3.1. Dry modes of free–free elastic plate

The added-mass matrix (2.16) describes interactions of the dry modes $\psi _n(x)$ through the fluid, see (2.9) and (2.14). A general solution of the fourth-order differential equation (2.9) has the form

(3.1)$$\begin{gather} \psi_n(x)=L_{n1}f_{n1}(x)+L_{n2}f_{n2}(x)+L_{n3}f_{n3}(x)+L_{n4}f_{n4}(x), \end{gather}$$

(3.2a–d)$$\begin{gather}f_{n1}(x)=\cos (\lambda_n x),\quad f_{n2}(x)=\sin(\lambda_n x),\quad f_{n3}(x)={\rm e}^{-\lambda_n(1+x)},\quad f_{n4}={\rm e}^{-\lambda_n (1- x)}, \end{gather}$$

where $L_{nj}$, $j=1, 2, 3, 4$, are coefficients specific for imposed edge conditions. Substituting (3.1) in the four edge conditions we arrive at four equations with respect to the coefficients $L_{nj}$. The right-hand sides of these linear equations are zero. Non-zero solutions of the system exist only for some special values of the spectral parameter $\lambda _n$, which are obtained by equating the determinant of the system to zero. These solutions are determined up to factors, which are obtained using the normalisation condition (2.10). For a free–free dry plate with the edge conditions shown in (2.9), we obtain two modes, $n=1$ and $n=2$, with $\lambda _1=\lambda _2=0$, which correspond to rigid motions of the plate,

(3.3a,b)\begin{equation} \psi_1(x)=\frac 1{\sqrt 2},\quad \psi_2(x)=\sqrt{\frac 32} x, \end{equation}

and normalised elastic modes (3.1), where

(3.4a–d)\begin{equation} L_{n1}=\frac{1}{\sqrt{2}\cos\lambda_n},\quad L_{n2}=0,\quad L_{n3}=\frac{1}{\sqrt{2}(1+{\rm e}^{{-}2\lambda_n})},\quad L_{n4}=L_{n3}, \end{equation}

for odd numbers, $n=2m+1$, $m\geq 1$, and

(3.5a–d)\begin{equation} L_{n1}=0,\quad L_{n2}=\frac{1}{\sqrt{2}\sin\lambda_n},\quad L_{n3}=\frac{-1}{\sqrt{2}(1-{\rm e}^{{-}2\lambda_n})},\quad L_{n4}={-}L_{n3}, \end{equation}

for even numbers, $n=2m+2$, $m\geq 1$. Here $\lambda _n$ are real positive roots of the equation,

(3.6)\begin{equation} \cosh(2\lambda_n) \cos(2\lambda_n)=1. \end{equation}

For large $n$ we have

(3.7a,b)\begin{equation} \lambda_n=\frac{\rm \pi}{4}(n-2)+O({\rm e}^{-({\rm \pi}/{2})n}),\quad |L_{n3}|=\frac 1{\sqrt 2}+O({\rm e}^{-({\rm \pi}/{2})n}), \end{equation}

see, for example, Khabakhpasheva & Korobkin (Reference Khabakhpasheva and Korobkin2021).

3.2. Added masses as bilinear forms of dry modes

The elements $S_{nm}$ of the added-mass matrix, where $n,m\geq 1$, are given by the integral (2.16) and depend on the mode $\psi _m(x)$ directly and on the mode $\psi _n(x)$ through the potential (2.14) in the interval $y=0$, $-1< x<1$. It is convenient to introduce a bilinear form

(3.8)\begin{equation} U[F(x),G(x)]=\int_{{-}1}^1 \varPsi(x,0)G(x)\,{\rm d}\kern0.7pt x, \end{equation}

where $F(x)$ and $G(x)$ are smooth functions defined in the interval $-1< x<1$, and the potential $\varPsi (x,y)$ is the solution of the following mixed boundary-value problem,

(3.9a,b)\begin{equation} \nabla^2 \varPsi=0\ \ (y<0),\quad\varPsi=0 \ \ (y=0,\ |x|>1),\quad\varPsi_y=F(x)\ \ (y=0, |x|<1), \end{equation}

which decays at infinity, $\varPsi \to 0$ as $x^2+y^2\to \infty$, and is continuous in $y\leq 0$. The bilinear form $U[F(x),G(x)]$ has the following properties:

1. (a) it is symmetric,

   (3.10)\begin{equation} U[F(x),G(x)] = U[G(x),F(x)], \end{equation}
   which follows from Green's second identity;

2. (b) it is linear with respect to each argument,

   (3.11)$$\begin{gather} U[C_1F_1(x)+C_2F_2(x),G(x)]=C_1 U[F_1(x),G(x)]+ C_2U[F_2(x),G(x)], \end{gather}$$
   (3.12)$$\begin{gather}U[F(x),C_1G_1(x)+C_2G_2(x)]=C_1 U[F(x),G_1(x)]+ C_2U[F(x),G_2(x)]; \end{gather}$$

3. (c) it is zero if the product $F(x)G(x)$ is an odd function of $x$;

4. (d)

   (3.13)\begin{equation} U[F({-}x),G(x)]=U[F(x),G({-}x)]={\pm} U[F(x),G(x)], \end{equation}
   where plus is for even $G(x)$ and minus is for odd $G(x)$;

5. (e)

   (3.14)\begin{equation} U[{\rm e}^{\alpha x},{\rm e}^{\beta x}]=\begin{cases} {\rm \pi}[ I_0(\alpha)I_1(\beta)+I_1(\alpha)I_0(\beta)]/(\alpha+\beta)\quad (\alpha\neq{-}\beta), \\ {\rm \pi}[ I^2_0(\alpha)-I^2_1(\alpha)-I_0(\alpha)I_1(\alpha)/\alpha]\quad (\alpha={-}\beta), \end{cases} \end{equation}
   where $\alpha$ and $\beta$ are complex numbers and $I_0(\alpha )$, $I_1(\alpha )$ are the modified Bessel functions of the first kind; formula (3.14) is derived in Appendix A.

Then $S_{nk}=U[\psi _k(x),\psi _n(x)]$, where $\psi _n(x)$ is given by (3.4) for $n\geq 3$ and by (3.3) for $n=1$ and $n=2$. Let us consider first the elastic modes, $n, k\geq 3$, and the corresponding elements of the added-mass matrix. Substituting (3.1) in the bilinear form, we find

(3.15)\begin{equation} S_{nk}=U[\psi_k(x),\psi_n(x)]=\sum_{j=1}^4 \sum_{i=1}^4 L_{kj} U[f_{kj}(x),f_{ni}(x)] L_{ni} =\boldsymbol{L}_{\boldsymbol{k}}\boldsymbol{\cdot}\boldsymbol{\mathsf{Se}}^{(\boldsymbol{\mathsf{nk}})}\boldsymbol{\cdot} \boldsymbol{L}_{\boldsymbol{n}},\end{equation}

where $\boldsymbol {L}_{\boldsymbol {n}}=(L_{n1},L_{n2},L_{n3},L_{n4})$ and $\boldsymbol{\mathsf{Se}}^{(\boldsymbol{\mathsf{nk}})}$ is a symmetric $4\times 4$ matrix with the elements

(3.16)\begin{equation} Se^{(nk)}_{ij} =U[f_{kj}(x),f_{ni}(x)]\quad (1\leq i,j\leq4,\ n, k\geq 3). \end{equation}

Note that $Se^{(nk)}_{12}=Se^{(nk)}_{21}=0$, which follows from (3.2) and property (d) of the bilinear form (3.13). As a result, calculations of the elastic part of the added-mass matrix are reduced to evaluation of eight integrals (3.16). Other elements of the matrix are obtained using the symmetry relation $Se^{(nk)}_{ij}=Se^{(kn)}_{ji}$. The elements $Se^{(nk)}_{ij}$ are evaluated using properties of the bilinear form and (3.2) and (3.14). For example,

(3.17)\begin{align} Se^{(nk)}_{34}&= U[{\rm e}^{-\lambda_k(1+x)},{\rm e}^{-\lambda_n(1-x)}]={\rm e}^{-\lambda_k-\lambda_n}U[{\rm e}^{-\lambda_kx},{\rm e}^{\lambda_nx}]\nonumber\\ & = \begin{cases} {\rm \pi}[\tilde{I}_1(\lambda_n)\tilde{I}_0(\lambda_k)- \tilde{I}_1(\lambda_k)\tilde{I}_0(\lambda_n)]/(\lambda_n-\lambda_k)\quad (n\neq k),\\ {\rm \pi}[\tilde{I}^2_0(\lambda_n)-\tilde{I}_1^2(\lambda_n)-\tilde{I}_0(\lambda_n)\tilde{I}_1(\lambda_n)/\lambda_n] \quad(n=k), \end{cases} \end{align}

where $\tilde {I}_0(x)=I_0(x) \,{\rm e}^{-x}$ and $\tilde {I}_1(x)=I_1(x) \,{\rm e}^{-x}$ for $x\geq 0$. We used in (3.17) that $I_0(x)$ is even and $I_1(x)$ is odd functions. Other elements for both elastic and rigid modes are calculated analytically in Appendix A.

3.3. Wet modes of free–free plate and their frequencies

The elastic $k$th wet mode, $k\geq 3$, is given by series (2.11), where the coefficients $W_{kn}$ are solutions of the algebraic system (2.15). The value $\varOmega _k$ is a solution of the (2.17). This solution depends on a single parameter $\alpha$. The solutions are real because the matrix of the system (2.15) is symmetric but not necessary positive. Only positive solutions of (2.17) provide the frequencies of the wet modes, $\omega _k=[\varOmega _k D/(\rho L^5)]^{1/2}$. We consider only positive solutions. We truncate the matrix $[\boldsymbol{\mathsf{D}}-\varOmega (\alpha \boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{S}})]$ retaining $N_{mod}$ terms and calculate its determinant as a function of $\varOmega$ with a certain step in $\varOmega$ starting from $\varOmega =0$. The intervals, where the determinant changes its sign, are identified and the bisection method is used within each interval to find the roots $\varOmega _k$, $1\leq k\leq N_{mod}$, with required accuracy of $10^{-4}$. Then the roots are determined for larger number of retained terms $N_{mod}$ to confirm convergence of the roots with respect to the number of retained equations. The system (2.15) is solved for each root $\varOmega _k$, convergence of which has been confirmed.

The only parameter of the problem, $\alpha$, is small in the theory of thin plates. The ratios of wet, $\omega _n$, and dry, $\omega _n^{(d)}$, elastic natural frequencies,

(3.18a–c)\begin{equation} \frac{\omega_n}{\omega^{(d)}_n}=\frac {\sqrt{\alpha\varOmega_n}}{\lambda_n^2}, \quad \omega_n=\left(\frac{\varOmega_n D}{\rho L^5}\right)^{1/2}, \quad \omega_n^{(d)}=\lambda_n^2 \left(\frac{D}{m L^4}\right)^{{1}/{2}}, \end{equation}

are shown in figure 2(a) for different numbers $n\geq 3$ and different $\alpha$. Note that this ratio is independent of the plate rigidity. The ratios are not monotonic for $n=3,4,5$, then monotonically increase with increase of $n$ for all values of $\alpha$, and approach 1 for large $n$. The ratios increase with increase of $\alpha$ for any $n$, and are always smaller than one. Wet frequencies are also defined for $\alpha =0$, where the ratios tend to zero. Scaled wet frequencies for small α are shown in figure 2(b). To confirm convergence of the results, calculations for $\alpha =0.1$ and $0.01$ were performed with $N_{mod}$ equal to 30 and 50. It was found that the absolute differences of the ratios $\omega _n/\omega _n^{(d)}$ calculated with $N_{mod}=30$ and $N_{mod}=50$ for $3\leq n\leq 20$ is less that $5\times 10^{-6}$ for both values of $\alpha$.

Figure 2. The ratio (3.18a–c) of the wet and dry frequencies (a) and the scaled ratios for small $\alpha$ (b) as functions of the frequency number for different values of the parameter $\alpha$.

For each root $\varOmega _k$, we set $W_{kk}=1$ in (2.15), truncate the system to $N_{mod}$ equations and $N_{mod}$ unknowns, move $k$th column of (2.15) to the right-hand side and cut $k$th row from the system. The obtained finite system with non-zero right-hand side is solved numerically. The elements $W_{kn}$ of the resulting vector $\boldsymbol {W}_{\boldsymbol {k}}$ are listed in table 1 for $\alpha =0.01$ with three significant figures. Red entries in this table are for $W_{kk}$ which are set to one. It is seen that the wet modes have approximately the same shapes as the corresponding dry modes, but the natural frequencies of the modes are very different, see figure 2. The main contribution to the $k$th wet mode comes from the $k$th dry mode, the second mode, which provides the highest contribution, is the $(k+1)$th mode but its contribution is less than $10\,\%$ for even modes and $20\,\%$ for odd modes.

Table 1. The coefficients $W_{kn}$ in the series (2.11) of the $k$th wet mode as superposition of the dry modes $\psi _n(x)$ for $\alpha =0.01$. Wet modes with odd numbers are listed in the top of the table and the wet modes with even numbers are listed in the bottom of the table.

4.1. Added-mass matrix of elastic plate with different edge conditions

For edge conditions different from those in (2.6), eigenmodes of an elastic dry plate still have the form (3.1), (3.2) but with the coefficients $L_{nj}$ and $\lambda _n$ being specific for these conditions. The elements of the added-mass matrix are still given by (3.15) with the $4\times 4$ matrix $\boldsymbol{\mathsf{Se}}^{(\boldsymbol{\mathsf{nk}})}$ being independent of the edge conditions. For example, for a simply supported plate in full contact with the fluid, the edge conditions in (2.6) should be changed for $W(\pm 1)=W''(\pm 1)=0$, which provide the following end conditions in the spectral problem (2.9), $\psi _n=\psi ''_n=0$ at $x=\pm 1$. Symmetric dry modes of such a plate and the corresponding eigenvalues read

(4.1a,b)\begin{equation} \psi_n(x)=\cos(\lambda_n x), \quad \lambda_n={-}\frac{\rm \pi}{2}+{\rm \pi} n \quad (n\geq 1). \end{equation}

Therefore, $L_{n1}=1$ and $L_{n2}=L_{n3}=L_{n4}=0$ in (3.1) and the elements of the added-mass matrix (3.15) for symmetric vibrations of the plate are given by (see Appendix A)

(4.2)\begin{align} S_{nk}&=U[\cos(\lambda_k x), \cos(\lambda_n x)]=Se^{(nk)}_{11}\nonumber\\ &=\begin{cases} {\rm \pi}[ \lambda_n J_1(\lambda_n)J_0(\lambda_k) -\lambda_k J_1(\lambda_k)J_0(\lambda_n)]/(\lambda_n^2 - \lambda_k^2)\quad (n\neq k), \\ {\rm \pi}[ J^2_0(\lambda_n)+J^2_1(\lambda_n)]/2 \quad (n=k). \end{cases} \end{align}

The added-mass matrix (4.2) corresponds to that in Korobkin (Reference Korobkin1998, equation (41)) with $c=1$, where the plate is completely wetted.

4.2. Added-mass matrix of an elastic plate in partial contact with fluid

An elastic plate corresponds to the interval $-1< x<1$ in the dimensionless variables (2.5). However, now only a part of the plate, $a< x< b$, is in contact with the fluid, where $-1\leq a< b\leq 1$. Then the boundary-value problem (2.14) for the potentials $\phi _n(x,y)$ and the definition of the elements of the added-mass matrix (2.16) should be changed as follows

(4.3)$$\begin{gather} \nabla^2\phi_n=0 \quad (y<0), \end{gather}$$

(4.4a,b)$$\begin{gather}\phi_n=0 \quad (y=0,\ x< a \ \mbox{and}\ x>b),\quad \frac{\partial \phi_n}{\partial y}=\psi_n(x)\quad (y=0, a< x< b), \end{gather}$$

(4.5)$$\begin{gather}S_{nm}=\int_{a}^b \phi_n(x,0)\psi_m(x)\,{\rm d}\kern0.7pt x. \end{gather}$$

The dry modes of the plate are given by (3.1) with the coefficients $L_{nj}$ and $\lambda _n$ being dependent on edge conditions. The elements $S_{nm}$ in (4.5) are evaluated using the results of § 3.

To this aim, we introduce new variables, $x=A\tilde {x}+B$, $y=A\tilde {y}$, $\phi _n=A\tilde {\phi }_n(\tilde {x}, \tilde {y})$, and $\tilde \lambda _n=\lambda _n A$, where $A=(b-a)/2$ and $B=(b+a)/2$. Substituting $x=A\tilde {x}+B$ in (3.1) and denoting $\psi _n(A\tilde {x}+B)=\tilde \psi _n(\tilde {x})$, one obtains that $\tilde \psi _n(\tilde {x})$ has the same form as (3.1) but with modified coefficients,

(4.6)\begin{equation} \tilde\psi_n(\tilde{x})=\tilde{L}_{n1}\tilde{f}_{n1}(\tilde{x})+\tilde{L}_{n2}\tilde{f}_{n2}(\tilde{x})+\tilde{L}_{n3}\tilde{f}_{n3}(\tilde{x})+\tilde{L}_{n4}\tilde{f}_{n4}(\tilde{x}), \end{equation}

where

(4.7a)$$\begin{gather} \tilde{L}_{n1}=L_{n1}\cos(\lambda_n B)+L_{n2}\sin(\lambda_n B), \quad \tilde{L}_{n2}=L_{n2}\cos(\lambda_n B)-L_{n1}\sin(\lambda_n B), \end{gather}$$

(4.7b)$$\begin{gather}\tilde{L}_{n3}=L_{n3}\,{\rm e}^{-\lambda_n (1+a)}, \quad \tilde{L}_{n4}=L_{n4}\,{\rm e}^{-\lambda_n (1-b)}, \end{gather}$$

(4.8)\begin{equation} \tilde{f}_{n1}(\tilde{x})=\cos (\tilde\lambda_n \tilde{x}),\ \ \tilde{f}_{n2}(x)=\sin(\tilde\lambda_n \tilde{x}),\ \ \tilde{f}_{n3}(\tilde{x})={\rm e}^{-\tilde\lambda_n(1+\tilde{x})},\ \ \tilde{f}_{n4}={\rm e}^{-\tilde\lambda_n (1- \tilde{x})}. \end{equation}

Equations (30) and (31) in the new variables read

(4.9a)$$\begin{gather} \tilde\nabla^2\tilde{\phi}_n=0 \quad (\tilde{y}<0), \quad \tilde{\phi}_n=0 \quad (\tilde{y}=0,\ |\tilde{x}|>1), \end{gather}$$

(4.9b)$$\begin{gather}\frac{\partial \tilde{\phi}_n}{\partial \tilde{y}}=\tilde\psi_n(\tilde{x})\quad (\tilde{y}=0,\ |\tilde{x}|<1), \end{gather}$$

(4.10)\begin{gather} S_{nm}=A^2\int_{{-}1}^1 \tilde{\phi}_n(\tilde{x},0)\tilde\psi_m(\tilde{x})\,{\rm d} \tilde{x}. \end{gather}

Equations (4.6) and (4.9) are identical to (3.1) and (2.14). The only difference of (4.10) from (2.16) is the factor $A^2$ in (4.10). Therefore, (3.15) and detailed equations from Appendix B for the elements of the added-mass matrix can be used directly after modification of the coefficients (4.7), changing the eigenvalues $\lambda _n$ for $\tilde \lambda _n=\lambda _n A$, and multiplying the final results by $A^2$. It can be checked directly that for a simply supported plate, which is in contact with the fluid in the interval $-c< x< c$, where $0< c<1$, the added-mass elements (4.2) treated by the procedure described in this subsection, provide equation (41) from Korobkin (Reference Korobkin1998).

It is important to note that the contact region $a< x< b$, in general, is not symmetric, $a\neq -b$. The approach of this section cannot be used if an elastic plate is in contact with the fluid in several disconnected intervals, see Korobkin & Khabakhpasheva (Reference Korobkin and Khabakhpasheva2006, § 5). In such a case, the problem (2.14) should be solved numerically and the integrals (2.16) for the added-mass elements should be evaluated numerically as well.

5.1. Dry modes of a compound elastic plate

Dry modes of a compound elastic plate, $\psi _m(x)$, where $m\geq 1$ and $-1< x<1$, in the dimensionless variables (2.5) are the non-zero solutions of (5.2) without the potential $\varPhi (x,0)$ for the plates, $1\leq j\leq N_p$, the compound plate is made of. The dry modes are not necessary continuous functions of $x$. They and/or their derivatives can be discontinuous at the connectors, $x=a_j$, $2\leq j\leq N_p$, between the elementary plates. It is convenient to write these equations for each plate separately, where $\psi _m(x)=\psi _{mj}(x)$ in $a_j< x< a_{j+1}$. The functions $\psi _{mj}(x)$ are smooth in their corresponding intervals. The function $\psi _{m1}(x)$ satisfies the edge condition at $x=-1$. The function $\psi _{m N_p}(x)$ satisfies the edge condition at $x=1$. The functions $\psi _{mj}(x)$ satisfy also the conditions at the connectors, $x=a_j$, $2\leq j\leq N_p$, between the elementary plates. The edge and connection conditions are not specified at this stage. We know that there are two edge conditions at $x=-1$, two edge conditions at $x=1$ and four connection conditions at each connector. Therefore, there are $4N_p$ conditions to be satisfied by the $N_p$ functions $\psi _{mj}(x)$. These functions satisfy the equations

(5.9)\begin{equation} \frac{{\rm d} ^4\psi_{mj}}{{\rm d}\kern0.7pt x^4}=\lambda_m^4 \mu_j^4\psi_{mj} \quad (a_k< x< a_{j+1}),\end{equation}

where $\lambda _m$ is a spectral parameter, $m\ge 1$, to be determined, and $\mu _j=(\tilde {D}_j m_j/m_1)^{{1}/{4}}$ are known parameters. The frequencies of the dry modes are given by $\omega _m^{(d)}=\lambda _m^2[D_1/(m_1 L^4)]^{{1}/{2}}$, where $2L$ is the length of the whole structure.

General solutions of the fourth-order differential equations (5.9) have the form

(5.10)\begin{gather} \psi_{mj}(x)=L_{mj1}f_{mj1}(x)+L_{mj2}f_{mj2}(x)+L_{mj3}f_{mj3}(x)+L_{mj4}f_{n4}(x),\end{gather}

(5.11a)$$\begin{gather} f_{mj1}(x)=\cos (\lambda_m \mu_j x),\quad f_{mj2}(x)=\sin(\lambda_m \mu_j x), \end{gather}$$

(5.11b)$$\begin{gather}f_{mj3}(x)={\rm e}^{-\lambda_m \mu_j(a_j-x)},\quad f_{mj4}={\rm e}^{-\lambda_m \mu_j (a_{j+1}- x)}, \end{gather}$$

where $L_{mji}$, $i=1, 2, 3, 4$, are coefficients specific for imposed edge and connection conditions (compare with the solution (3.1) and (3.2) for a homogeneous plate). There are $4N_p$ coefficients in (5.10), which are obtained by using the $4N_p$ edge conditions and the conditions at the connectors. There are no forcing at the edges and connectors. Therefore, we arrive at $4N_p$ linear equations with zero right-hand sides for $4N_p$ coefficients $L_{mji}$. Equating the determinant of the matrix of this algebraic system to zero, we obtain equation for the spectral parameter $\lambda$. In general, the solutions $\lambda _m$ of this nonlinear equation are complex. Calculations of the determinant for complex values of $\lambda$ are straightforward because the elements of the matrix are given by the analytical functions (5.11) and their first, second and third derivatives. In the simplest case, the solutions of the equation are real and positive as for a free–free floating homogeneous plate, see dispersion relation (3.6). Equations (5.9) yield

(5.12)\begin{align} &(\lambda_n^4-\lambda_m^4) \frac{1}{\alpha_1}\sum_{j=1}^{N_p} \alpha_j \int_{a_j}^{a_{j+1}}\psi_{nj}(x)\psi_{mj}(x)\,{\rm d}\kern0.7pt x \nonumber\\ &\quad =U_{nm}(a_{N_p+1})-U_{nm}(a_1)- \sum_{j=1}^{N_p}[U_{nm}(a_j)], \end{align}

where

(5.13a)$$\begin{gather} [U_{nm}(a_j)]=U_{nm}(a_j+0)-U_{nm}(a_j-0), \end{gather}$$

(5.13b)$$\begin{gather}U_{nm}(x)=\dfrac{1}{\tilde{D}_j }\{\psi_n'''(x)\psi_m(x)-\psi_m'''(x)\psi_n(x)+\psi_m''(x)\psi_n'(x)-\psi_n''(x)\psi_m'(x)\}. \end{gather}$$

Therefore, the dry modes with $\lambda _n\neq \lambda _m$ are orthogonal,

(5.14)\begin{equation} \sum_{j=1}^{N_p} \alpha_j \int_{a_j}^{a_{j+1}}\psi_{nj}(x)\psi_{mj}(x)\,{\rm d}\kern0.7pt x =0,\end{equation}

if the edge conditions are such that $U_{nm}(a_{N_p+1})=0$, $U_{nm}(a_1)=0$, and the connectors are such that $[U_{nm}(a_j)]=0$ for $2\leq j\leq N_p$. The edge and connection conditions, which do not provide orthogonal modes, are not considered in this study. The dry modes are normalised as

(5.15)\begin{equation} \sum_{j=1}^{N_p} \alpha_j \int_{a_j}^{a_{j+1}}\psi_{nj}^2(x)\,{\rm d}\kern0.7pt x =\alpha_1.\end{equation}

Such normalisation is selected to match the case of a homogeneous plate with the normalisation (2.10).

The coefficients $C_{mk}$ in (5.3) are obtained by multiplying both sides of (5.3) by $\bar {\psi }_n(x)$, integrating the result with respect to $x$ from $-1$ to $1$ using (2.10), and changing finally $n$ to $k$,

(5.16)\begin{equation} C_{mn}=\int_{{-}1}^1 \psi_m(x)\bar{\psi}_n(x) \,{\rm d}\kern0.7pt x = \sum_{j=1}^{N_p}\int_{a_j}^{a_{j+1}}\psi_{mj}(x)\bar{\psi}_{n}(x)\,{\rm d}\kern0.7pt x,\end{equation}

where the integrals over the intervals $(a_j, a_{j+1})$ are evaluated by multiplying both sides of (5.3) by $\bar {\psi }_n(x)$ and integrating the result by parts,

(5.17)\begin{equation} \int_{a_j}^{a_{j+1}}\psi_{mj}(x)\bar{\psi}_{n}(x)\,{\rm d}\kern0.7pt x = \tilde{D}_j \frac{\tilde{U}_{nm}(a_{j})-\tilde{U}_{nm}(a_{j+1})}{\lambda_m^4\mu_j^4-\bar{\lambda}_n^4}.\end{equation}

In (5.17), $\bar {\lambda }_n$ is the value of the spectral parameter corresponding to the dry mode $\bar {\psi }_{n}(x)$, and $\tilde {U}_{nm}(x)$ is equal to $U_{nm}(x)$, where $\psi _{n}(x)$ is changed to $\bar {\psi }_{n}(x)$.

5.2. Wet modes of a compound elastic plate

The wet modes $W_k(x)$, $k\geq 1$, which are the solutions of (5.2), are sought as the series (2.11), where $\psi _n(x)$ are the dry modes of the complex plate and the coefficients $W_{kn}$ are to be determined. Dividing both sides of (5.2) written for the wet mode $W_k(x)$ by $\tilde {D}_j$, substituting (2.11), and using (5.9), we find

(5.18)\begin{equation} \frac{1}{\alpha_1}\sum_{n=1}^{\infty} W_{kn} \lambda_n^4 \alpha_j \psi_{nj}(x) = \varOmega_k \alpha_j \sum_{n=1}^{\infty} W_{kn} \psi_{nj}(x) + \varOmega_k \varPhi_k(x,0).\end{equation}

Using the series (2.12) for $\varPhi _k(x,0)$, multiplying both sides of (5.18) by $\psi _m(x)$, and integrating the result with respect to $x$ from $-1$ to $1$ using (2.16) and (5.14), we obtain

(5.19)\begin{equation} \lambda_m^4W_{km}=\varOmega_k \alpha_1 W_{km}+\varOmega_k \sum_{n=1}^{\infty} W_{kn} S_{nm}. \end{equation}

Equation (5.19) in the matrix form reads

(5.20)\begin{equation} [\boldsymbol{\mathsf{D}}-\varOmega_k (\alpha_1 \boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{S}})]\boldsymbol{W_k} =0, \end{equation}

compare with the (2.15) for a homogeneous plate. Here $\boldsymbol {W_k}=(W_{k1}, W_{k2}, W_{k3},\ldots )^{\rm T}$ is the vector of the coefficients in the series (2.11), $\boldsymbol{\mathsf{D}}$ is diagonal matrix, $D_{nm}=0$ for $n\ne m$, $D_{nn}=\lambda ^4_n$, $\boldsymbol{\mathsf{I}}$ is the unit matrix, $I_{nm}=\delta _{nm}$, and $\boldsymbol{\mathsf{S}}$ is the added-mass matrix given by (5.8).

5.3. Wet modes of two elastic plates connected by a torsional spring

The approach of this section is illustrated for two identical floating plates of different length connected by a torsional spring of rigidity $K_T$. In this example, $N_p=2$, $a_1=-1$, $a_2=\ell$ and $a_3=1$, where $-1<\ell <1$. The elastic characteristics of the plates are identical, $\mu _1=\mu _2=1$ and $\tilde {D}_1 = \tilde {D}_2 =1$ and $\alpha _1=\alpha _2=\alpha$. The edges of the plate, $x=\pm 1$, are free of stresses and shear forces, see figure 3. The (5.9) for the dry modes in the dimensionless variables (2.5) together with the edge conditions and conditions at the spring connector provide the following spectral problem

(5.21a)$$\begin{gather} \dfrac{{\rm d} ^4\psi}{{\rm d}\kern0.7pt x^4}=\lambda^4\psi\quad ({-}1< x<1), \end{gather}$$

(5.21b)$$\begin{gather}\dfrac{{\rm d} ^3\psi}{{\rm d}\kern0.7pt x^3}= \dfrac{{\rm d} ^2\psi}{{\rm d}\kern0.7pt x^2}=0 \quad(x=\pm1), \end{gather}$$

(5.21c)$$\begin{gather}\left[\dfrac{{\rm d} ^3\psi}{{\rm d}\kern0.7pt x^3}\right]=\left[\dfrac{{\rm d} ^2\psi}{{\rm d}\kern0.7pt x^2}\right]=[\psi]=0, \quad\dfrac{{\rm d} ^2\psi}{{\rm d}\kern0.7pt x^2} =k_T\left[\dfrac{{\rm d} \psi}{{\rm d}\kern0.7pt x}\right] \quad(x=l), \end{gather}$$

where $k_T=K_T L/D$ is the dimensionless stiffness of the torsional spring and $[w]=w(\ell +0,t)-w(\ell -0, t)$. The condition, $\psi ''(l)=k_T[\psi ']$, comes from the dimensional condition for the bending moments at the spring,

(5.22a)$$\begin{gather} M'(\ell',t')=K_T\left(\frac{\partial w'}{\partial x'}(\ell'+0,t')-\frac{\partial w'}{\partial x'}(\ell'-0,t')\right), \end{gather}$$

(5.22b)$$\begin{gather}M'(\ell',t')=EJ\frac{\partial^2 w'}{\partial x'^2}(\ell', t'), \end{gather}$$

where $M'(\ell ',t')$ is the bending moment at the spring. The modes of the problem (5.21) converge to the modes of the homogeneous plate when $k_T\to \infty$ and to the modes of two plates with hinged connection when $k_T\to 0$.

Figure 3. Sketch of two elastic plates connected by a torsional spring.

A general solution of the fourth-order differential equation of the spectral problem (5.21) reads

(5.23a)$$\begin{gather} \psi(x)=A_{1}\cos(\lambda x)+B_{1}\sin(\lambda x)+C_{1} \,{\rm e}^{-\lambda(1+x)}+D_{1}\,{\rm e}^{-\lambda(l-x)} \quad ({-}1< x< l), \end{gather}$$

(5.23b)$$\begin{gather}\psi(x)=A_{2}\cos(\lambda x)+B_{2}\sin(\lambda x)+C_{2}\,{\rm e}^{-\lambda_n(x-l)}+D_2 \,{\rm e}^{-\lambda_n(1-x)} \quad (l< x<1), \end{gather}$$

where the eight coefficients are solutions of the algebraic system of eight equations, which are obtained by substituting (5.23) in the eight edge and connection conditions of (5.21). Non-zero solutions of the system exist if the determinant of the system ${\rm det}(\lambda )$ is zero. The determinant is a function of the spectral parameter $\lambda$ with two dimensionless parameters $\ell$ and $k_T$. The system and its determinant are not displayed here. The determinant is calculated as a function of $\lambda$ starting from $\lambda =0$ with step $\Delta \lambda =0.01$. The intervals, where the determinant ${\rm det}(\lambda )$ changes its sign, are identified and the roots of the determinant are determined for each interval with a required accuracy by the bisection method. In the present calculations, the bisections stop when the interval of a root is shorter than $10^{-10}$. The calculations are terminated when a required number of positive roots, $N_{mod}$, is obtained. The positive roots correspond to elastic modes of the dry plate. In our calculations, $N_{mod}=49$. Note that $\lambda =0$ is a double root of the determinant with the corresponding modes being modes of the rigid motions, heave and pitch, of the compound plate. The spectral values $\lambda _n(k_T)$ of the lowest elastic modes, $3\leq n\leq 7$ as functions of the dimensionless rigidity $k_T$ for $\ell =0$ and $\ell =0.25$ are shown in figure 4. Here $\lambda _n(0)$ are the spectral values for the hinged connection, and $\lambda _n(k_T)\to \bar {\lambda }_n$ for $k_T\to \infty$, where $\bar {\lambda }_n$ are the spectral values of the corresponding homogeneous plate. We could not prove that the equation ${\rm det}(\lambda )=0$ for particular values of the parameters $k_T$ and $\ell$ has only real and positive solutions. We are unaware of such deep analysis by others.

Figure 4. The spectral values $\lambda _n(k_T)$ of the elastic dry modes with $3\leq n\leq 7$ as functions of the dimensionless rigidity $k_T$ for the spring at the centre of the plate, $\ell =0$ (a), and for the spring at $\ell =0.25$ (b).

The linear homogeneous system of eight equations for the coefficients in (5.23) is solved for each computed spectral parameter $\lambda _n$. The obtained coefficients provide the elastic modes of the compound plate, $\psi _n(x)$, $n\geq 3$, which depend on the location of the spring and its rigidity. Equation (5.14) yields that the dry modes $\psi _n(x)$ are orthogonal in a standard sense, because $\alpha _1=\alpha _2$, and can be normalised as, see (5.14) and (5.15),

(5.24)\begin{equation} \int_{{-}1}^1 \psi_n(x)\psi_m(x) \,{\rm d}\kern0.7pt x=\delta_{nm}.\end{equation}

Note that the elastic modes with odd numbers are even, and the modes with even numbers are odd with respect to the coordinate $x$ along the plate. Therefore, for a spring at the centre of the plate, $\ell =0$, the modes with even numbers are not affected by the spring because the deflections of these modes and their second derivatives are zero at $x=0$. This implies, see the conditions at $x=\ell$ in the spectral problem (5.21), that the first derivatives of the odd modes are continuous at the place of the spring and the dry modes are the same as for the plate without a spring. Figure 3(a) shows that the calculated spectral values of the odd modes, $\lambda _4(k_T)$ and $\lambda _6(k_T)$, are indeed constant.

Figure 4 shows that $\lambda _3(k_T)\to 0$ as $k_T\to 0$, which means that the lowest elastic mode, $\psi _3(x)$, converges to a rigid mode, see figure 4(a), when the spring becomes weak. This third rigid, flapping, mode is such that both the left and right parts of the plate move as rigid plates with hinged connection. The shapes of the elastic modes, $\psi _3(x)$, $\psi _4(x)$, $\psi _5(x)$ and $\psi _6(x)$, for different rigidity of the spring are shown in figure 5 for $\ell =0.25$. For this location of the spring, the mode $\psi _5(x)$ is not sensitive to the spring rigidity, see figure 5(c). The corresponding spectral value $\lambda _5(k_T)$ is almost constant, see figure 4(b). We conclude that some elastic mode can be weakly dependent on the spring rigidity if the spring is at a certain location. It was shown above that, if the spring is at the plate centre, the odd modes are independent of the spring rigidity. In the following, we show how to find modes weakly dependent on the spring rigidity for other locations of the spring. Figures 4 and 5 indicate that the presence of a spring can be approximately disregarded for spring dimensionless rigidity $k_T$ being greater than seven for any location of the spring.

Figure 5. Dry elastic modes $\psi _n(x)$, $n=3, 4, 5, 6$, for $\ell =0.25$ and $k_T=0, 1, 5, \infty$.

Equations (5.16) and (5.17) provide the coefficients $C_{mk}$ in the presentation (5.3) of the dry elastic mode $\psi _m(x)$, $m\geq 3$, as a superposition of the modes $\bar {\psi }_k(x)$, $k\geq 1$, of the corresponding homogeneous plate,

(5.25)\begin{equation} C_{mk}=\frac{\psi_m''(\ell)\bar{\psi}_k ''(\ell)}{k_T (\bar{\lambda}_k^4-\lambda_m^4)}. \end{equation}

It is seen that, if the spring is placed at $x=\ell$ where $\bar {\psi }_k ''(\ell )=0$ for a certain $k$, then $\bar {\psi }_k(x)$ is also a mode of the compound plate for any rigidity of the spring.

The dry modes $\psi _3(x)$ and $\psi _6(x)$ for $\ell =0.25$ and $k_T=0$ are shown in figure 6 together with their approximations by series (5.3), where 10, 20 and 50 terms are retained. It is seen that 50 terms in (5.3) accurately approximate these 2 dry modes with 10 terms providing a reasonable approximation everywhere except the vicinity of the spring.

Figure 6. The rigid mode, $\psi _3(x)$ (a), and the elastic mode, $\psi _6(x)$ (b), shown by red lines for the hinged connection, $k_T=0$, at $\ell =0.25$ approximated by linear superposition of $N_{approx}$ modes of the homogeneous plate, where $N_{approx}=10, 20, 50$.

The added-mass matrix $\boldsymbol{\mathsf{S}}$ is calculated by formula (5.8). Note that $\bar {\boldsymbol{\mathsf{S}}}$ in this formula depends only on the type of the edge conditions, and the transformation matrix $\textbf {C}$, in addition to that, depends on the position of the spring, $\ell$, and the spring rigidity, $k_T$. The dimensionless wet frequencies $\varOmega _k$, $k\geq 3$, of the wet elastic modes are obtained by solving the equation ${\rm det}[\boldsymbol{\mathsf{D}}-\varOmega _k (\alpha \boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{S}})]=0$, where the matrix $[\boldsymbol{\mathsf{D}}-\varOmega _k (\alpha \boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{S}})]$ is truncated to $N_{mod}\times N_{mod}$. The wet frequencies depend on three parameters $\ell$, $k_T$, and $\alpha$.

The ratios of wet, $\omega _n$, and dry, $\omega _n^{(d)}$, elastic natural frequencies,

(5.26a–c)\begin{equation} \frac{\omega_n}{\omega^{(d)}_n}=\frac {\sqrt{\alpha\varOmega_n}}{\lambda_n^2}, \quad \omega_n=\left(\frac{\varOmega_n D_1}{\rho L^5}\right)^{1/2}, \quad \omega_n^{(d)}=\lambda_n^2 \left(\frac{D_1}{m_1 L^4}\right)^{{1}/{2}},\end{equation}

are shown in figure 7(a,b) for elastic modes, $3\leq n\leq 20$, with $\alpha =0.1$ and $\alpha =0.01$, $\ell =0.25$, and different rigidity of the spring, $k_T$. Even for the limiting cases, where $k_T\to \infty$ and $k_T\to 0$, the ratios are close to each other. The relative differences of the ratios,

(5.27)\begin{equation} \Delta\omega_n=\left(\frac{\bar{\omega}_n}{\bar{\omega}^{(d)}_n}-\frac{\omega_n}{\omega^{(d)}_n}\right)\left/\, \frac{\bar{\omega}_n}{\bar{\omega}^{(d)}_n}\right. \times 100\,\%, \end{equation}

where a bar stands for the frequencies of the corresponding homogeneous plate, are shown in figure 7(c) for a weak spring, $k_T=10^{-4}$, and different $\alpha$. It is seen that $|\Delta \omega _n|\leq 3\,\%$ for $n\geq 3$. Therefore, the ratio $\omega _n/\omega ^{(d)}_n$ is weakly dependent on the spring rigidity. It is important to note that, at the same time, the relative differences of the dry frequencies,

(5.28)\begin{equation} \Delta\omega^{(d)}_n=\frac{\bar{\omega}^{(d)}_n-\omega^{(d)}_n}{\bar{\omega}^{(d)}_n}\times 100\,\%, \end{equation}

shown in figure 7(d), are not small and can be as large as 30 %. Here $\Delta \omega ^{(d)}_3$ for very small $k_T$ is not shown because $\omega ^{(d)}_3\to 0$ as $k_T\to 0$ and then $\Delta \omega ^{(d)}_3=100\,\%$. In calculations $\Delta \omega ^{(d)}_3=98\,\%$ for $k_T=10^{-4}$.

Figure 7. The ratios of wet and dry frequencies for $\ell =0.25$ and different rigidity $k_T$ with $\alpha =0.1$ (a), $\alpha =0.01$ (b). The relative differences of these ratios (c) and the relative differences of the dry frequencies (d).

Similar analysis is performed for $k_T=1$ and different positions of the spring, see figure 8. It is seen that the ratio $\omega _n/\omega ^{(d)}_n$ is weakly dependent on the position of the spring. We can conclude that the ratio $\omega _n/\omega ^{(d)}_n$ is weakly dependent on details of the structure and can be approximated by the ratio calculated for homogeneous plate,

(5.29)\begin{equation} \frac{\omega_n}{\omega^{(d)}_n}\approx \frac{\bar{\omega}_n}{\bar{\omega}^{(d)}_n},\end{equation}

and, then,

(5.30)\begin{equation} \omega_n\approx \bar{\omega}_n \frac{\lambda_n^2}{\bar{\lambda}_n^2}.\end{equation}

The formula (5.30) can be used for preliminary estimates of wet frequencies of a compound plate using the known frequencies of the dry plate.

Figure 8. The ratios of wet and dry frequencies for $k_T=1$ and different positions of the spring $\ell$: $\alpha =0.1$ (a), $\alpha =0.01$ (b). The relative differences of these ratios (c) and the relative differences of the dry frequencies (d).

The wet modes are obtained by solving the truncated system (5.20) with $W_{kk}=1$. The elements $W_{kn}$ of the resulting vector $\boldsymbol {W_k}$ are listed for $\alpha =0.1$, $\ell =0.25$ in table 2 for $k_T=1$ and in table 3 for $k_T=10^{-4}$ with three significant figures. Red entries in this table are for $W_{kk}$ which are set to one. It is seen that the wet modes have approximately the same shapes as the corresponding dry modes. The main contribution to the $k$th wet mode comes from the $k$th dry mode. Only the dry modes $\psi _n(x)$ with numbers such that $|k-n|\leq 2$ provide contributions to the wet mode $W_k(x)$ greater than 2 %. This result depends on the position of the spring, its rigidity and the mass parameter $\alpha$. The wet elastic mode $W_3(x)$ for a weak spring, see the first column in table 3, behaves differently with the main contribution coming from the rigid heave mode $\psi _1(x)$. Except this special case the wet modes are close to the corresponding dry modes, the natural frequencies of the wet and dry modes are very different, see figure 7(c,d), but their ratios are weakly dependent on the position and rigidity of the spring, see figure 7(a,b) and (5.29).

Table 2. The coefficients $W_{kn}$ in the expansion (2.11) of the $k$th wet mode $W_k(x)$ as a superposition of the dry modes ${\psi }_n(x)$ for $\alpha =0.1$, $\ell =0.25$ and $k_T=1$.

Table 3. The coefficients $W_{kn}$ in the expansion (2.11) of the $k$th wet mode $\psi _k(x)$ as a superposition of the dry modes $\psi _n(x)$ for $\alpha =0.1$, $\ell =0.25$ and $k_T=10^{-4}$.

6.1. Dry modes of elastic plate with linear thickness

A general solution of the differential equation (6.6) with $h(x)=1+\beta x$, $|\beta |<1$, is given through Bessel functions (see, for example, Li Reference Li2000; Batyaev & Khabakhpasheva Reference Batyaev and Khabakhpasheva2022),

(6.9a)$$\begin{gather} \psi_n(x)=\frac{1}{\xi}M_n(\xi), \quad \xi=\frac{2}{\beta}\lambda_n\sqrt{1+\beta x}, \end{gather}$$

(6.9b)$$\begin{gather}M_n(\xi)=A_n J_1(\xi)+B_n Y_1(\xi)+C_n I_1(\xi)+D_n K_1(\xi), \end{gather}$$

where $J_1(\xi )$ and $Y_1(\xi )$ are the Bessel functions of order one of the first and second kind correspondingly, and $I_1(\xi )$ and $K_1(\xi )$ are the modified Bessel functions of order one of the first and second kind correspondingly. The free–free edge conditions in (6.6) are written in terms of the function $M_n(\xi )$ as

(6.10a,b)\begin{equation} \xi\frac{{\rm d} ^3 M_n}{{\rm d} \xi^3}=\frac{{\rm d} ^2 M_n}{{\rm d} \xi^2}, \quad \frac{1}{3}\xi\frac{{\rm d} ^2 M_n}{{\rm d} \xi^2}+\frac{1}{\xi} M_n= \frac{{\rm d} M_n}{{\rm d} \xi}\quad (\xi=\xi_n^{{\pm}}), \end{equation}

where $\xi _n^{\pm }=2\lambda _n\sqrt {1\pm \beta }/\beta$, and provide four equations for the coefficients $A_n$, $B_n$, $C_n$, $D_n$,

(6.11a)\begin{align} & A_n\left\{\left(\frac{8}{\xi^2}-3\right) J_1(\xi) +\left(\xi-\frac{4}{\xi}\right) J_0(\xi)\right\} + B_n\left\{\left(\frac{8}{\xi^2}-3\right) Y_1(\xi) +\left(\xi-\frac{4}{\xi}\right) Y_0(\xi)\right\} \nonumber\\ &\quad +C_n\left\{\left(\frac{8}{\xi^2}+3\right) I_1(\xi) -\left(\xi+\frac{4}{\xi}\right) I_0(\xi)\right\}\nonumber\\ & \quad + D_n\left\{\left(\frac{8}{\xi^2}+3\right) K_1(\xi) +\left(\xi+\frac{4}{\xi}\right) K_0(\xi)\right\} = 0, \end{align}

(6.11b)\begin{align} & A_n\left\{\left(2-\frac{\xi^2}{4}\right) J_1(\xi) -\xi J_0(\xi)\right\} + B_n\left\{\left(2-\frac{\xi^2}{4}\right) Y_1(\xi) -\xi Y_0(\xi)\right\}\nonumber\\ & \quad + C_n\left\{\left(2+\frac{\xi^2}{4}\right) I_1(\xi) -\xi I_0(\xi)\right\} + D_n\left\{\left(2+\frac{\xi^2}{4}\right) K_1(\xi) +\xi K_0(\xi)\right\} =0, \end{align}

where $\xi =\xi _n^{\pm }$. Non-zero solutions of the homogeneous system (6.11) exist only if the determinant of the system is equal to zero, which gives the dispersion equation with respect to $\lambda _n$ as a function of the parameter $\beta$. The solution of the system (6.11) is determined up to a constant factor, which is obtained using the normalisation condition (6.7). This condition written in terms of the functions $M_n(\xi )$ reads

(6.12)\begin{equation} \frac{\beta^3}{8\lambda_n^4}\int_{\xi_n^-}^{\xi_n^+} \xi M_n^2(\xi) \,{\rm d} \xi =1. \end{equation}

Substituting $M_n(\xi )$ from (6.9) in (6.12), we obtain several integrals of the form

(6.13)\begin{equation} \int \xi Z_1(\xi) B_1(\xi)\,{\rm d}\xi, \end{equation}

where $Z_1$ and $B_1$ are any functions $J_1$, $Y_1$, $I_1$ and $K_1$. Such integrals are evaluated analytically, see the tables of integrals by Gradstein & Ryzhik (Reference Gradstein and Ryzhik1965, formula 5.54).

Equations (6.8) provide the parameters of the second rigid dry mode, $\psi _2(x)$, for the linear plate thickness, $c=\beta /3$ and $a=3/\sqrt {2(3-\beta ^2)}$. It is seen that $\psi _1(x)=\bar {\psi }_1(x)=1/\sqrt {2}$ and $\psi _2(x)$ tends to $\bar {\psi }_2(x)= \sqrt {3/2}x$ as $\beta$ tends to $0$. It is expected that $\lambda _n(\beta )\to \bar {\lambda }_n$ and $\psi _n(x)\to \bar {\psi }_n(x)$ for elastic modes, $n\geq 3$, as $\beta \to 0$, where $\bar {\psi }_n(x)$ are solutions of the spectral problem (2.9) for elastic plate of constant thickness with the corresponding eigenvalues $\bar {\lambda }_n$. However, the limit of the dry modes (6.9) as $\beta \to 0$ is singular, because $\xi \to \infty$ and, therefore, at least the coefficients $A_n$ and $B_n$ tend to infinity as $\beta \to 0$.

To calculate wet modes and the added-mass matrix for a floating plate with linear thickness, the dry modes, $\psi _n(x)$, are represented as superposition of the dry modes of the plate with constant thickness,

(6.14)\begin{equation} \psi_n(x)=\sum_{k=1}^{k_{max}} C_{nk}\bar{\psi}_k(x). \end{equation}

The coefficients $C_{nk}$ are calculated numerically by the formula

(6.15)\begin{equation} C_{nk}=\int_{{-}1}^{1} \psi_n(x)\bar{\psi}_k(x) \,{\rm d}\kern0.7pt x,\end{equation}

where the orthogonality condition (2.10) was used. Accuracy of the approximation (6.14) is investigated in Appendix B.

6.2. Wet modes of elastic plates with linear thickness

The wet rigid modes, $W_1(x)=1/\sqrt 2$ and $W_2(x)=a(x-c)$, where $c=\beta /3$ and $a=3/\sqrt {2(3-\beta ^2)}$, and their frequencies, $\omega _1=\omega _2=0$, are the same as for the dry plate with constant thickness. The wet elastic modes $W_k(x)$, $k\ge 3$, are sought as superposition (2.11) of the dry modes $\psi _n(x)$ of the same plate. Substituting the series (2.11) in the plate (6.4), using the differential equation (6.6) for the dry modes, multiplying both sides of the resulting equation by $\psi _m(x)$, and integrating with respect to $x$ from $-1$ to 1 using the orthogonality relation (6.7), we obtain the infinite system of algebraic equations for the coefficients $W_{kn}$ in the series (2.11),

(6.16)\begin{equation} [\boldsymbol{\mathsf{D}}-\varOmega_k (\alpha \boldsymbol{\mathsf{I}}+\boldsymbol{\mathsf{S}})]\boldsymbol{W_k} =0. \end{equation}

This system has the same form as the corresponding system (2.15) for a homogeneous plate. Here $\boldsymbol {W_k}=(W_{k1}, W_{k2}, W_{k3},\ldots )^{\rm T}$ is the vector of the coefficients in the series (2.11), $\boldsymbol{\mathsf{D}}$ is the diagonal matrix, $D_{nm}=0$ for $n\ne m$, $D_{nn}=\lambda ^4_n$, where $\lambda _n$ are the spectral values from (6.6), $\boldsymbol{\mathsf{I}}$ is the unit matrix, $I_{nm}=\delta _{nm}$. The added mass matrix $\boldsymbol{\mathsf{S}}$ in (6.16) is given by (5.8), where $\bar {\boldsymbol{\mathsf{S}}}$ is the added-mass matrix for the plate of constant thickness, see § 5, and $\textbf {C}$ is the transformation matrix with the elements (6.15). We conclude that the coefficients of the wet modes $W_k(x)$ and the dimensionless frequencies of these modes $\varOmega _k$ can be obtained numerically using the same algorithm as for the plate of constant thickness, see § 3.3.

The ratios of the wet, $\omega _n$, and dry, $\omega _n^{(d)}$, elastic natural frequencies,

(6.17a–c)\begin{equation} \frac{\omega_n}{\omega^{(d)}_n}=\frac {\sqrt{\alpha\varOmega_n}}{\lambda_n^2}, \quad \omega_n=\left(\frac{\varOmega_n D_c}{\rho L^5}\right)^{1/2}, \quad \omega_n^{(d)}=\lambda_n^2 \left(\frac{D_c}{\rho_ph_c L^4}\right)^{{1}/{2}},\end{equation}

are shown in figure 10(a,b) for $3\le n\le 20$, different values of the parameters $\beta$, and $\alpha =0.1$ in (a) and $\alpha =0.01$ in (b). The ratios for $\beta =0$, which is for the plate of constant thickness, are shown by black dots. Calculations are performed with $k_{max}=40$ in (6.14). Note that the ratios $\omega _n/\omega _n^{(d)}$ are independent of the plate rigidity. The ratios decrease with increase of the parameter of the plate inclination, $\beta$, see figure 10(a,b). Figure 10(c,d) present relative differences $\Delta \omega _n$, of the ratios (6.17) and the ratios calculated for the corresponding plate with constant thickness $h_c$. The differences $\Delta \omega _n$, $n\ge 3$, are defined by (5.27).

Figure 10. (a) The ratios of the wet and dry frequencies for $\beta =0$, 0.25, 0.5 and 0.75 and for $\alpha =0.1$ (a) and $\alpha =0.01$ (b). Relative differences $\Delta \omega _n$ for the same values of $\beta$ and $\alpha =0.1$ (c) and $\alpha =0.01$ (d).

It is seen that the ratios (6.17) can be approximated by the corresponding ratios for the plate with the mean thickness, if $\beta \le 0.5$. The differences $\Delta \omega _n$ increase with increase of $\beta$. The lowest wet modes, $n=3,4,5$, are more sensitive to the thickness variation than the higher modes. The differences $\Delta \omega _n$ increase with decrease of the mass parameter $\alpha$, which is for lighter plates. For a heavy plate with $\alpha =0.1$, and $\beta =0.5$ the differences $\Delta \omega _n$ are less than $4.2\,\%$ for $n\ge 3$ and less than $1\,\%$ for $n\ge 7$. For the same plate and $\beta =0.75$ the difference $\Delta \omega _n$ are less than $12\,\%$ for $n\ge 3$ and less than $5\,\%$ for $n\ge 9$. The frequency ratios for lighter plates, see figure 10(d), are more sensitive to the variation of the plate thickness. We may conclude, similar to the results of § 5.3 for compound plates, that the wet frequencies $\omega _n$ of the plate with linear thickness can be approximated by the formula (5.30), where $\bar \omega _n$ are the wet frequencies of the plate with the mean thickness $h_c$. This approximation is less accurate for large $\beta$ and lowest modes.

The solutions of (6.16) for $3\le k\le 12$ are listed in table 4 for $\alpha =0.01$ and $\beta =0.5$. Note that we set $W_{kk}=1$ for each $k\ge 3$, delete the $k$th equation, move the $k$th column to the right-hand side, and solve the resulting non-homogeneous equations with respect to the coefficients $W_{kn}$, where $1\le n\le k-1$ and $k+1\le n\le k_{max}$. The coefficients are shown with three significant figures. Red entries in this table are for $W_{kk}$ which are set to one.

Table 4. The coefficients $W_{kn}$ in the series (6.14) of the $k$th wet mode $\psi _k(x)$ as a superposition of the dry modes $\bar {\psi }_n(x)$ for $\alpha =0.01$ and $\beta =0.5$.

It is seen that the wet modes have approximately the same shapes as the corresponding dry modes, but the natural wet and dry frequencies are very different, see figure 10. The main contribution to the $k$th wet mode comes from the $k$th dry mode, the next mode, which provides the highest contribution, is the $(k+1)$th mode but its contribution is less than $15\,\%$ for $3\le k\le 8$ and less than $23\,\%$ for $9\le k\le 12$.

6.3. Dry and wet modes of elastic plate with piecewise linear thickness

To find the dry modes of a plate with variable continuous plate thickness $h(x)$, the thickness can be approximated as a piecewise linear function. Let the plate length be divided into $N_p$ intervals, $a_j< x< a_{j+1}$, $1\leq j\leq N_p$, $a_1=-1$ and $a_{N_p+1}=1$ in the dimensionless variables. The plate thickness is approximated as

(6.18a–c)\begin{equation} h(x)\approx h_j+T_j (x-a_j), \quad T_j=\frac{h_{j+1}-h_j}{a_{j+1}-a_j} \quad (a_j< x< a_{j+1}), \end{equation}

where $h_j=h(a_j)$ and $h_j>0$. For each interval $a_j< x< a_{j+1}$ of the plate a general solution of the differential equation (6.6) can be obtained using the solution (6.9), which is for $h(x)=1+\beta x$, with four undetermined coefficients for each interval. These coefficients are determined using the four edge conditions at $x=\pm 1$ and four matching conditions at the ends of each interval. The matching conditions require that the function $\psi _n(x)$ is continuous together with its first and second derivatives at the ends of the intervals, and the shear force ${\rm d}[h^3(x)\psi ''(x)]/{\rm d}\kern0.7pt x$ is also continuous there, which gives

(6.19a,b)\begin{equation} \left[\frac{{\rm d} ^2\psi}{{\rm d}\kern0.7pt x^2}\right]=\left[\frac{{\rm d} \psi}{{\rm d}\kern0.7pt x}\right]=[\psi]=0, \quad h\left[\frac{{\rm d} ^3\psi}{{\rm d}\kern0.7pt x^3}\right]+ 3\frac{{\rm d} ^2\psi}{{\rm d}\kern0.7pt x^2}\left[\frac{{\rm d} h}{{\rm d}\kern0.7pt x}\right]=0 \quad (x=a_j),\end{equation}

$2\leq j\leq N_p$. Indeed, for the interval $a_j< x< a_{j+1}$ and $T_j\neq 0$, (6.6) using (6.18) reads

(6.20)\begin{equation} \frac{{\rm d} ^2}{{\rm d}\kern0.7pt x^2}\left([h_j+T_j(x-a_j)]^3\frac{{\rm d} ^2\psi_n^{(j)}}{{\rm d}\kern0.7pt x^2}\right)=\lambda_n^4 [h_j+T_j(x-a_j)] \psi_n^{(j)}, \end{equation}

where $\psi _n^{(j)}(x)$ is the function $\psi _n(x)$ in the interval under consideration. We limit ourselves to the case where $h_j-T_j a_j\neq 0$. Then (6.20) can be written as

(6.21)\begin{equation} \frac{{\rm d} ^2}{{\rm d}\kern0.7pt x^2}\left((1+\beta_j x)^3\frac{{\rm d} ^2\psi_n^{(j)}}{{\rm d}\kern0.7pt x^2}\right)=\tilde\lambda_{nj}^4 (1+\beta_j x) \psi_n^{(j)}\quad (a_j< x< a_{j+1}), \end{equation}

where

(6.22a,b)\begin{equation} \beta_j=T_j/(h_j-T_j a_j),\quad\tilde\lambda_{nj}=\lambda_n/|h_j-T_j a_j|^{1/2}. \end{equation}

It is not required now that $|\beta _j|<1$ as in § 6.1. A general solution of the differential equation (6.21) is given by (6.9),

(6.23a,b)$$\begin{gather} \psi_n^{(j)}(x)=\frac{1}{\xi}M_{nj}(\xi),\!\quad M_{nj}(\xi)=A_{nj} J_1(\xi)+B_{nj} Y_1(\xi)+C_{nj} I_1(\xi)+D_{nj} K_1(\xi), \end{gather}$$

(6.24a,b)$$\begin{gather}\xi=b_{nj}\sqrt{1+\beta_j x}, \quad b_{nj}=\frac{2\lambda_n}{T_j}|h_j-T_j a_j|^{1/2}\,{\rm sgn}(h_j-T_ja_j), \end{gather}$$

where ${\rm sgn}(x)$ is the sign function, $x=|x|\,{\rm sgn}(x)$. There are $4 N_p$ coefficients $A_{nj}$, $B_{nj}$, $C_{nj}$, $D_{nj}$ to be determined, $4 (N_p-1)$ matching conditions (6.19) and four edge conditions as in (6.6). Therefore, we have $4 N_p$ linear equations with zero right-hand sides for the coefficients $A_{nj}$, $B_{nj}$, $C_{nj}$ and $D_{nj}$. This system has non-zero solutions only for $\lambda _n$ which are roots of the determinant of the system. Next the system is solved numerically for each $\lambda _n$ up to a constant factor, which is determined using the normalisation condition (6.7). The obtained dry modes $\psi _n(x)$ of the plate with piecewise linear thickness are presented as the superpositions (6.14) of the dry mode, $\bar {\psi }_k(x)$, of the plate with constant thickness $h_c$ defined by (6.3), where the coefficients $C_{mk}$ are given by the integrals (6.15). Finally, the added-mass matrix of the plate with a variable thickness is calculated using (5.8), and the coefficients of the wet modes, $W_{kn}$ in (6.14), and the corresponding dimensionless frequencies of these modes, $\varOmega _k$, are obtained solving the system (6.16).

If $T_j=0$ for an interval $a_j< x< a_{j+1}$, then the general solution (6.22) for this interval should be replaced by the solution (3.1), (3.2), where $\lambda _n$ is changed to $\lambda _n/\sqrt {h_j}$. If $T_j\neq 0$ but $h_j-T_j a_j=0$, then (6.20) takes the form

(6.25)\begin{equation} \frac{{\rm d} ^2}{{\rm d}\kern0.7pt x^2}\left(x^3\frac{{\rm d} ^2\psi_n^{(j)}}{{\rm d}\kern0.7pt x^2}\right)=\frac{\lambda_n^4 }{T_j^2}x \psi_n^{(j)}, \end{equation}

with its general solution being of the form (6.23), where $\xi =2\lambda _n |T_j|^{-\frac 12}x$.

Numerical calculations are performed for a plate with two intervals of linear thickness and one interval of a constant thickness. The plate thickness $h(x)$ is given in the dimensionless variables by the formula

(6.26)\begin{equation} h(x)=\left\{\begin{array}{ll} h_* \tilde{h}_1+h_* \tilde{T}_1(x-a_1) & (a_1 < x \leq a_2),\\ h_* & (a_2 \leq x \leq a_3),\\ h_*+h_* \tilde{T}_3(x-a_3) & (a_3 \leq x < a_4), \end{array}\right.\end{equation}

see figure 11. The parameters in (6.18) and (6.26) are related by $a_1=-1$, $a_4=1$, $h_1=h_*\tilde {h}_1$, $\tilde {h}_2=\tilde {h}_3=h_*$, $T_1=h_*\tilde {T}_1$, $T_2=0$ and $T_3=h_*\tilde {T}_3$. The values of the parameters in the present calculations for three cases are given in table 5. The average thickness of a plate should be equal to 1 in the dimensionless variables, see (6.3). This conditions gives

(6.27)\begin{equation} h_*=\left[1+\frac{(\tilde{h}_1-1)(1+a_2)}4+\frac{(\tilde{h}_4-1)(1-a_3)}4\right]^{{-}1}. \end{equation}

The expressions $h_j-T_ja_j$, $j=1,2,3$, are not zero and positive for all the three cases of the table 5. Therefore, general solutions of the (6.20) for the intervals $(-1, a_2)$ and $(a_3,1)$ are given by (6.23), where the values $\beta _j$ and $b_{nj}$, for $n=1$ and $n=3$, are calculated by formulae (6.22), (6.24) and (6.26). In the interval $a_2< x< a_3$, where $T_2=0$, a general solution of (6.20) is given by (3.1), (3.2), where $\lambda _n$ should be changed to $\lambda _n/\sqrt {h_*}$. Here the spectral parameter $\lambda _n$ is the same in all three intervals.

Figure 11. Shape of the plate with variable thickness.

Table 5. Dimensionless parameters of calculations.

Substituting the solutions (6.23) written for the intervals $(-1,a_2)$ and $(a_3,1)$ with $\beta _1$, $\beta _3$, $b_{n1}$ and $b_{n3}$ given by (6.22) and (6.24), and the solution (3.1), (3.2) for the interval $(a_2,a_3)$, where $\lambda _n$ is replaced by $\lambda _n/\sqrt {h_*}$, in the matching conditions (6.19) at $x=a_2$ and $x=a_3$, and the edge conditions (6.2), we arrive at a system of twelve linear equations with zero right-hand sides for the eight coefficients $A_{nj}$, $B_{nj}$, $C_{nj}$, $D_{nj}$, where $j=1,3$, in (6.23), and four coefficients $L_{n1}$, $L_{n2}$, $L_{n3}$, $L_{n4}$ in (3.1). A non-zero solutions of this system exist only if the determinant of the system is equal to zero, which gives the equation for the spectral parameter $\lambda _n$. Then the system is solved for each root of the determinant, $\lambda _n$, with the solution being uniquely determined using the orthogonality condition (6.7).

To calculate the added-mass matrix $\boldsymbol{\mathsf{S}}$ for the complex plate (6.26), the dry modes, $\psi _n(x)$, are presented as superposition of the dry modes, $\bar \psi _k(x)$, of the plate with a constant thickness by formulae (6.14), (6.15). The coefficients (6.15) in the superposition (6.14) make a matrix $\boldsymbol{\mathsf{C}}$, which together with the added-mass matrix $\bar {\boldsymbol{\mathsf{S}}}$ of the plate constant thickness $h_c$, provide the added-mass matrix of the plate (6.26) by formula (5.8). Next the dimensional frequencies $\varOmega _k$ of the complex floating plate, and the vector of the coefficients $\boldsymbol {W_k}$ in the series (2.11) representing $k$th wet mode $W_k(x)$ as a series of dry modes $\psi _k(x)$, are obtained by solving the algebraic system (6.16).

The ratios of wet, $\omega _n$, and dry, $\omega _n^{(d)}$, elastic natural frequencies of the plate with thickness (6.26), which are defined by (6.17), are shown in figure 12(a,b) for $3\le n\le 20$, three different cases from table 5, and $\alpha =0.1$ in (a) and $\alpha =0.01$ in (b). Calculations are performed with 40 terms in series (6.14). Correspondingly, the matrices in (6.16) are $40\times 40$. It is seen that the ratios $\omega _n/\omega _n^{(d)}$ are close to each other, even the plate thicknesses in cases I, II and III are rather different. Figure 12(c,d) present the relative differences $\Delta \omega _n$ of the ratios $\omega _n/\omega _n^{(d)}$ for the plate (6.26) and for the plate with constant thickness $h_c$. The differences $\Delta \omega _n$, $n\ge 3$, are defined by (5.27). The differences $\Delta \omega _n$ are smaller than 10 %. They decay with increase of $n$.

Figure 12. The ratios of the wet and dry frequencies for cases I, II and III and $\alpha =0.1$ (a) and $\alpha =0.01$(b). Black circles are for the plate with constant mean thickness $h_c$. Relative differences $\Delta \omega _n$ for the same cases and for $\alpha =0.1$ (c) and $\alpha =0.01$ (d).

The solutions of (6.16) for $3\le k \le 12$ are listed in table 6 for case III and $\alpha =0.1$. Red entries in this table are for $W_{kk}$ which are set to one. It is seen that the wet modes $W_k(x)$ have approximately the same shapes as the corresponding dry modes $\psi _k(x)$. Contributions of other dry modes are smaller than 8 %.

Table 6. The coefficients $W_{kn}$ in the series (6.14) of the $k$th wet mode $\psi _k(x)$ as a superposition of the dry modes $\bar {\psi }_n(x)$ for case III and $\alpha =0.1$.