upper-layer speed $U$ occurs at wavenumber $p_{KH}$

(4.17) \begin{align} p_{KH} = \frac {g(1-s^2)}{s U^2}. \end{align}

Equation (4.17) implies that $p_{KH}$ approaches zero for large values of $U$ for given values of $s$ . Besides, $p_{KH}$ approaches zero when the densities of the two layers are close to each other, $s \to 1$ . The occurrences of KH instability are shown in figure 4 for different values of upper-layer speed. The parameters of the floating plate and liquid are considered the same as in figure 2. A detailed study on KH instability can be found in Cushman-Roisin (Reference Cushman-Roisin2005) for surface gravity waves and in Boral et al. (Reference Boral, Sahoo and Stepanyants2021b ) for capillary-gravity waves in a two-layer density-stratified fluid with tangential discontinuity of velocity across the mean free interface. The dispersion relation (4.16) for the interfacial waves with $|p|\lt p_{KH}$ reduces to

(4.18) \begin{align} \Omega _I^{\pm } = \frac {U s p \pm U \sqrt {s|p|(p_{KH} - |p|)}}{1+s}. \end{align}

Without loss of generality, the upper-layer velocity, which moves from right to left, is considered negative. Therefore, $pU\gt 0$ for $p\lt 0$ , which implies that $\Omega _{I}^-(p)\gt 0$ . Consequently, the phase speed $c_I^-(p) = \Omega _I^-(p)/p$ is negative. It is checked that $\Omega _I^-(p) = 0$ for $p_c = p_{KH}/(1+s)$ , which are marked with orange squares in figure 4. For $U\lt 0$ . Equation (4.18) gives that $\Omega _{I}^-\lt 0$ for $p_c\lt p\lt p_{KH}$ . Note that $\Omega _{I}^+(p) = \Omega _I^-(-p)$ , which implies that $\Omega _I^+\gt 0$ for waves in the range $-p_{KH}\lt p\lt -p_c$ with $p$ being negative. These waves are exhibited by dashed lines in figure 4. For interfacial waves within $0\lt p\lt p_{KH}$ , the group velocity is positive for $p\lt p^*$

(4.19) \begin{align} p^* = \frac {p_{KH}}{2} \left (1 + \sqrt {\frac {s}{1+s}}\right ), \end{align}

and it is negative for $p^*\lt p_{KH}$ . This implies that the interfacial wave propagates in the opposite direction to the current, whilst the corresponding wave energy propagates in the direction of the current. Similarly, for $-p_{KH}\lt p\lt 0$ , it can be checked that the group velocity is negative for $p_*\lt -p$ and it is positive for $-p_{KH}\lt p_*$ , where

(4.20) \begin{align} p_* = -\frac {p_{KH}}{2}\left (1 + \sqrt {s/(1+s)}\right ). \end{align}

In figure 4, the waves with zero group speed are marked with brown circles, where wave blocking occurs. Figure 4 reveals that no interfacial wave propagates in the opposing direction to the upper-layer speed for frequencies beyond the blocking point. Additionally, two interfacial waves of positive and negative group velocity exist for any frequency before the occurrence of wave blocking for $p\gt 0$ . However, two interfacial waves propagate in the direction of the upper-layer speed for frequencies prior to the occurrence of KH instability at $\Omega _I^- \lt \Omega _I^-(p_{KH})$ . It is found that, for frequencies within the range $\Omega _I^-(p_{KH})\lt \Omega _I^-\lt \Omega _I^-(p_*)$ , two interfacial waves propagate in the direction of upper-layer speed. However, their group velocities are of opposite signs.

Figure 4.

Short interfacial wave dispersion curve (4.16) in a two-layer fluid having a tangential discontinuity of velocity across the interface for different values of upper-layer speed $U$ .

4.2.1. Time-domain simulation

Here, we simulate the flexural-gravity and internal gravity wave profiles when the upper layer is moving over a stationary lower layer in the time domain. Deep-water approximations are considered in both the upper and lower layers. The parameters are taken the same as in figure 4. Waves propagating in the direction opposite to the current are only considered for clarity. Equations (4.15) and (4.16) reveal that the flexural-gravity wave on the plate-covered surface and the interfacial wave on the interface are independent of each other under the assumption of deep-water approximation.

Subsequently, using the Fourier integral, the plate deflection can be written (see Das et al. Reference Das, Sahoo and Meylan2018b )

(4.21) \begin{align} \eta _1(x,t) = {\mathrm{Re}\,}\left \{\int _0^\infty \hat {f}(p) e^{{\mathrm{i}}(p x - \Omega _s t)} {\rm d}p\right \}, \end{align}

where $\Omega _s$ is the frequency obtained from (4.15) for a given incident wave with wavenumber $p$ . The function $\hat {f}(p)$ is the Gaussian wavepacket in the transform space, which is defined as

(4.22) \begin{align} \hat {f}(p) = 2 A \sqrt {\pi b}\, e^{-b(p - p_a)^2}, \end{align}

where $p_a$ is the centre wavenumber, $A = 0.3$ m is the amplitude and $b = 2000$ m $^2$ is the spreading function.

From (3.7), the amplitude ratio is obtained

(4.23) \begin{align} a(p) \equiv \frac {\eta _{20}}{\eta _{10}} = \frac {2s\Omega _{r1}^2}{e^{ph} \left \{\Omega _I^2 + s \Omega _{r1}\Omega _I - g(1 -s )p + s pU \Omega _{r1}\right \}}. \end{align}

Thus, the displacement of the interfacial wave is written as

(4.24) \begin{align} \eta _2(x, t) = {\mathrm{Re}\,}\left \{\int _0^\infty a(p) \hat {f}(p) e^{{\mathrm{i}}(p x - \Omega _I^+ t)} {\rm d}p\right \}. \end{align}

We consider the upper-layer speed $U = -0.5$ m s⁻¹ and two different values of centre wavenumber, $p_a = 0.1$ 1 m⁻¹ (before the occurrence of wave blocking) and $p_a =$ $0.2$ 1 m⁻¹ (after the occurrence of wave blocking), see figure 6(a).

Figure 5.

(a) Dispersion curve, and (b–f) time-domain simulation of the waves for two different times are shown. In (b) and (c), the flexural-gravity wave profiles are exhibited and in (d), (e) and (f) the interfacial wave profiles are shown for $p_a = 0.1$ 1 m⁻¹. Note that here non-dimensional time is set as $\bar {t} = U p_a t$ .

In figures 5(b)–5(f), the time-domain simulation of the flexural-gravity waves in blue and interfacial waves in red are exhibited for $p_a = 0.1$ 1 m⁻¹. It is observed that the group velocity of the flexural-gravity wave is higher than that of the interfacial wave, and both waves are propagating to the right. Figures 6(a) and 6(b) are taken from ‘Movie FGW’, and figures 5(c)–5(f) are taken from ‘Movie IGW figures 5, which are available at https://doi.org/10.1017/jfm.2025.10268. On the other hand, in figure 6, only the time-domain simulation of the interfacial wave for $p_a = 0.2$ 1 m⁻¹ is shown. These figures are taken from ‘Movie IGW figure 6’, which is also available in the supplementary material. Figure 6 reveals that the interfacial wave packet moves to the left due to negative group velocity, whereas the phase velocity is positive. This is obvious from the dispersion curve, see figure 6(a), since the slope of the interfacial wave dispersion curve is negative beyond the blocking point.

Figure 6.

Time-domain simulation of the interfacial waves for two different times with $p_a = 0.2$ 1 m⁻¹.

5.1. Uniform current in the upper and lower layers

Here, the speed of the upper layer is considered to be the same as that of the lower layer, which corresponds with a uniform current with velocity $U$ beneath the thin floating elastic plate in a two-layer fluid. Note that the current velocity $U$ is considered negative, which means that physical phenomena where the wave and current propagate in opposite directions are exhibited in the first quadrant of the $\Omega -p$ plane. One considers $U$ as positive, providing a mirror image of the subsequent results about the line $p=0$ without changing the physical meaning. The dispersion relation (3.10) is analysed by taking $\tau = 0$ to determine the criteria for the occurrences of wave blocking in flexural-gravity and interfacial waves in finite water depth. For clarity, the dispersion curves for positive frequencies are presented. Note that high current speeds, $U$ , are considered in this section, although uncommon in real ocean conditions, which is crucial for theoretical analysis. This helps explore the wave dynamics under extreme scenarios, providing insights into interactions between wave modes, current shear and the interface’s response. This theoretical framework can be applied to practical situations, such as studying the response of a floating plate to a moving load with appropriate modification in the mathematical formulation. High-speed currents simulate rapid moving forces on floating structures, allowing a better understanding of energy transfer, shear-induced instabilities and wave interactions, with relevance to dynamic loading in engineering and oceanography.

Figure 7.

Dispersion curves (3.10), (4.2) and (4.3) for (a) flexural-gravity waves with $U = -5$ m s⁻¹ and (b) interfacial waves with $U = -0.7$ m s⁻¹.

Figure 7 demonstrates the dispersion curves (3.10), (4.2) and (4.3) for a flexural-gravity wave with $U = -5$ m s⁻¹ and interfacial wave with $U = -0.7$ m s⁻¹. These figures depict the changes in the wave characteristics with changes in the depths of the upper and lower layers. Figure 7(a) reveals that the variations in the flexural-gravity wave characteristics in the direction of the current are minimal in deep water at finite depth. However, a strong interaction between the opposing current and flexural-gravity wave is observed in deep water at finite water depth for wavenumbers within the range $0\lt ph\lt 1.5$ , leading to conditions where the wave energy is redistributed or dissipated. It is found that wave blocking occurs in short flexural-gravity waves with opposing currents for certain values of frequency and wavenumber, representing a physical phenomenon where the wave energy cannot overcome the opposing flow. However, no wave blocking occurs in the flexural-gravity wave at finite water depth. Thus, the occurrence of wave blocking and interaction of multiple waves on the plate-covered surface depends on the depths of the upper and lower layers for given values of uniform current. On the other hand, figure 7(b) exhibits a minimal variation for the interfacial waves in deep water at finite water depth in the direction of current. This suggests that the stratified layers allow smooth propagation of the interfacial waves in the direction of the current, with negligible resistance or energy redistribution. Conversely, a pronounced variation in the interfacial wave with an opposing current is observed with changes in upper- and lower-layer depths for wavenumbers within the range $0\lt ph\lt 3$ . Figure 7(b) reveals that wave blocking occurs for lower values of frequency and higher values of wavenumber in the interfacial wave with opposing current in the case of finite water depth compared with deep water. This implies that the frequency beyond which interfacial waves do not propagate in the opposite direction to the current reduces as the depth of the upper and lower layers is reduced.

Figure 8.

Dispersion curve (3.10) for waves on the plate-covered surface and interface in finite water depth for (a) $U = 0,\,-0.6$ and $-0.8$ m s⁻¹, and (b) $U = -5.1,\,-6.2$ and $U = -6.65$ m s⁻¹.

Figure 8 depicts the frequency $\Omega$ as a function of wavenumber $p$ given by (3.10) for different values of current speed $U$ . In figures 8(a) and 8(b), solid and dashed lines represent the flexural-gravity waves and interfacial waves, respectively. Figure 8(a) reveals that the dispersion curves associated with the flexural-gravity and interfacial waves are symmetric about the vertical line $p=0$ in the absence of current. This symmetry reflects that waves behave identically in both directions. However, the dispersion curves become asymmetric in the presence of uniform current velocity $U(\lt 0)$ . Figure 8(a) demonstrates that variations in the flexural-gravity waves are minimal in both the downstream (with the current) and upstream (against the current) directions, indicating that the current has a negligible effect on their propagation characteristics. Additionally, no wave blocking occurs in the flexural-gravity waves since no extrema exists in the dispersion curve. In contrast, wave blocking occurs in the interfacial wave where the dispersion curve possesses a maximum, which shifts to lower values of frequency and wavenumber $p$ for higher values of current speed $U$ . This indicates that stronger opposing currents reduce the range of frequencies and wavelengths for which interfacial waves can propagate upstream. On the other hand, a comparison of figures 8(b) and 2 reveals that the saddle point occurs for higher values of uniform current, $U = -5.1$ m s⁻¹, with lower values of frequency, $\Omega = 0.3953$ 1 s⁻¹, compared with deep upper and lower layers. In addition, the critical current speed $U_c$ where the phase velocity attains zero minimum is reduced from $-6.73$ to $-6.67$ m s⁻¹ in finite water depth $H=10$ m compared with deep water. For $U =$ $-6.2$ m s⁻¹, primary and secondary wave blocking occurs, and multiple real roots exist within a range of frequencies, indicating complex wave interactions. These findings are consistent with the results shown in figure 2. Figure 8(b) exhibits that interfacial waves cannot propagate in the upstream but are instead carried downstream. This is not the case observed in short interfacial waves, since $p_c$ in (4.11) approaches zero for infinitely large values of $U$ , as shown in figure 3. Two interfacial waves propagate only in the direction of the current, irrespective of the values of frequency $\Omega$ . Notably, figures 8(a) and 8(b) illustrate that wave blocking does not occur simultaneously in the plate-covered surface and interface for given values of uniform current $U$ and frequency $\Omega$ . This suggests that, when multiple waves are present, they may propagate either at the interface or on the plate-covered surface in the upstream direction, depending on the current speed and frequency.

Figure 9.

Dispersion curves (3.10) for waves on the plate-covered surface and interface in finite water depth for (a) different values of plate thickness $d$ and (b) density ratio $s$ for $U=-5$ m s⁻¹.

In figure 9, the dispersion curves (3.10) for waves on the plate-covered surface and interface are presented for different values of plate thickness $d$ and density ratio $s$ in the presence of uniform current $U=-5$ m s⁻¹. Figure 9(a) demonstrates that the primary and secondary blocking points occur in the flexural-gravity wave with opposing current for $d = 0.05$ m. Note that a plate thickness of $d = 0.05$ m was used to represent grease ice in the laboratory experiment described by Carr et al. (Reference Carr2019). However, no wave blocking occurs in the flexural-gravity waves propagate in the upstream direction for $d= 0.1$ and $d = 0.15$ m. This means that a thinner plate makes it easier for the opposing current to interact with the waves, leading to situations where the energy of the waves is trapped, causing wave blocking within certain frequency ranges. Besides, negligible variations are observed in the flexural-gravity wave and interfacial wave dispersion curve when it propagates in the direction of current $U$ . Similar to figure 8(b), in figure 9(a), interfacial waves do not propagate opposite to the current for each value of frequency $\Omega$ and plate thickness $d$ with $U = -5$ m s⁻¹. This means that, at higher opposing current speeds, interfacial waves cannot move upstream because the current pushes the wave energy strongly downstream. On the other hand, figure 9(b) illustrates that wave blocking occurs in the flexural-gravity wave with opposing current $U = -5$ m s⁻¹ for two closely situated frequencies $\Omega$ for density ratio $s = 0.95$ and $d = 0.1$ m. This implies that flexural-gravity wave blocking occurs for certain values of plate thickness, $d$ , with a change in the values of density ratio $s$ . The variations in the flexural-gravity and interfacial wave dispersion curves in the direction of current are similar to those of figure 9(a) with changes in the values of $s$ . Figures 9(a) and 9(b) reveal that the interfacial wave with opposing current is not influenced by a change in the values of plate thickness $d$ and density ratio $s$ in the presence of higher values of uniform current $U$ .

5.2. Different current speed in the upper and lower layers

This section delves into the analysis of wave blocking that occurs in the flexural-gravity wave and interfacial wave in a two-layer density-stratified fluid with different current speeds between the upper and lower layers. The upper layer is moving at a speed $U_1$ , and the lower layer moves at a speed $U_2$ such that $U_1 \ne U_2$ . Note that the case for $U_1 = U_2$ is studied in sub § 5.1. The speed of the lower layer is considered to be lower than that of the upper layer since the lower layer is heavy. The associated dispersion relation (3.8) for $\tau = 0$ is analysed. Emphasis is given to analysing the dispersion curves when the upper and lower layers are moving in opposite directions. This section provides deep insights into the occurrences of wave blocking in flexural-gravity waves and interfacial waves. Furthermore, the section provides the range of frequencies for which multiple wave interactions take place either on the plate-covered surface or on the interface, depending on the parameters. The occurrences of KH instability due to tangential discontinuity of velocity across the interface and its influence on the plate-covered surface are analysed. Figure 10 demonstrates frequency $\Omega$ as a function of wavenumber $p$ given by dispersion relation (3.8) for different values of upper-layer speed $U_1$ over a stationary lower layer with $U_2 = 0$ . Figure 10 reveals negligible variations among the flexural-gravity waves in the upstream and downstream directions of upper-layer velocities $U_1$ . Wave blocking occurs in the interfacial waves propagating in the opposite direction of $U_1$ for certain values of wavenumber $p$ , beyond which no interfacial waves propagates. Besides, KH instability occurs for certain values of wavenumber $p$ in the interfacial wave, depending on the values of $U_1$ , when waves propagate in the direction of upper-layer velocity. This instability arises from the mean flow discontinuity at the interface, where the shear between the layers creates a mechanism for energy exchange that destabilises the system. Before this instability occurs, wave blocking is observed, which highlights the complex interplay between the wave dynamics and shear-induced instabilities. A comparison of figures 10 and 4 depicts that the values of the wavenumber at which various physical phenomena like wave blocking, KH instability and zero phase speed occur shift to slightly lower values in finite water depth. This implies that the interaction of the flexural-gravity wave with the interfacial wave reduces the interfacial wave speed in the following and opposing directions of upper-layer speed. Unlike the scenario shown in figure 8, where no blocking occurs for interfacial waves with the following uniform current, figure 10 demonstrates that wave blocking occurs even for short interfacial waves with higher frequencies when propagating in the direction of the upper-layer velocity. Physically, this implies that the interface acts as a barrier that modifies the energy propagation pathways and leads to instability in a two-layer fluid with a stationary lower layer.

Figure 10.

Dispersion curves (3.15) for waves on the plate-covered surface (solid lines) and interface (dashed lines) for different values of upper-layer speed $U$ over a stationary lower layer.

Figure 11.

Dispersion curves (3.8) for waves on the plate-covered surface (solid lines) and interface (dashed lines) for different values of upper-layer speed $U_1$ with $U_2 = -0.5$ m s⁻¹.

In figure 11, a similar result as in figure 10 is plotted for $U_2 = -0.5$ m s⁻¹. Here, the reference case of moving upper and lower layers in the same direction from right to left is discussed. Figure 11 depicts an analogous observation with figure 10 for flexural-gravity wave propagation in the following and opposing directions of the upper-layer velocity. A comparison of figures 11 and 10 reveals that the values of positive wavenumber $p$ , at which wave blocking and zero phase speed occurs in the interfacial wave for $U_1 = -1$ and $-1.5$ m s⁻¹, are significantly lower in the presence of lower-layer speed $U_2 = -0.5$ m s⁻¹. Physically, this indicates that the additional motion of the lower layer reduces the relative velocity difference experienced by interfacial waves propagating in the direction opposite to the upper-layer flow. As a result, the wavelength and phase speed of these interfacial waves decrease, reflecting the enhanced influence of the moving lower layer in modulating the wave dynamics. On the other hand, KH instability and wave blocking occur for lower values of negative wavenumber $p$ in the case of $U_2 = -0.5$ m s⁻¹ than that of $U_2 = 0$ for $U_1 = -1$ and $-1.5$ m s⁻¹. This suggests that when the upper and lower layers flow at different speeds in the same direction, the velocity gradient at the interface becomes more pronounced, leading to increased shear effects. These effects allow shorter interfacial waves, which are associated with higher frequencies, to propagate with the flow.

Figure 12.

Dispersion curves (3.8) for waves on the plate-covered surface (solid lines) and interface (dashed lines) for different values of upper-layer speed $U_1$ with $U_2 = 0.5$ m s⁻¹.

In figure 12, similar results as in figure 11 are plotted by changing the direction of the lower-layer velocity with $U_2 = 0.5$ m s⁻¹. It is observed that the flexural-gravity wave in the following and opposing directions of the upper-layer speed is not affected by the change in direction of the lower-layer speed. A comparison of figures 12 and 11 shows that wave blocking and KH instability occur at lower frequencies and wavenumbers associated with the interfacial wave propagating in the direction of upper-layer speed. This is due to the increased shear at the interface, which amplifies instability and disrupts energy transfer, leading to earlier blocking of shorter, high-frequency interfacial waves. On the other hand, figure 12 illustrates that wave blocking occurs for slightly higher values of frequencies and wavenumbers when the interfacial waves propagate in the opposite direction to the upper-layer speed and in the same direction as the lower-layer speed compared with that of figure 11. Additionally, the wavenumbers at which phase speeds become zero in the interfacial waves shift slightly to higher wavenumber values in figure 12 for $U_2 = 0.5$ m s⁻¹ compared with figure 11 with $U_2 = -0.5$ m s⁻¹. Figures 11 and 12 show that, when interfacial waves attempt to travel upstream (against the upper-layer speed), against the flow of the lower layer, they encounter significant resistance due to the opposing motion. However, when interfacial waves move downstream in the same direction as the lower layer’s flow, the supporting motion of the current reduces the relative shear, allowing these waves, especially those with higher frequencies and shorter wavelengths, to propagate more effectively.

Figure 13.

Dispersion curves (3.8) for waves on the plate-covered surface (solid lines) and interface (dashed lines) for different values of lower-layer speed $U_2$ with $U_1 = -6$ m s⁻¹.

In figure 13, the frequency $\Omega$ as a function of wavenumber $p$ given by (3.8) is studied for different values of lower-layer speed $U_2$ with the upper-layer speed $U_1 = -6$ m s⁻¹. The lower layer is assumed to be moving in the direction of the upper-layer speed. Here, the current speeds are considered larger than those typically observed in the ocean, and this theoretical framework is crucial for understanding extreme interactions that may inform practical scenarios, such as the response of a floating plate to high-speed moving loads in a two-layer fluid. Physically, this figure shows that wave blocking for flexural-gravity waves with opposing upper-layer speed occurs at lower frequencies and wavenumbers when the shear across the interface is weaker. This suggests that a weaker velocity gradient at the interface reduces the energy required for blocking, making it easier for these waves to lose momentum. On the other hand, as the shear current strengthens, the primary and secondary blocking points shift to higher frequencies. Furthermore, the negligible variations in the flexural-gravity waves propagating in the direction of the upper-layer speed, irrespective of the shear strength, highlight that downstream propagation is less sensitive to current speed differences across the interface. Additionally, the interfacial waves cannot travel in the opposite direction to the upper-layer speed $U_1$ when the current speed is high, regardless of the speed of the lower layer $U_2$ . This shows that the upper-layer current has a stronger influence on the wave movement at the interface.

Figure 14.

Dispersion curves (3.8) for waves on the interface with upper and lower layers moving in the same direction (dashed lines) and in the opposite direction (solid lines) for different values of density ratios $s$ with $U_1 = -1.5$ m s⁻¹ and $|U_2| = 0.5$ m s⁻¹.

Figure 14 demonstrates the variations in the dispersion curves associated with the interfacial waves for different values of the density ratio $s$ . Here, the upper layer flows from right to left with $U_1 = -1.5$ m s⁻¹, while the lower layer is considered to move either in the same direction as the upper layer with $U_2 = -0.5$ m s⁻¹ (dashed lines) or in the opposite direction with $U_2 = 0.5$ m s⁻¹ (solid lines). The variations in the flexural-gravity waves with changes in density ratio $s$ are negligible and are not presented here for clarity. Figure 14 reveals a qualitatively similar observation to that of figures 11 and 12. Figure 14 depicts that KH instability and wave blocking occur in the interfacial waves for smaller values of frequency $\Omega$ and wavenumber $p$ with an increase in the values of density ratio $s$ , regardless of the direction of the lower-layer velocity with respect to the upper-layer velocity. This can be interpreted as the higher values of the density ratio, $s$ , implying weaker stratification, reducing the wavelength and phase speed of the interfacial wave for given values of frequency and wavenumber in a two-layer fluid with tangential discontinuity of velocity. Moreover, for higher values of $s$ , particularly at $s = 0.95$ , no interfacial wave propagation is observed in the direction opposite to the upper-layer flow, irrespective of the direction of the lower layer. This can be explained by the diminished stratification at high $s$ , which suppresses the ability of the interfacial wave to propagate against the current. It is noteworthy that the wavenumbers at which the phase speed vanishes depend on the density ratio $s$ but are independent of the direction of the lower-layer velocity relative to the upper-layer velocity.

Figure 15.

Dispersion curves (3.8) for waves on the interface with upper and lower layers moving in the same direction (dashed lines) and in the opposite direction (solid lines) for different values of water depth ratios $h/H$ with $U_1 = -1.5$ m s⁻¹ and $|U_2| = 0.5$ m s⁻¹.

Similar to figure 14, in figure 15, the role of water depth ratio $h/H$ in the variations of the interfacial wave dispersion curve is demonstrated. Figure 15 reveals that negligible variations are observed in the interfacial wave propagation in the direction of the moving upper and lower layers for $h/H = 0.25$ and $h/H = 0.75$ . However, when the interfacial wave propagates in the same direction as the upper layer and opposite to the lower layer, KH instability and wave blocking occur at lower frequencies. For $h/H = 0.25$ , the figure shows that interfacial waves do not propagate in the opposing direction of the upper-layer velocity. This indicates that, when the interface is close to the plate-covered surface, the shallow water depth (small $h/H$ ) limits the interfacial wave’s ability to move opposite to the upper layer’s current. On the other hand, for $h/H = 0.75$ , the interfacial wave propagates in the opposite direction to the upper-layer flow within a certain range of frequency and wavenumber, regardless of the direction of the lower-layer flow. This is due to the increased water depth, which provides more space for the wave to propagate, reducing the influence of the plate-covered surface and allowing interfacial waves to propagate upstream. Figure 15 suggests that the wavelength and phase speed of the interfacial wave propagating with the upper-layer flow are highly influenced by the direction of the lower-layer flow in a two-layer fluid with tangential discontinuity of velocity across the interface.