2.1. Static boundary conditions of the FFMB

Figure 1 shows the sketch of an FFMB, in which the $x$ axis is the still water surface and the $z$ axis is upward. The FFMB is composed of membrane and internal fluid, in which the membrane density is $\rho _s$ and the membrane thickness is $\varDelta$. The fluid density inside and outside of the membrane are $\rho _I$ and $\rho _O$, respectively, satisfying $\rho _I<\rho _O$. As a result, the FFMB floats on the water surface due to the buoyancy. Here $\theta$ represents the angle between the tangent of the membrane and the positive direction of the $x$ axis; $z = z_h$ and $z = -z_b$ denote the vertical positions of the highest and lowest points of the FFMB, respectively; $S$ represents the length of the membrane extending clockwise from the apex $(0,z_h)$; $p_{i0}$ represents the internal membrane pressure of the FFMB and $T_S$ represents the initial static tension of the FFMB. The static shape of the FFMB depends on the filling ratio $\gamma$ and the relative density $\rho _I/\rho _O$ of the fluid inside and outside the membrane.

Figure 1. Schematic diagram of wave with the FFMB.

The filling ratio $\gamma$ is defined as

(2.1)\begin{equation} \gamma = A_0/A_{max}, \end{equation}

where $A_0$ is the area surrounded by the membrane and $A_{max}$ is the largest possible area for which the membrane shape is circular. When the filling ratio is $\gamma = 1$, the geometry of the membrane is circular (as shown in figure 3, blue line) and the radius of this circle is called the membrane radius $R_C$.

This paper briefly introduces the method of obtaining the static shape of the FFMB as follows. For more details, please refer to Zhao (Reference Zhao1995). As shown in figure 2, assuming that a small element of the membrane is ${\rm d} S$, the initial membrane tension is $T$, the resultant force of internal and external fluid pressures is $p_n$, the tangential force on the membrane is $p_t$ and the deflection angle of the element is ${\rm d} {\theta }$. Since the membrane thickness is infinitely thin, the mass and gravity of the membrane can be neglected. The force balance equation of the membrane element can be written as

(2.2)\begin{equation} \left. \begin{array}{@{}c@{}} {{p_n}\, {\rm d} S = 2T\sin ({\rm d} \theta /2) + {\rm d} T\sin ({\rm d} \theta /2)}, \\ {{\rm d} T\cos ({\rm d} \theta /2) + {p_t}\, {\rm d} S = 0,} \end{array} \right\} \end{equation}

where

(2.3)\begin{equation} \left. \begin{array}{@{}ll@{}} {p_n} = {p_{i0}} + {\rho _I}g ({z_h} - z), & z \ge 0, \\ {p_n} = {p_{i0}} + {\rho_I}g ({z_h} - z) + {\rho_O}gz, & z < 0. \end{array} \right\} \end{equation}

Figure 2. Stress analysis of the FFMB.

Since the fluid is assumed to be inviscid, the friction force on the membrane is zero, that is, ${{\partial T}}/{{\partial S}} = 0$. According to (2.2), the differential equation of the membrane can be expressed as

(2.4)\begin{equation} {{p_n}\,{\rm d} S = T\,{\rm d} \theta \Rightarrow \frac{1}{R} = \frac{{{\rm d} \theta }}{{{\rm d} S}} = \frac{{{p_n}}}{T},} \end{equation}

where $R$ is the radius of curvature.

The static shape of the FFMB is symmetrical to the $z$ axis. Therefore, for the highest $(0,z_h)$ and lowest points $(0,-z_b)$, the membrane needs to satisfy the following boundary conditions:

(2.5)\begin{equation} \left. \begin{array}{@{}ll@{}} \theta (S) = 0, & S = 0, \\ \theta (S) = {\rm \pi}, & S = {L_m}/2. \end{array} \right\} \end{equation}

Here $L_m$ is the total length of the membrane.

Applying Taylor expansion to the membrane angle $\theta (S)$ and substituting (2.4) into it, we can obtain

(2.6)\begin{equation} \theta ({S_j} + \Delta S) = \theta ({S_j}) + \frac{{{\rm d} \theta ({S_j})}}{{{\rm d} S}}\Delta S + \frac{{{{\rm d}^2}\theta ({S_j})}}{{{\rm d} {S^2}}}{(\Delta S)^2} + \cdots\simeq \theta ({S_j}) + \frac{{{p_n}}}{T}\Delta S. \end{equation}

According to (2.5) and (2.6), when the internal membrane pressure $p_{i0}$ and the highest point $(0,z_h)$ of the FFMB are known, the curve of the membrane boundary can be obtained by arbitrarily assuming the membrane tension $T$ and membrane difference length $\Delta S$. However, because the tension is unique when the membrane is in equilibrium, the curve will not satisfy (2.5) when the assumed tension $T$ does not match the actual tension. Assuming that the actual tension is $T_S$, (2.5) is adopted as the discriminant condition, and the tension $T$ is iterated continuously until $T = T_S$, then the static shape of the membrane can be obtained.

Figure 3 shows the static shape of the FFMB with different filling ratio $\gamma$ (2.1) and relative density $\rho _I/\rho _O$. It is observed that the shape strongly depends on the filling ratio $\gamma$. With the decrease of $\gamma$, the FFMB becomes more and more flat. The relative density mainly affects the relative position of the FFMB on the water surface, and has little influence on its static shape.

Figure 3. Static shape of the FFMB, $R_C=0.2$ m. (a) Effects of filling ratio $\gamma$, $\rho _I/\rho _O = 0.8$ and (b) effects of relative density $\rho _I/\rho _O$, $\gamma = 0.9$.

2.2. Dynamic boundary conditions of the FFMB

Similar to the equation under the condition of still water, the motion equation of the FFMB under dynamic action can be written as

(2.7)\begin{equation} \left. \begin{array}{@{}c@{}} {P'\, {\rm d} S - 2 T'\sin ({\rm d} \theta '/2) = {\rm d} m{a_n}}, \\ {\dfrac{{\partial T'}}{{\partial S}} = \rho_s \Delta{a_t}}, \end{array} \right\} \end{equation}

where $a_n$ is the normal acceleration and $a_t$ is the tangential acceleration; $P' = {P_S} + {P_D}$, $T' = {T_S} + {T_D}$, $\theta ' = {\theta _S} + {\theta _D}$; $P_S$, $T_S$ and $\theta _S$ are the pressure, membrane tension and membrane angle in still water, respectively, which can be obtained from § 2.1. Here $P_D$, $T_D$ and $\theta _D$ are the dynamic pressure, membrane tension and membrane angle caused by the external load; $\textrm {d} m = \rho _s \Delta \,\textrm {d} S$ is the mass of the membrane.

Because the mass force of the membrane is much smaller than the additional mass force generated by the fluid motion, and the mass of the membrane is small enough to be ignored, the tangential equation of the FFMB can be written as (2.8)

(2.8)\begin{equation} {\frac{{\partial T'}}{{\partial S}} = \frac{{\partial ({T_S} + {T_D})}}{{\partial S}} \approx 0 \Rightarrow \frac{{\partial {T_D}}}{{\partial S}} = 0}. \end{equation}

Simplifying (2.7) and ignoring the second-order small quantity, the dynamic boundary condition of the FFMB can be expressed as:

(2.9)\begin{equation} {{P_D} - {T_D}\frac{{{\rm d} {\theta_S}}}{{{\rm d} S}} - {T_S}\frac{{{\rm d} {\theta_D}}}{{{\rm d} S}} = {\rho _s}\Delta a_n}. \end{equation}

As $T_D$, $\textrm {d} {\theta _D}/ \textrm {d} S$ and $a_n$ are all unknowns, it is difficult to solve (2.9) directly and generally it can only be solved by finite element method (Phadke & Cheung Reference Phadke and Cheung1999, Reference Phadke and Cheung2001). Considering that the membrane exhibits simple harmonic motion under the action of waves, its normal displacement can be expressed as $w\,{\textrm {e}^{-\textrm {i}\sigma t}}$, where $w$ is the amplitude of the membrane element in the external normal direction, $\sigma$ is the circular frequency, $\textrm {i} = \sqrt { - 1}$ and $t$ is the time. The corresponding normal acceleration $a_n$ can be expressed as

(2.10)\begin{equation} {a_n ={-}\sigma^2w\,{{\rm e}^{-{\rm i}\sigma t}}}. \end{equation}

For convenience of description, the time factor $\textrm {e}^{-\textrm {i} \sigma t}$ is omitted hereinafter. According to (2.8), the tangential displacement of the membrane can be ignored. Under the linear theory, the effects of displacement in different directions can be linearly superimposed, and figure 4 shows the curvature change ($\textrm {d}{\theta _D}/ \textrm {d} S$) caused by the displacement. It can be seen that the blue line represents the shape of the membrane after moving, the red line represents the initial shape, the black dotted line represents the curve after the red line moves along its normal vector by a distance of $w_j$, while $w_j$ is positive in the direction pointing to the outer normal. It can be seen that $\textrm {d} {\theta _D}/ \textrm {d} S$ consists of two parts, one is the curvature change of the whole circumferential membrane ($\textrm {d} {\theta _{D1}}/ \textrm {d} S$) caused by the normal motion of the membrane, the other is the angle change ($\textrm {d} {\theta _{D2}}/ \textrm {d} S$) caused by the relative displacement of the membrane. Therefore, $\textrm {d} {\theta _D}/ \textrm {d} S$ can be expressed as

(2.11)\begin{equation} {\frac{{{\rm d} {\theta_D}}}{{{\rm d} S}} = \frac{{{\rm d} {\theta_{D1}}}}{{{\rm d} S}} + \frac{{{\rm d} {\theta _{D2}}}}{{{\rm d} S}} ={-} \frac{w}{{{R^2}}} - \frac{{{{\rm d}^2}w}}{{{\rm d} {S^2}}}}. \end{equation}

Figure 4. Motion diagram of the membrane element.

For a two-dimensional plane strain problem in cylindrical polar coordinates, the linear strain–displacement and stress–strain relation are given by

(2.12)$$\begin{gather} {\varepsilon ' = {\varepsilon_D} + {\varepsilon _S} = \frac{1}{R}\left(\frac{{{\rm d} u}}{{{\rm d} \theta }} + w\right) + {\varepsilon_S},} \end{gather}$$

(2.13)$$\begin{gather}{{\varepsilon '} = \frac{1}{E}({\sigma_\theta } - \nu {\sigma_R }),} \end{gather}$$

where $\varepsilon '$ is the total hoop strain of the membrane, $\varepsilon _S$ is the hoop strain in initial state, $\varepsilon _D$ is the hoop strain caused by the dynamic action, $u$ is the tangential displacement of the membrane, $E$ is elastic modulus, $\nu$ is the Poisson's ratio, $\sigma _{\theta }$ represents hoop stress and $\sigma _R$ represents normal stress. For a thin membrane, the normal stress is zero.

Using (2.12) and (2.13), the dynamic tension of the membrane $T_D$ can be expressed as

(2.14)\begin{equation} {{T_D} = E{\varepsilon_D}\varDelta = E\Delta \frac{1}{R}\left(\frac{{{\rm d} u}}{{{\rm d} \theta }} + w\right) = E\Delta \left(\frac{{{\rm d} u}}{{{\rm d} S}} + \frac{w}{R}\right)}. \end{equation}

To eliminate the influence of tangential displacement strain, integrate (2.14) and the dynamic tension of the membrane can be written as

(2.15) \begin{align} {\int_0^{{L_m}} {{T_D}} \, {\rm d} S = \int_0^{{L_m}} {E\Delta \left(\frac{{{\rm d} u}}{{{\rm d} S}} + \frac{w}{R}\right)} {\rm d} S \Rightarrow {T_D}{L_m} = E\Delta ({u_{{L_m}}} - {u_0}) + E\Delta \int_0^{{L_m}} {\frac{w}{R}} \, {\rm d} S}. \end{align}

Because the membrane is annular, the starting point and the ending point are the same point. The tangential displacement of the starting point $u_0$ and the ending point $u_{L_0}$ satisfies the equation $u_{L_m} = u_0$. The dynamic tension of the membrane can be written as

(2.16)\begin{equation} {{T_D}{L_m} = E\Delta \int_0^{{L_m}} {\frac{w}{R}} \, {\rm d} S}. \end{equation}

In summary, the dynamic boundary condition of the FFMB can be expressed as (2.17), the dynamic tension of the membrane $T_D$ can be written as (2.18) and the dynamic pressure $P_D$ can be written as (2.19). Since the gravity of the membrane is much smaller than the pressure inside and outside the membrane and the membrane tension, the gravity of the membrane can be ignored. When the structure moves, the inertial force of the membrane is taken into account without loss of generality. In fact, ignoring the inertial force of the structure is only a special case of (2.17), when the membrane density $\rho _s$ is assumed to be zero, the inertial force is zero.

(2.17)$$\begin{gather} {P_D} - {T_D}\frac{{{\rm d} {\theta _S}}}{{{\rm d} S}} + {T_S}\left( {\frac{w}{{{R^2}}} + \frac{{{{\rm d}^2}w}}{{{\rm d} {S^2}}}} \right) = \rho_s \Delta \frac{{{{\rm d}^2}w}}{{{\rm d} {t^2}}}, \end{gather}$$

(2.18)$$\begin{gather}{T_D} = \frac{{E\Delta }}{{{L_m}}}\int_0^{{L_m}} {\frac{w}{R}} \, {\rm d} S, \end{gather}$$

(2.19)$$\begin{gather}{P_D} = \left\{ \begin{array}{@{}ll@{}} {\rm i}\sigma {\rho_I}{\phi_I} - {\rho_I}gw\cos {\theta_S}, & z \ge 0, \\ {\rm i}\sigma ({\rho _I}{\phi_I} - {\rho_O}{\phi_O}) + ({\rho_O} - {\rho_I})gw\cos {\theta_S}, & z < 0, \end{array} \right. \end{gather}$$

where $\theta _S$ is the angle between the membrane element and the $x$ axis in the static equilibrium state, $g$ is the gravitational acceleration, and $\phi _I$ and $\phi _O$ are the fluid velocity potentials inside and outside the membrane, respectively. For the problems of hydrodynamics, the normal displacement $w$ can also be expressed by the velocity of the object surface ${{{\partial {\phi _I}}}/{{\partial \boldsymbol {n}}}}$

(2.20)\begin{equation} {w = \frac{{\rm i}}{\sigma }\frac{{\partial {\phi_I}}}{{\partial {\boldsymbol{n}}}}}. \end{equation}

Here ${\boldsymbol {n}}$ is the normal vector.

Using (2.17), combined with (2.18)–(2.20) and the corresponding boundary conditions of the flow field, the solution of the membrane motion and flow field can be realized. Note that the motion equation of the linear membrane can be written as (Kim & Kee Reference Kim and Kee1996; Karmakar & Sahoo Reference Karmakar and Sahoo2008; Sahoo Reference Sahoo2012; Koley et al. Reference Koley, Behera and Sahoo2015, Reference Koley, Vijay, Nishad and Sundaravadivelu2022)

(2.21)\begin{equation} {{P_D} + {T_S}\frac{{{{\rm d}^2}w}}{{{\rm d} {S^2}}} = \rho_s \Delta\frac{{{{\rm d}^2}w}}{{{\rm d} {t^2}}}.} \end{equation}

Because the radius of curvature of a straight line is infinite, (2.21) is merely a particular instance of (2.17), which further verifies the correctness of the motion equation deduced in this paper. Besides, it shows that the motion equation has universal significance and can replace the existing linear membrane motion equation.

2.3. Governing equations and boundary conditions

As shown in figure 1, assuming that $A_{i}$ is the amplitude of the incident wave, $H_{i}$ is the height of the incident wave, $T_{i}$ is the period of the incident wave, $h$ is the water depth, $H_{r}$ and $H_t$ are the heights of the reflected wave and transmitted wave, respectively. In addition, $\varGamma _{L}$ set at $x=-l$ is the boundary of the wave inlet and $\varGamma _{R}$ set at $x= r$ is the boundary of the wave outlet; $\varGamma _{B}$ is the bottom boundary, $\varGamma _{M_1}$ is the membrane boundary above the external fluid and $\varGamma _{M_2}$ is the membrane boundary below the external fluid.

Considering that the problem satisfying linear wave theory and the velocity potential can be expressed as $\varPhi (x,z,t) = {\textrm {Re}} [\phi (x,z){\, \textrm {e}^{ -\textrm {i}\sigma t}}]$, where the symbol $\textrm {Re}[]$ denotes the real part of a complex number. For instance, if a complex number is defined as $x=A+Bi$, the real part of $x$ is $A (\textrm {Re}[x]=A)$. Here $\phi (x,z)$ represents the complex amplitude of the velocity potential. The total fluid domain is divided into four subregions ($\varOmega _{1}, \varOmega _{2}, \varOmega _{3}, \varOmega _{4}$) and the velocity potential in each subregion is assumed to be $\phi _{1}, \phi _{2}, \phi _{3}, \phi _{4}$, where $\varOmega _{1}$ and $\varOmega _{3}$ are the outer region for the wave inlet and outlet, $\varOmega _{2}$ is the inner region outside the FFMB and $\varOmega _{4}$ is the inner region for the FFMB. Under the linear wave theory, velocity potential $\phi$ should satisfy the following governing equation and boundary conditions

(2.22)\begin{equation} {{\nabla^2}{\phi_s}(x,z) = 0\quad {\rm for}\ s = 1,2,3,4.} \end{equation}

The linearized free surface boundary condition is given by

(2.23)\begin{equation} {\frac{{\partial {\phi_s}}}{{\partial {z}}} = K{\phi_s}\quad \text{for }s = 1,2,3,\ {\rm on}\ z = 0,} \end{equation}

where $K = {\sigma ^2}/g$, $g$ is the gravitational acceleration and the impermeable bottom boundary condition can be expressed as

(2.24)\begin{equation} {\frac{{\partial {\phi_s}}}{{\partial {z}}} = 0 \quad \text{for }s = 1,2,3,\ {\rm on}\ z ={-} h}.\end{equation}

For the membrane boundary, the dynamic boundary condition of the FFMB can be expressed as (2.25), and the specific derivation can be found in § 2.2

(2.25)$$\begin{gather} {P_D} - {T_D}\frac{{{\rm d} {\theta_S}}}{{{\rm d} S}} + {T_S}\left( {\frac{w}{{{R^2}}} + \frac{{{{\rm d}^2}w}}{{{\rm d} {S^2}}}} \right) = \rho_s \Delta \frac{{{{\rm d}^2}w}}{{{\rm d} {t^2}}},\quad {\rm on}\ {\varGamma_{M1}}\ {\rm and}\ {\varGamma_{M2}}, \end{gather}$$

(2.26)$$\begin{gather}{{T_D} = \frac{{E\Delta }}{{{L_m}}}\int_0^{{L_m}} {\frac{w}{R}} \, {\rm d} S,} \end{gather}$$

(2.27)$$\begin{gather}{P_D} = \left\{ \begin{array}{@{}ll@{}} {\rm i}\sigma {\rho_I}{\phi_4} - {\rho_I}gw\cos {\theta_S}, & {\rm on}\ {\varGamma_{M1}}, \\ {\rm i}\sigma ({\rho_I}{\phi_4} - {\rho_O}{\phi_2}) + ({\rho_O} - {\rho_I})gw\cos {\theta_S}, & {\rm on}\ {\varGamma_{M2}}, \end{array} \right. \end{gather}$$

(2.28)$$\begin{gather}w = \frac{\rm i}{\sigma}\frac{{\partial {\phi _4}}}{{\partial {\boldsymbol{n}}}}, \end{gather}$$

where $\boldsymbol {n}$ represents the normal vector pointing out from the corresponding fluid domain enclosed by the boundaries. Since there is no separation between the fluid and the membrane boundary, at the boundary $\varGamma _{M2}$, it is necessary to meet the following equation:

(2.29)\begin{equation} {\frac{{\partial {\phi_2}}}{{\partial {\boldsymbol{n}}}} ={-} \frac{{\partial {\phi _4}}}{{\partial {\boldsymbol{n}}}}\quad {\rm on}\ {\varGamma _{M2}}}. \end{equation}

The radiation conditions in the far fields can be expressed as

(2.30)\begin{equation} \left. \begin{array}{@{}ll@{}} \dfrac{{\partial ({\phi _1}-{\phi_0})}}{{\partial x}} + {\rm i}{k_0}({\phi _1}-{\phi_0}) = 0, & {\rm as}\ x \to - \infty, \\ \dfrac{{\partial {\phi_3}}}{{\partial x}} - {\rm i}{k_0}{\phi_3} = 0, & {\rm as}\ x \to + \infty , \end{array} \right\} \end{equation}

where ${\phi _0} = {\textrm {e}^{\textrm {i}{k_0}(x + l)}}{f_0}({k_0},z)$ is the incident wave, $k_0$ is the wavenumber and ${f_0}({k_0},z)$ is expressed as (2.35).

The continuity of pressure and velocity on $\varGamma _L$ and $\varGamma _R$ are written as

(2.31)\begin{equation} {\phi_2} = \left\{ \begin{array}{@{}ll@{}} {\phi_1} & {\rm on}\ {\varGamma_L}, \\ {\phi_3} & {\rm on}\ {\varGamma_R}, \end{array} \right.\quad \frac{{\partial {\phi _2}}}{{\partial {\boldsymbol{n}}}} = \left\{ \begin{array}{@{}ll@{}} - \dfrac{{\partial {\phi_1}}}{{\partial {\boldsymbol{n}}}} ={-} \dfrac{{\partial {\phi_1}}}{{\partial x}} & {\rm on}\ {\varGamma_L}, \\ - \dfrac{{\partial {\phi_3}}}{{\partial {\boldsymbol{n}}}} = \dfrac{{\partial {\phi _3}}}{{\partial x}} & {\rm on}\ {\varGamma_R}. \end{array} \right. \end{equation}

2.4. Solution of the EEBEM

Since there is a second-order differential term in (2.25), it is difficult to solve it by a theoretical method. The EEBEM (Lv et al. Reference Lv, Zhao and Li2022) is a numerical method combining the eigenfunction expansion method and the multi-domain boundary element method, which can solve the problem of wave-structure interaction. The complex potentials $\phi _1$ and $\phi _3$ in the outer region can be obtained by using the eigenfunction expansion method, and the boundary value problem in the inner region can be transformed into an integral equation by using the multi-domain boundary element method. To reduce the computational cost and improve the accuracy, $\varGamma _L$ and $\varGamma _R$ are situated $h/5$ away from the FFMB, and the influence of evanescent wave modes is considered.

The complex potential $\phi _1$ and $\phi _3$ in the outer region can be written as

(2.32)$$\begin{gather} {\phi_1} = {{\rm e}^{{\rm i}{k_0}(x + l)}}{f_0}({k_0},z) + \sum_{m = 0}^\infty {{D_m}{\, {\rm e}^{ -{\rm i}{k_m}(x + l)}}{f_m}({k_m},z)} , \end{gather}$$

(2.33)$$\begin{gather}{\phi_3} = \sum_{m = 0}^\infty {{T_m}\,{{\rm e}^{{\rm i}{k_m}(x - r)}}{f_m}({k_m},z)} , \end{gather}$$

where $D_m$ and $T_m (m=0, 1, 2,\ldots )$ are the undetermined coefficients, $m$ is the velocity potential expansion terms, $k_m (m=0, 1, 2,\ldots )$ satisfies

(2.34)\begin{equation} {\sigma^2} = g{k_m}\tanh {k_m}{h}, \quad m = 0,1,2\ldots,\end{equation}

where $k_0$ is a real number and $k_m (m = 1, 2,\ldots )$ are imaginary numbers. It may be noted that $k_m (m = 1, 2,\ldots )$ are in the form of $k_m = k'_{mi}$, where $k'_m$ are the real positive roots of $\sigma ^2 = -g{k'_m}\tan {k'_m}{h}$.

The eigenfunctions $f(k_0,z)$ and $f(k_m,z)$ may be expressed as

(2.35)\begin{equation} {f}({k_m},z) ={-} \frac{{{\rm i}gA_i}}{\sigma} \frac{{\cosh {k_m}(z + {h})}}{{\cosh {k_m}{h}}}, \quad m = 0,1,2\ldots . \end{equation}

Besides, $\langle {\cdot } , {\cdot } \rangle$ is denoted the inner product. According to orthogonality, the inner product of $f(k_m,z)$ and $f(k_n,z)$ can be expressed as

(2.36)\begin{equation} \langle\, {f({k_m},z),f({k_n},z)}\rangle = \int_{{-}h}^0 f({k_m},z)f({k_n},z)\,{\rm d} z = {S_n}{\delta_{mn}}, \end{equation}

where ${S_n} = - {( {{{gA_i}}/{\sigma }})^2} [ {({{\sinh ( {2{k_n}{h}} ) + 2{k_n}{h}}})/{{4 {{k_n}}{{\cosh }^2}({k_n}{h})}}}]$ and $\delta$ is the Kronecker function.

According to (2.32) and (2.36), the unknown coefficient $D_m$ can be expressed as

(2.37)\begin{equation} {D_m} + {\delta_{m0}} = \frac{1}{{{S_m}}}\langle {\widehat{{\phi _2}},{f_m}({k_m},z)} \rangle , \end{equation}

where $\widehat {{\phi _2}}$ represents the velocity potential when the expansion term $m$ is determined.

According to (2.32), the normal derivative of $\phi _1$ can be written as

(2.38)\begin{equation} \frac{{\partial {\phi_1}}}{{\partial \boldsymbol{n}}} = {\rm i}{k_0}\,{{\rm e}^{{\rm i}{k_0}(x + l)}}{f_0}({k_0},z) - \sum_{m = 0}^\infty {D_m}i{k_m}{\, {\rm e}^{ -{\rm i}{k_m}(x + l)}}{f_m}({k_m},z). \end{equation}

By substituting (2.37) and (2.38) into (2.31), the boundary condition on $\varGamma _{L}$ can be written as

(2.39)\begin{equation} \left. {\frac{{\partial {\phi_2}}}{{\partial \boldsymbol{n}}}} \right|_{x ={-} l} = \sum_{m = 0}^M \frac{{{\rm i}{k_m}{f_m}({k_m},z)}}{{{S_m}}}\langle {\widehat{{\phi_2}},{f_m}({k_m},z)} \rangle - 2\,{\rm i}{k_0}\,{f_0}({k_0},z)\quad {\rm on}\ {\varGamma_L}. \end{equation}

Similarly, the boundary condition on $\varGamma _{R}$ can be expressed as

(2.40)$$\begin{gather} {T_m} = \frac{1}{{{S_m}}}\langle {\widehat {{\phi_2}},{f_m}({k_m},z)} \rangle, \end{gather}$$

(2.41)$$\begin{gather}{{\left. {\frac{{\partial {\phi_2}}}{{\partial \boldsymbol{n}}}} \right|_{x = r}} = \sum_{m = 0}^M {\frac{{{\rm i}{k_m}{\,f_m}({k_m},z)}}{{{S_m}}}\langle {\widehat {{\phi_2}},{f_m}({k_m},z)} \rangle }\quad {\rm on}\ {\varGamma_R}}. \end{gather}$$

As can be seen from the (2.25), (2.28) and (2.29), all the boundary conditions of the membrane ($\varGamma _{M1}$ and $\varGamma _{M2}$) are the functions of $\phi$ and ${{{\partial {\phi }}}/{{\partial \boldsymbol {n}}}}$. To calculate $\phi$ and ${{{\partial {\phi }}}/{{\partial \boldsymbol {n}}}}$, $\varGamma _{M1}$ and $\varGamma _{M2}$ are divided into $J_1$ and $J_2$ finite elements, respectively. Because the differential equation involves the second derivative of normal displacement $w$, it needs to be solved by central difference. As shown in figure 4, the motion equation (2.25) of any element $\varGamma {(j)}$ on the membrane can be discretized as

(2.42)\begin{gather} \begin{aligned} & {T_S}\frac{{\rm i}}{\sigma}\left[ \begin{array}{l} \dfrac{{\partial {\phi_{j + 1}}}}{{\partial \boldsymbol{n}}}\dfrac{2}{{{\rm d} {S_j}({\rm d} {S_{j + 1}} + {\rm d} {S_j})}} + \dfrac{{\partial {\phi_{j - 1}}}}{{\partial \boldsymbol{n}}}\dfrac{2}{{{\rm d} {S_j}({\rm d} {S_j} + {\rm d} {S_{j - 1}})}} \\ +\dfrac{{\partial {\phi_j}}}{{\partial \boldsymbol{n}}}\left( {\dfrac{1}{{R_j^2}} - \dfrac{{2({\rm d} {S_{j + 1}} + 2\, {\rm d} {S_j} + {\rm d} {S_{j - 1}})}}{{{\rm d} {S_j}({\rm d} {S_{j + 1}} + {\rm d} {S_j})({\rm d} {S_j} + {\rm d} {S_{j - 1}})}}} \right) \end{array} \right] \\ & \quad +{\rm i}\sigma {\rho_I}{\phi_j} + \left( {{\rm i}\sigma {\rho_s}\Delta - \dfrac{{{\rm i}g\cos {\theta _{Sj}}{\rho_I}}}{\sigma}} \right)\dfrac{{\partial {\phi_j}}}{{\partial \boldsymbol{n}}} - \dfrac{T_D}{R_j} = 0 \end{aligned} \quad {\rm on} {\varGamma_{M_1}}.\end{gather}

(2.43)\begin{gather} \begin{aligned} & {T_S}\frac{\rm i}{\sigma} \left[ \begin{array}{l} \dfrac{{\partial {\phi_{j + 1}}}}{{\partial \boldsymbol{n}}} \dfrac{2}{{{\rm d} {S_j}({\rm d} {S_{j + 1}} + {\rm d} {S_j})}} + \dfrac{{\partial {\phi_{j - 1}}}}{{\partial \boldsymbol{n}}} \dfrac{2}{{{\rm d} {S_j}({\rm d} {S_j} + {\rm d} {S_{j - 1}})}}\\ + \dfrac{{\partial {\phi_j}}}{{\partial \boldsymbol{n}}} \left( {\dfrac{1}{{R_j^2}} - \dfrac{{2({\rm d} {S_{j + 1}} + 2\, {\rm d} {S_j} + {\rm d} {S_{j - 1}})}}{{{\rm d} {S_j} ({\rm d} {S_{j + 1}} + {\rm d} {S_j})({\rm d} {S_j} + {\rm d} {S_{j - 1}})}}} \right) \end{array} \right]\\ & \quad + {\rm i}\sigma ({\rho_I}{\phi_j} - {\rho_O}{\phi_{Oj}}) + \left( {{\rm i}\sigma {\rho_s}\Delta + \frac{{{\rm i}g\cos {\theta_{Sj}}({\rho_O} - {\rho_I})}}{\sigma }} \right)\frac{{\partial {\phi_j}}}{{\partial \boldsymbol{n}}} - \frac{T_D}{R_j} = 0 \end{aligned} \quad {\rm on}\ {\varGamma_{M_2}}, \end{gather}

(2.44)\begin{gather} {T_D} = \frac{{E\Delta }}{{{L_m}}}\sum_{q = 1}^{{J_1} + {J_2}} {\frac{{{w_q}}}{{{R_q}}}} {\rm d} {S_q} = \frac{{{\rm i}E\Delta }}{{\sigma {L_m}}}\sum_{q = 1}^{{J_1} + {J_2}} \frac{1}{{{R_q}}}\frac{{\partial {\phi_q}}}{{\partial \boldsymbol{n}}} \,{\rm d} {S_q}, \end{gather}

where $\textrm {d} S_j$ represents the length of $\varGamma {(j)}$, $R_j$ represents the radius of curvature of $\varGamma {(j)}$, $w_j$ represents the normal displacement outside $\varGamma {(j)}$ and $\theta _{Sj}$ represents the deflection angle when the element is static; $\phi _j$ and ${{{\partial {\phi _j}}}/{{\partial \boldsymbol {n}}}}$ represent the velocity potential and its normal derivative in the subdomain $\varOmega _{4}$; $\phi _{Oj}$ represents the velocity potential in the subdomain $\varOmega _{2}$.

Through (2.42) and (2.43), a total of $(J_1+J_2)$ equations can be established, and $J_2$ equations can be established from (2.29). Other boundaries $\varGamma _{L}$, $\varGamma _{B}$, $\varGamma _{R}$, $\varGamma _{F1}$ and $\varGamma _{F2}$ are divided into $J_3$ elements, and according to the corresponding boundary conditions ((2.23), (2.24), (2.39) and (2.41)), $J_3$ equations can be established. For more details of computation length and mesh discretization, refer to § 2.5.1. Then, according to the above boundary conditions, the $(J_1+2J_2+J_3)$ equation system can be established. Besides, for inner region $\varOmega _{2}$ and $\varOmega _{4}$, ($J_1+2J_2+J_3)$ boundary integral equations can be established by Green's second theorem. Finally, the value of $\phi$ and ${{{\partial {\phi }}}/{{\partial \boldsymbol {n}}}}$ at the boundary can be obtained through the $2(J_1+2J_2+J_3)$ equation system.

The boundary integral equations of $\varOmega _{2}$ and $\varOmega _{4}$ are expressed as

(2.45)$$\begin{gather} \lambda (\xi ,\eta )\phi (\xi ,\eta ) = \int_\varGamma \left[ {\phi (x,z)\frac{{\partial G (x,z;\xi ,\eta )}}{{\partial \boldsymbol{n}}} - G (x,z;\xi ,\eta )\frac{{\partial \phi (x,z)}}{{\partial \boldsymbol{n}}}} \right] {\rm d} \varGamma (x,z), \end{gather}$$

(2.46)$$\begin{gather}\lambda (\xi ,\eta ) = \left\{ \begin{array}{@{}ll@{}} 0.5 & {\rm if}\ (\xi ,\eta ) \text{ on the }{\varGamma}, \\ 1 & {\rm if}\ (\xi ,\eta ) \in \varOmega, \text{but not on the }\varGamma, \end{array} \right. \end{gather}$$

where ($\varepsilon, \eta$) is the source point; $\varGamma$ represents the boundary of $\varOmega _{2}$ or $\varOmega _{4}; G(x,z;\xi,\eta )$ is the Green's function and it is written as

(2.47)\begin{equation} G(x,z;\xi ,\eta ) = \frac{{\ln (r)}}{{2{\rm \pi} }},\quad r = \sqrt{{{(x - \xi )}^2} + {{(z - \eta )}^2}} \quad \text{for }(x,z) \ne (\xi ,\eta ). \end{equation}

Actually, the boundary integral equations of $\varOmega _{2}$ and $\varOmega _{4}$ are independent, and they are coupled together by (2.25) and (2.29). Then, through (2.23)–(2.25), (2.29), (2.39), 2.41 and 2.45, $\phi$ and ${{{\partial {\phi }}}/{{\partial \boldsymbol {n}}}}$ at the boundary of $\varOmega _{2}$ and $\varOmega _{4}$ can be obtained. The wave transmission coefficient $K_t$ and reflection coefficient $K_r$ can be expressed as

(2.48)$$\begin{gather} {{K_r} = | {{D_0}} | = \left| {\frac{1}{{{S_0}}}\langle {\widehat {{\phi_2}},{f_0}({k_0},z)} \rangle - 1} \right| = \left| {\frac{1}{{{S_0}}}\int_{ - {h}}^0 {\widehat {{\phi_2}},{f_0}({k_0},z)}\, {\rm d} z - 1}\right|\quad {\rm on}\ {\varGamma_L},} \end{gather}$$

(2.49)$$\begin{gather}{{K_t} = | {{T_0}} | = \left| {\frac{1}{{{S_0}}}\langle {\widehat {{\phi_2}},{f_0}({k_0},z)} \rangle} \right| = \left| {\frac{1}{{{S_0}}}\int_{ - {h}}^0 {\widehat {{\phi_2}},{f_0}({k_0},z)} \, {\rm d} z} \right|\quad {\rm on}\ {\varGamma_R}}. \end{gather}$$

Also, the wave energy dissipation coefficient $K_d$ is defined as

(2.50)\begin{equation} {{K_d} =1 - K_t^2 - K_r^2.} \end{equation}

Wave force $\boldsymbol {F}$ on the FFBM can be written by integrating the pressure on the membrane surface $\varGamma _{M2}$:

(2.51)\begin{equation} {\boldsymbol{F} = ({F_x},{F_z}) = \int_{{\varGamma_{M2}}} {{\rm i}\sigma \rho_O \phi \boldsymbol{n}} \, {\rm d} \varGamma}. \end{equation}

Here $\boldsymbol {n}=(n_x,n_z)$ is the unit normal vector of the membrane surface, $F_x$ is the horizontal wave force and $F_z$ is the vertical wave force.

2.5. Verifications of the EEBEM

2.5.1. Convergence tests of the EEBEM

The selection of the expansion term $M$ of the velocity potential $\phi$ and the boundary element length $\textrm {d} S$ will affect the calculation accuracy of the EEBEM. Appropriate $M$ and $\textrm {d} S$ must be selected to ensure the accuracy of the model and to reduce the computation time. In the following, the wavenumber $k_0$ of a linear wave is simplified to $k$ and $kh$ represents the dimensionless water depth.

Table 1 shows the convergence tests of the wave transmission coefficient $K_t$ with different expansion terms $M$. It is found that as $M$ increases, the results converge when $M \ge 4$. However, when $M$ is small, there is a large difference between the results with different $M$. For example, when $kh=1.5$, the difference between $M=0$ and $M=4$ is three times, indicating that the evanescent mode has a large influence and cannot be ignored in the calculation. Table 2 shows the convergence test of the transmission coefficient $K_t$ under different boundary element lengths $\textrm {d} S$. It is observed that the result is accurate to three decimal places when $\textrm {d} S \le 0.01$ m. Since the increase in computational cost with increasing $M$ is small, $\textrm {d} S = 0.01$ m and $M = 10$ are chosen in the following calculations to improve the accuracy of the numerical results.

Table 1. Convergence tests of $K_t$ with different expansion terms $M$. Here $h = 0.7$ m, $R_C = 0.2$ m, ${\rho _I = 800\ \textrm {kg}\ \textrm {m}^{-3}}$, $\rho _O = 1000\ \textrm {kg}\ \textrm {m}^{-3}$, $\rho _s = 1000\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.98$, $E=10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m, ${\textrm {d} S=0.01}$ m.

Table 2. Convergence tests of $K_t$ with different boundary element length $\textrm {d} S$ (m). Here $h = 0.7$ m, $R_C = 0.2$ m, $\rho _I = 800\ \textrm {kg}\ \textrm {m}^{-3}$, $\rho _O = 1000\ \textrm {kg}\ \textrm {m}^{-3}$, $\rho _s =1000\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.98$, $E=10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m, $M = 5$.

2.5.2. Verifications of the EEBEM

The motion equation considering the dynamic tension of the membrane and the explicit integral expression of the dynamic tension $T_D$ (2.25) and (2.26) are the major contributions of this paper. To verify the accuracy and reliability of the model in solving wave–membrane interactions, the model is applied below to solve wave interactions with floating and submerged fluid-filled membrane structures, respectively, and compared with the corresponding numerical and experimental results.

Zhao & Aarsnes (Reference Zhao and Aarsnes1998) measured the membrane tension of a floating liquid-filled membrane structure under a regular wave by the model experiments and compared it with the numerical results. Figure 5 represents the corresponding experimental set-up for measuring the membrane tension in the wave flume. The model was made of an elastic coated fabric and the fluid inside the membrane was a mixture of fresh water and alcohol. The stern and bow sections are equal, i.e. the model is doubly symmetric about the $x$ and $y$ axes. The filling ratio $\gamma$ of the model is 0.9; the density of the liquid inside the membrane is 914 kg m$^{-3}$; the volume of the liquid inside the membrane is 0.935 m$^3$; the length over the whole model is 6 m; the length of the parallel part of the model is 4.44 m; and the circumferential length of the parallel part of the model is 1.566 m. The dynamic tension $T_D$ was obtained by measuring the stretch of the membrane and was located at $x=2.1$ m. The sampling frequency for these recordings was 50 Hz.

Figure 5. Experimental set-up for measuring the tension of a floating liquid-filled membrane structure in the wave flume (Zhao & Aarsnes Reference Zhao and Aarsnes1998). (a) Experimental set-up of the model. (b) Horizontal section of the model and (c) longitudinal section of the model.

Figure 6 shows the comparisons of $T_D$ between the EEBEM results, the experiments and the numerical results of Zhao (Reference Zhao1995), Zhao & Aarsnes (Reference Zhao and Aarsnes1998). In their numerical results, the governing equations of the membrane are based on the equilibrium of the circumferential tension and pressure, and the dynamic response of the membrane is obtained by high-order polynomial fitting due to the unknown dynamic tension and curvature variation of the membrane. It can be seen that the EEBEM results are in good agreement with those obtained by Zhao (Reference Zhao1995), Zhao & Aarsnes (Reference Zhao and Aarsnes1998), indicating that the explicit integral expression of the dynamic tension derived in this paper is reliable.

Figure 6. Comparisons of dynamic tension $T_D$ between the EEBEM results, the experiments (Zhao & Aarsnes Reference Zhao and Aarsnes1998) and the numerical results (Zhao Reference Zhao1995). Results are shown for (a) $\rho _I /\rho _O = 0.7$, $R_C/h=0.2$, $E=+\infty \ \textrm {N}\ \textrm {m}^{-2}$; (b) $\rho _I /\rho _O = 0.9$, $\gamma = 0.9$, $R_C/h=0.133$, $E = + \infty \ \textrm {N}\ \textrm {m}^{-2}$.

Ohyama et al. (Reference Ohyama, Tanaka, Kiyokawa, Uda and Murai1989) investigated the hydrodynamic performance of a SFMB by model experiments and compared it with the numerical results. Figure 7 shows a schematic diagram of a wave interaction with an SFMB. In their experiments, the model was made of a rubber membrane with $E=58\ \textrm {kN}\ \textrm {m}^{-2}$, membrane density $\rho _S=1.26\ \textrm {ton}\ \textrm {m}^{-3}$, membrane thickness $\varDelta =1.65$ mm and it was filled with water. In addition, the model width $B=$ 1.6 m, water depth $h= 0.8$ m, incident wave height $H_i= 0.04$ m, internal membrane pressure $p_{i0}$ and the structure height $e$ were varied.

Figure 7. Schematic diagram of wave interaction with the FFMB (Ohyama et al. Reference Ohyama, Tanaka, Kiyokawa, Uda and Murai1989).

Figure 8 shows the comparisons of the wave transmission coefficient $K_t$ between the EEBEM results, the experiments and the numerical results (Ohyama et al. Reference Ohyama, Tanaka, Kiyokawa, Uda and Murai1989) under different structure heights and internal membrane pressures. In their numerical results, the wave–membrane interaction is solved by the displacement coordination equation. It can be seen that the present results are close to the experimental results, and the resonant frequency of the structure can be captured more accurately, indicating that the present model is correct and reliable.

Figure 8. Comparisons of wave transmission coefficient $K_t$ between the EEBEM results, the experiments and the numerical results (Ohyama et al. Reference Ohyama, Tanaka, Kiyokawa, Uda and Murai1989). Results are shown for (a) $e/h = 0.537$, $p_{i0}/\rho _Ogh = 0.065$; (b) $e/h = 0.556$, $p_{i0}/\rho _Ogh = 0.208$.

3.1. Set-up of experiments

Physical model experiments are conducted in a large cross-section wave flume at the Port Engineering Hall of Zhejiang University. Figure 9 depicts a schematic diagram of the experimental set-up. The wave flume is 75 m in length, 1.8 m in width and 2.0 m in height, with a maximum test depth of 1.5 m, a test wave period of 0.5 s to 5.0 s and a test wave height of 0.02 m to 0.60 m. The horizontal direction is defined as the $x$ axis, with positive values indicating a direction from the head to the end of the flume. An active absorption push plate wavemaker is installed in front of the flume at the point $x=0$ m, which is capable of generating unidirectional two-dimensional regular waves and absorbing the reflected wave. In addition, a wave absorber is installed at the end of the flume to absorb transmitted and reflected waves.

Figure 9. Schematic diagram of the experiments.

Eight wave gauges, named G1–G8, are located at $x=20$, 25.2, 25.5, 26, 31, 35.5, 41 and 41.5 m. These gauges are employed to collect data on wave height with a sampling frequency of 50 Hz. In accordance with the two-point method proposed by Goda & Suzuki (Reference Goda and Suzuki1976), the wave reflection coefficient is calculated using G1, G2 and G3, while the wave transmission coefficient is determined by G7 and G8.

The FFMB is placed at $x = 34$ m, with a length of 1.75 m, a radius $R_C$ of 0.2 m, a thickness $\varDelta$ of 0.001 m and an elasticity modulus $E$ of $10^7\ \textrm {N}\ \textrm {m}^{-2}$. Since the length of the FFMB is approximately equal to the width of the flume, the wave energy that passes through freely at the end of the FFMB is minimal, and the axial direction of the FFMB is always parallel to the width direction of the flume when subjected to a unidirectional regular wave. Moreover, eight wave gauges are positioned on the central axis of the tank, which serves to minimize the impact of transmitted waves near the flume wall on the data acquisition process. Consequently, the corresponding end effect can be disregarded, as it will not have a significant impact on the FFMB.

Furthermore, a high-speed camera with a sampling frequency of 50 Hz is positioned in front of the FFMB ($x = 34$ m) to observe the motion response of the FFMB at different times. Given that the motion response of the FFMB under the action of a regular wave is periodic, when the interaction between the FFMB and wave achieves a dynamic balance, the motion response of the FFMB under the same phase at different times is the same. Consequently, the subsequent figures illustrating the structural motion response, such as figures 13, 15 and 17, depict the instantaneous membrane cross-section shape in one period.

Figure 10 shows the schematic diagram of the FFMB. It is made of polyvinyl chloride (PVC) and comprises two air bags and a water bag, with the air bag situated within the water bag and located on either side of the water bag. The air bag contains only air and no water, while the water bag contains only water and no air. The water inside the water bag is tightly enclosed by the membrane, with no free surface. Since the maximum volume of the water bag is fixed and the air bag is located inside the water bag, the density change of the FFMB can be realized by adjusting the volume ratio of water to air.

Figure 10. Schematic diagram of FFMB. (a) Schematic diagram of the FFMB. (b) Model of the FFMB (side view). (c) Model of the FFMB (top view).

The method of adjusting the density and filling ratio of the FFMB in the experiment is as follows. Firstly, the water bag is filled with water through the water inlet and the air bag is exhausted through the air inlet, ensuring the filling ratio of the FFMB is 1 and the density is 1000 kg m$^{-3}$. Subsequently, the volume of water that should be contained in the water bag can be calculated according to the expected filling ratio and density. Then, the excess water in the FFMB can be released through the water inlet, and it is ensured that outside air does not enter during operation. Finally, the expected filling ratio and density of the FFMB can be obtained by inflating the air bags on both sides with the inflatable tube.

Since it is difficult to choose a fluid with a low density in practice, the fluid employed in the experiments differs slightly from that used in the numerical model. The fluid employed in the numerical model is uniform, whereas that used in the experiments is non-uniform, comprising water and air. Nevertheless, when the fluid filling ratio $\gamma$ is large (greater than 0.97), the hydrodynamic performance of the two is basically the same because the structural deformation is small.

Table 3 illustrates the conditions of the experiments, with a total of 228 cases. During the experiments, the data of the stable section are selected to calculate the corresponding transmission coefficient and reflection coefficient. The experiments were conducted in triplicate for each case and the mean of the three resulting data sets was calculated as the final result. The experiments are primarily designed to examine the influence of filling ratio and density on the hydrodynamic performance of the FFMB, as well as the dynamic response of the FFMB. Moreover, the influence of wave nonlinearity, which cannot be investigated through the EEBEM, is also included in the experiments.

Table 3. Conditions of experiments.

Note: in all case, the membrane radius $R_C = 0.2$ m, $\rho _O = 1000$ kg m$^{-3}$, $T_i= 0.9{-}3.5$ means it can be selected as 0.9, 0.95, 1.0, 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.6, 1.8, 2.0, 2.4, 3.0, 3.5.

3.2. Comparisons of the EEBEM and experimental results

Figures 11 and 12 show the comparisons of wave transmission coefficient $K_t$ and reflection coefficient $K_r$ between the experimental and EEBEM results, under the conditions of different filling ratio $\gamma$, density $\rho _I$ and water depth $h$. The point where $K_t=0$ is defined as the first-order resonant point, and the corresponding wave frequency is the first-order resonant frequency.

Figure 11. Comparisons of wave transmission coefficient $K_t$ and reflection coefficient $K_r$ between the EEBEM results and the experiments. Here $\rho _O = 1000\ \textrm {kg}\ \textrm {m}^{-3}$, $R_C=0.2$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m, ${H_i = 0.03}$ m. Results are shown for (a) $\rho _I = 550\ \textrm {kg}\ \textrm {m}^{-3}$, ${\gamma = 0.999}$, $h = 0.7$ m; (b) $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $h = 0.7$ m; (c) $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.988$, $h = 0.7$ m; (d) $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $h = 1.0$ m.

Figure 12. Comparisons of wave transmission coefficient $K_t$ and reflection coefficient $K_r$ between the EEBEM and experimental results at different internal fluid densities. Here $\rho _O = 1000\ \textrm {kg}\ \textrm {m}^{-3}$, $R_C=0.2$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m, $H_i = 0.03$ m. Results are shown for (a) $\rho _I = 645\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, ${h = 0.7}$ m; (b) $\rho _I = 735\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $h = 0.7$ m; (c) $\rho _I = 800\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $h = 0.7$ m; (d) $\rho _I = 880\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $h = 0.7$ m.

It is found that the EEBEM results are in good agreement with the experimental results, indicating that the corresponding motion equation and the dynamic tension expression of the membrane, as well as the numerical model proposed in this paper are accurate and reliable. As the wavelength decreases, the numerical results become increasingly smaller than the experimental results due to the dissipation of energy caused by fluid viscosity and collision. Experimental results indicate that the FFMB exhibits an effective short-wave attenuation performance, with a greater effect observed at lower densities. As the density of the FFMB increases, the first-order resonant frequency shifts to a lower frequency. When the relative density $\rho _I/\rho _O$ is between 0.735 and 0.835, and the relative water depth $kh>1.45$, the FFMB can achieve a satisfactory effect on wave attenuation. Furthermore, with the decrease of the filling ratio $\gamma$, the wave attenuation effect of the FFMB becomes worse, so the FFMB should have a higher filling ratio in practical application.

Figure 13 illustrates the comparison of the motion response of the FFMB between the EEBEM and experimental results when $T_i=1.0$ s (short wave). It is observed that the FFMB is dominated by horizontal swing and rolling, with minimal deformation. Figure 14 shows the wave surface and spectrum curve of the experimental results under the condition of figure 13. Figure 14(a) illustrates that the transmitted wave is obviously lower than the incident wave, indicating that the FFMB can effectively attenuate short waves. Figure 14(b) reveals that the second-order amplitude is relatively small and that the first-order energy at different wave gauges (G1–G5) in front of the FFMB is different.

Figure 13. Comparisons of the motion response of the FFMB between the EEBEM and experimental results when $T_i=1.0$ s (short wave). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, $H_i = 0.03$ m, ${E = 10^7\ \textrm {N}\ \textrm {m}^{-2}}$, $\varDelta = 0.001$ m. Results are shown for (a) $T_i=1.0$ s, EEBEM; (b) $T_i=1.0$ s, Exp.

Figure 14. Experimental wave surface and wave amplitude spectra when $T_i=1.0$ s (short wave). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m. (a) Wave surface and (b) wave amplitude spectra.

The above phenomenon can be attributed to the fact that when the wave period is relatively short, the majority of the wave is reflected by the FFMB, with only a minor portion of energy transferred to the leeside of the FFMB. Then, the superposition of the incident and reflected waves results in the formation of standing waves in front of the FFMB. In particular, when the wave is totally reflected, a complete standing wave is formed in front of the FFMB, while when the wave is partially reflected, the front of the FFMB is the superposition of a standing wave and a travelling wave. Therefore, when the wave period is short (figure 14), the wave in front of the FFMB is predominantly by a standing wave, with varying wave heights at different positions. However, when the wave period is long (figure 18), the wave in front of the FFMB is dominated by the travelling wave, while the reflection wave and the corresponding standing wave is small, and the wave height at different wave gauges is essentially uniform.

For the transmitted wave behind the FFMB, since the wave absorber is effective in absorbing the transmitted wave, the reflection wave from the end of the flume is small. The waveforms of the different wave gauges (G6–G8) are essentially similar, exhibiting the characteristics of travelling waves. In addition, due to the dissipation and conversion of different order waves, with the increase of propagation distance, the first-order wave energy gradually decreases.

Similar to figures 13 and 14, figures 15 and 16 demonstrate the corresponding variations when $T_i=1.4$ s (first-order resonance, see figure 11b).

Figure 15. Comparisons of the motion shape of the FFMB between the EEBEM and the experimental results when $T_i=1.4$ s (first-order resonant mode). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, ${H_i = 0.03}$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m. Results are shown for (a) $T_i=1.4$ s, EEBEM; (b) $T_i=1.4$ s, Exp.

Figure 16. Experimental wave surface and wave amplitude spectra when $T_i=1.4$ s (first-order resonant mode). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m. (a) Wave surface and (b) wave amplitude spectra.

As shown in figure 15, when the FFMB is in resonance, the FFMB is dominated by heaving, with small swaying and almost no rolling. Figure 16(a) demonstrates that the transmitted wave height is significantly lower than the incident wave height, indicating that the FFMB has a good effect on wave attenuation at the resonant frequency. At the same time, the transmitted wave exhibits a sub-peak phenomenon, which signifies that the radiated wave can be superimposed with the scattered wave due to the vibration of the FFMB. As illustrated in figure 16(b), the waveform comprises both first-order and second- and third-order wave components. The reflected wave in front of the FFMB is mainly composed of first-order and second-order waves. As the distance from the FFMB is reduced, the proportion of higher-order waves increases. Moreover, the energy of the transmitted wave in the first order remains relatively constant, whereas the energy of the third order decreases and that of the second order increases with increasing propagation distance. This indicates that the energy of the transmitted wave in the first- and second-order frequencies is more stable, while the higher-order wave energy is more susceptible to dissipation or transformation.

Similar to figures 13 and 14, figures 17 and 18 show the corresponding variations when $T_i=2.4$ s (long wave).

Figure 17. Comparisons of the motion response of the FFMB between the EEBEM and the experimental results when $T_i=2.4$ s (long wave). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, $H_i = 0.03$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m. Results are shown for (a) $T_i=2.4$ s, EEBEM; (b) $T_i=2.4$ s, Exp.

Figure 18. Experimental wave surface and wave amplitude spectra when $T_i=2.4$ s (long wave). Here $\rho _I = 835\ \textrm {kg}\ \textrm {m}^{-3}$, $\gamma = 0.999$, $R_C=0.2$ m, $h = 0.7$ m, $E = 10^7\ \textrm {N}\ \textrm {m}^{-2}$, $\varDelta = 0.001$ m. (a) Wave surface and (b) wave amplitude spectra.

Figure 17 shows the motion response of the FFMB and the trajectory of the peak point, it is found that the motion response of the FFMB is similar to that of the water particle. The roll of the FFMB is negligible and the whole structure moves elliptically around its equilibrium position. As illustrated in figure 18, at the beginning of the interaction between the FFMB and wave, the transmitted wave is basically the same as the incident wave, which shows that FFMB has poor wave attenuation performance under the action of a long wave. Furthermore, as time progresses, the waveforms at different positions are gradually variable. This is due to the fact that the wave absorber is unable to fully absorb the transmitted long wave, resulting in a superimposition of the reflected waves with the transmitted and incident waves. The spectrum provides a more intuitive explanation of the phenomenon described above. It is observed that the energy of different wave gauges is essentially identical, indicating that the FFMB has a minimal shielding effect on long waves. Consequently, the majority of the energy can pass through the structure and is transferred directly to the rear of the structure.