Wave interaction with a floating and submerged elastic plate system

Abstract

Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal mode-coupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems.

Appendix

1.1 Generalized expansion formulae

Under the assumption of the linearized theory of water waves, the spatial velocity potential $\phi (x,y)$ satisfies the Laplace equation (1). On the surface at $y=0$, it is assumed that the velocity potential $\phi$ satisfies a general type of boundary condition of the form

$$\begin{aligned} \mathcal{L }\bigg (\frac{\partial }{\partial x}\bigg )\frac{\partial \phi }{\partial y}+\mathcal{M }\bigg (\frac{\partial }{\partial x}\bigg )\phi =0, \end{aligned}$$

(96)

where $\mathcal L$ and $\mathcal M$ are the linear differential operators of the form

$$\begin{aligned} \mathcal{L }\bigg (\frac{\partial }{\partial x}\bigg )=\sum \limits _{n=0}^{l_0}c_n\frac{\partial ^{2n}}{\partial x^{2n}}, \quad \mathcal{M }\bigg (\frac{\partial }{\partial x}\bigg )=\sum \limits _{n=0}^{m_0}d_n\frac{\partial ^{2n}}{\partial x^{2n}}, \end{aligned}$$

(97)

with the $c_n$s and $d_n$s being known constants (as in [29]). It is assumed that $l_0\ge m_0$, so that $l_0$ determines the highest-order derivative in the boundary condition at $y=0$. The linearized condition on the flexible submerged plate at $y=h$ and the bottom boundary condition are the same as defined in Sect. 2. However, in case of finite water depth, the $p_n$s satisfy the dispersion relation in $p$ given by

$$\begin{aligned} \mathcal{G }(p;l_0,m_0)&\equiv KQ(p;m_0)\big \{1+\coth p(H-h)\coth ph\big \}-K\big \{T_2({\mathrm{i}}p)-1\big \}p\coth ph\nonumber \\&\quad -pP(p;l_0)Q(p;m_0)\big \{\coth ph+\coth p(H-h)\big \}+p^2P(p;l_0)\big \{T_2({\mathrm{i}}p)-1\big \}=0, \end{aligned}$$

(98)

where

$$\begin{aligned} P(p;l_0)=\sum \limits _{n=0}^{l_0}(-1)^{n}c_np^{2n}; \quad Q(p;m_0)=\sum \limits _{n=0}^{m_0}\big (-1)^{n}d_np^{2n}\big ) \end{aligned}$$

(99)

are the characteristic polynomials associated with the linear differential operators $\mathcal{L }({\partial }/{\partial x})$ and $\mathcal{M }({\partial }/{\partial x})$, respectively. The general form of the velocity potential $\phi (x,y)$ satisfying Eq. (1) along with the boundary conditions in Eqs. (3), (10), (12), and (96) in case of infinite depth is of the form

$$\begin{aligned} \phi (x,y)=\sum \limits _{n={\mathrm{I}}}^{2l_0+2}A_nF_n(y){\mathrm{e}}^{{\mathrm{i}}\mu _n x}+\int \limits _{0}^{\infty }A(\xi )L(\xi ,y){\mathrm{e}}^{-\xi x}\,{\mathrm{d}}\xi \quad {\mathrm{for}}\; x>0, \end{aligned}$$

(100)

where the $F_n(y)$s and $L(\xi , y)$ are given by

$$\begin{aligned}&F_n(y)= \left\{ \begin{array}{ll} - \dfrac{{\mathrm{i}} L_1({\mathrm{i}}\mu _n,y;l_0,m_0)}{{K\mathcal D }({\mathrm{i}}\mu _n, h;l_0,m_0)}, &{}\quad 0<y<h,\\ {\mathrm{e}}^{-\mu _n(y-h)}, &{}\quad h<y<\infty , \end{array}\right. \\&L(\xi ,y;l_0,m_0) = \left\{ \begin{array}{ll} L_1(\xi ,y;l_0,m_0) &{}\quad {\mathrm{for}}\; 0<y<h, \\ L_2(\xi ,y;l_0,m_0) &{}\quad {\mathrm{for}}\;h<y<\infty ,\end{array}\right. \end{aligned}$$

with

$$\begin{aligned}&L_1(\xi ,y;l_0,m_0)= K\big \{\xi P({\mathrm{i}}\xi ;l_0)\cos \xi y-Q({\mathrm{i}}\xi ;m_0)\sin \xi y\big \}, \\&L_2(\xi ,y;l_0,m_0)= L_1(\xi ,y;l_0,m_0)-W(\xi ,h;l_0,m_0)\cos \xi (y-h),\\&W(\xi ,h;l_0,m_0)= -\xi \{T_2( \xi )-1\} \mathcal{D }(\xi , h; l_0,m_0), T_2(\xi )=D_2\xi ^4+N_2\xi ^2+1,\\&\mathcal{D }(\xi , h;l_0,m_0) = \xi P({\mathrm{i}}\xi ;l_0) \sin \xi h + Q({\mathrm{i}}\xi ;m_0) \cos \xi h. \end{aligned}$$

The eigenfunctions $\mu _n$s satisfy the dispersion relation in $\mu$ given by

$$\begin{aligned} \mathcal{G }(\mu ;l_0,m_0) \equiv \frac{P(\mu ; l_0)\big [\mu Q(\mu ;m_0)( \coth \mu h +1)- \mu ^2\{T_2({\mathrm{i}}\mu )-1\}\big ]}{Q(\mu ;m_0)\big \{1 +\coth {\mu h}\big \}- \big \{T_2({\mathrm{i}}\mu )-1\big \}\mu \coth \mu h} - K =0. \end{aligned}$$

(101)

It is assumed that the dispersion relation in Eq. (101) has two distinct real positive roots and $2l_0$ number of complex roots of the forms $a\pm {\mathrm{i}}b$ and $-c\pm {\mathrm{i}}d$, with $\mu _{\mathrm{I}}$ and $\mu _{\mathrm{II}}$ being the real roots and $\mu _{\mathrm{III}}$, $\mu _{\mathrm{IV}},\ldots ,\mu _{{2l_0+2}}$ being the complex roots. However, out of the $2l_0+2$ roots of the dispersion relation, only roots leading to boundedness of the velocity potential will be considered in the expansion formulae in Eq. (100). It may be noted that $\mathcal{D }({\mathrm{i}}\mu _n, h) \ne 0$ for all $0 < h < \infty$. Further, the eigenfunctions $F_n(y)$s and $L(\xi , y;\,l_0,\,m_0)$ satisfy the generalized orthogonal mode-coupling relation as given by

$$\begin{aligned} \langle F_m,F_n \rangle = \langle F_m,F_n \rangle _{_1}+\langle F_m,F_n \rangle _{_2} = E_n \delta _{mn}\quad {\mathrm{for}}\quad m, n = {\mathrm{I, II}},\ldots ,2{l}_0+2 \end{aligned}$$

(102)

and

$$\begin{aligned} \left\langle F_m, L(\xi , y;l_0,m_0) \right\rangle = 0 \quad {\mathrm{for}} \quad m = {\mathrm{I,\,II}},\ldots ,2{l}_0+2, \xi >0, \end{aligned}$$

(103)

where $\delta _{mn}$ is the Kronecker delta function, and the $E_n$s are given by

$$\begin{aligned}&E_n =- \frac{\sinh \mu _n h [\psi _n^{'}(0)]^2 \mathcal{G }{'}(\mu _n;l_0,m_0) \mathcal{D } ({\mathrm{i}}\mu _n, h; l_0,m_0)}{2K^2 \mu _n^2},\\&\langle F_m,F_n \rangle _{_1}=\int \limits _0^{h}F_m(y)F_n(y)\,{\mathrm{d}}y+\sum \limits _{j=1}^{l_0}(-1)^{j}\frac{c_j}{Q(\mu _n;m_0)} \sum \limits _{k=1}^{j}F_m^{2k-1}(0)F_n^{2j-(2k-1)}(0)\\&\qquad \qquad \qquad +\sum \limits _{j=1}^{m_0}(-1)^{j+1}\frac{d_j}{P(\mu _n;l_0)} \sum \limits _{k=1}^{j}F_m^{2k-2}(0)F_n^{2j-2k}(0),\\&\langle F_m,F_n \rangle _{_2}=\lim _{\epsilon \rightarrow 0} \int \limits _{h}^{\infty }{\mathrm{e}}^{-\epsilon y} F_m(y)F_n(y)\,{\mathrm{d}}y+\frac{D_2}{K}\bigg \{F_m'(h)F_n'''(h)+ F_m'''(h)F_n'(h)\bigg \}-\frac{N_2}{K}F_m'(h)F_n'(h). \end{aligned}$$

Using Eqs. (102) and (103), the unknowns $A_n$s and $A(\xi )$ in Eq. (100) are obtained as

$$\begin{aligned} A_n=\frac{\langle \phi (x,y),F_n(y)\rangle }{E_n}, \quad \quad A(\xi )=\frac{2\langle \phi (x,y),L(\xi ,y;l_0,m_0)\rangle }{\pi \Delta (\xi ,h;l_0,m_0)}, \end{aligned}$$

(104)

with

$$\begin{aligned} \Delta (\xi ,h;l_0,m_0)&= W^2(\xi ,h;l_0,m_0)-2W(\xi ,h;l_0,m_0)L_1(\xi , h;l_0,m_0)\\&\quad +K^2\big \{\xi ^2P^2({\mathrm{i}}\xi ;l_0)+Q^2({\mathrm{i}}\xi ;m_0)\big \}.\nonumber \end{aligned}$$

This completes the derivation of the expansion formulae in case of infinite water depth. On the other hand, the general form of the velocity potential $\phi (x,y)$ satisfying Eq. (1) along with the boundary conditions (2), (10), (12), and (96) in case of finite depth is

$$\begin{aligned} \phi (x,y)=\sum \limits _{n={\mathrm{I}}}^{2l_0+2}A_n \psi _n(y){\mathrm{e}}^{{\mathrm{i}} p_n x}+\sum \limits _{n=1}^{\infty }A_n \psi _n(y){\mathrm{e}}^{-p_n x}\quad {\mathrm{for}}\; x>0, \end{aligned}$$

(105)

where the eigenfunction $\psi _n$s can be obtained as

$$\begin{aligned} \displaystyle \psi _n(y)=\left\{ \begin{array}{ll} -\displaystyle \frac{{\mathrm{i}} \tanh p_n(H-h)L_1({\mathrm{i}}p_n, y;l_0,m_0)}{K\mathcal{D }({\mathrm{i}}p_n,h; l_0,m_0) } &{} \quad 0<y<h,\\ \displaystyle \frac{\cosh p_n(H-y)}{\cosh p_n(H-h)} &{} \quad h<y<H,\\ \end{array}\right. \end{aligned}$$

(106)

with the eigenvalues $p_n$s satisfying the dispersion relation in Eq. (98). Similar to the case of infinite water depth, in this case also, it is assumed that the dispersion relation in Eq. (98) has two distinct positive real roots $p_{\mathrm{I}}$, $p_{\mathrm{II}}$, $p_n=a\pm {\mathrm{i}}b,\;-c\pm {\mathrm{i}}d$ for $n={\mathrm{III}},\ldots ,2{\mathrm{l_0}} +2$, and an infinite number of purely imaginary roots $p_n, \; n=1,2,\ldots$ of the form $p_n = {\mathrm{i}} \nu _n$. In the expansion formulae, terms leading to unboundedness of the velocity potential will not be included. In this case, the eigenfunctions $\psi _n(y)$s satisfy the orthogonal mode-coupling relation in Eq. (102) with $\langle \psi _m,\,\psi _n \rangle _{_2}$ and $E_n$s being replaced by

$$\begin{aligned}&\langle \psi _m,\psi _n \rangle _{_2}= \int \limits _{h}^{H} \psi _m(y)\psi _n(y)\,{\mathrm{d}}y+ \frac{D_2}{K}\Big \{\psi _m'(h)\psi _n'''(h) + \psi _m'''(h)\psi _n'(h)\Big \}-\frac{N_2}{K} \psi _m'(h)\psi _n'(h),\\&E_n = - \frac{\sinh p_n h [\psi _n'(0)]^2 \mathcal{G }'(p_n;l_0,m_0) \mathcal{D } ({\mathrm{i}}p_n, h;l_0,m_0)}{2K^2 p_n^2}. \end{aligned}$$

Thus, the unknowns $A_n$s in Eq. (105) are obtained as

$$\begin{aligned} A_n=\frac{\langle \phi (x,y),\psi _n(y)\rangle }{E_n} \quad {\mathrm{for }} \;n={I},\ldots ,2{l}_{0}+2,\,1,\,2,\ldots . \end{aligned}$$

Thus, the generalized expansion formulae in case of finite water depth are derived completely.