1 Introduction

Numerous hypothetical investigations have been performed to dissect the diffraction and radiation of water waves on a structure. Bharatkumar et al. [1] computed the first order forces and pressure on a pair of circular cylinders in a channel by using the Green’s function method and the method of images. Bhatta and Rahman [2] solved the problem of diffraction and radiation to the water waves by a circular cylinder in uniform water depth. Bhattacharjee and Soares [4] calculated wave induced force acting on a floating rectangular structure placed in front a vertical wall in water of step type bottom. Buffer and Thomas [3] used multipole method to compute reflection and transmission coefficients for an array of cylinders in a channel. Hassan and Bora [5, 6] analyzed the diffraction problem by a couple of vertical cylinders. They evaluated the forces acting on the cylinders for various draft and radii of the cylinders. Kashiwagi [7] used three dimensional Green’s function method to solve the problem of water waves by offshore structure in a channel. Kashiwagi [7] has also processed mean second-order drift forces on four truncated cylinders orchestrated in a square and contrasted his outcomes and exploratory information. Linton [8], Linton and Evans [9] and Mclver and Bennett [12] applied multipole expansions method to solve the diffraction along with radiation problems of water wave by a cylinder in a channel. Martins-rivas and Mei [10] given linearized theory of an oscillating water column (OWC) on a straight coast. They evaluated the coefficients of apparent mass and radiation damping, and the chamber pressure. Martins-rivas and Mei [11] given the theoretically investigation of a single oscillating water column (OWC) situated at the tip of a long and thin breakwater. Neelamani et al. [14] approach the same problem who compared the experimental results with the theoretical results for a specific case of two circular cylinder. Thomas [15] utilized the image method to solve scattering and radiation problems for a single vertical circular cylinder in a channel. Thorne [16] derived multipoles in two and three dimensions which allow the clear arrangement of numerous issues. MacCamy and Fuchs [13] solved the diffraction problem by a vertical cylinder of discretionary water depth and derived analytical solution of diffracted potential. This solution is as yet utilized as a building configuration code. Wu et al. [17, 18] discussed the hydrodynamic coefficients and wave exciting force on a float over an arched body whose radius was smaller or larger than that of the float. Yeung and Sphaier [19] used the image method to solve scattering and radiation problems for a vertical circular cylinder in a channel, at times stretching out all through the whole water profundity. Zhu and Mitchell [20] solved the problem of diffraction of ocean wave by a hollow cylinder in an ocean of uniform depth and they used Galkerin’s method to find the analytical solution of this problem. Zheng et al. [21] investigated the fulfilment of an oscillating water column(OWC) over a vertical structure in uniform water depth. the have valodated their results and then model is used to study the effect of the thickness of the chamber wall and the radius.

The study of the diffraction problem by a fixed structure can provide fundamental information about wave force on the structure. Our present investigation is also dealing with the diffraction problem by cylinderical stuctures which are placed on the centreline between the channel walls. It is thus important to understand the elucidation of the results of the wave tank testing of the offshore structures. Also it is necessary to evaluate how the tank walls impacts on quantities such as hydrodynamic forces on a fixed structure. In this paper, we consider two coaxial cylindrical structures in a channel of finite width, the upper one is partially submerged floating hollow cylinder which can be considered as oscillating water column (OWC) and the lower one is solid bottom-mounted obstacle which can be assumed solid cylindrical structure. The method to solve this problem is based on the construction of suitable multipoles for the channel problem which was developed by Linton and Evans [9]. Thus, when we consider a channel of water of uniform depth H and width 2d of channel walls, then suitable multipoles satisfy the Laplace’s equation in our identified fluid region.

2 Mathematical model

We consider a pair of coaxial cylinders placed in an infinitely long channel of finite width 2d and depth of water is H. The pair of cylinders consists of one hollow cylinder which is partially merged in water and one solid cylinder which is bottom mounted. We also consider the radius of hollow cylinder is \(r_1\) and the radius of bottom-mounted solid cylinder is \({r_2} \, ({r_1}\le {r_2}< d)\). Cartesian axes \((x,\,y,\,z)\) are considered with x-axis taken parallel to the channel walls and \(z-\) axis taken vertically upward. The origin is chosen at O which lies on the centreline of the channel walls in the mean of the undisturbed free surface of water as shown in Fig. 1. The polar coordinates r and \(\theta\) are defined by

$$\begin{aligned} x=r\cos \theta \,\, \text {and}\,\, y=r\sin \theta . \end{aligned}$$

The regions are occupied by the hollow cylinder is given by \(r< {r_1}, \, 0\le \theta < 2\pi ,\, -e_1\le z\le 0\) and occupied by the bottom-mounted cylinder is given by \(r< {r_2}, \, 0\le \theta < 2\pi ,\, -{H}\le z\le -h_2\).

Fig. 1
figure 1

Schematic of the device

By considering the linear water wave theory, the time harmonic velocity potential \(\varPhi (x,\, y,\, z,\, t)\) can be written as

$$\begin{aligned} \varPhi (x,y,z,t)=Re[\phi (x,y,z)e^{-i\omega t}], \end{aligned}$$

where i is an imaginary unit which is defined by its property \(i=\sqrt{-1}\), t represents the time, \(\omega\) is the angular frequency whereas Re represents the real part of the complex quantity in bracket and \(\phi (x,y,z)\) be the spatial portion of the total potential satisfied to the following Laplace equation:

$$\begin{aligned} \dfrac{\partial ^2\phi }{\partial x^2}+\dfrac{\partial ^2\phi }{\partial y^2}+\dfrac{\partial ^2\phi }{\partial z^2}=0. \end{aligned}$$

Then the total velocity potential can be written as

$$\begin{aligned} \phi =\phi _{inc}+\phi _d, \end{aligned}$$

where \(\phi _{inc}\) and \(\phi _d\) are velocity potential due to incident wave and diffracted wave with the cylindrical structure, respectively. We divide the whole fluid region into three physical regions, namely \(I,\, II\) and III as indicated in Fig. 1. Let the velocity potentials in regions I, II and III are \(\phi _{d}^{I}\), \(\phi _{d}^{II}\) and \(\phi _{d}^{III}\), respectively.

Using separation of variable method, we obtain the vertical eigenfunctions for each region as shown in Fig. 1 and can be expressed as follows

$$\begin{aligned} f_m(z)&=M_m^{-1/2}{\cos } [\lambda _{m}(z+{H})],\,\,\text {for the region } I, \end{aligned}$$
$$\begin{aligned} g_m(z)&=N_m^{1/2}{\cos } [\alpha _{m}(z+h_2)], \,\,\text {for the region } II \text { and } III, \end{aligned}$$

where \(M_m=\dfrac{1}{2}\left( 1+\dfrac{\sin 2\lambda _m {H}}{2\lambda _m {H}}\right)\) and \(N_m=\dfrac{1}{2}\left( 1+\dfrac{\sin 2\alpha _mh_2}{2\alpha _mh_2}\right)\) and the eigenvalues \(\lambda _m\) and \(\alpha _m\) can be determined from the following dispersion relations

$$\begin{aligned}&\left\{ \begin{array}{l l} \lambda _{0}=-ik ,\quad \omega ^{2}=gk \tanh (k{H}) , &{} \quad m=0 \\ \omega ^{2}=- g \lambda _{m} \tan (\lambda _{m}{H}), &{} \quad \; m=1,2,3,\ldots , \\ \end{array} \right. \end{aligned}$$
$$\begin{aligned}&\left\{ \begin{array}{l l} \alpha _{0}=-ik' ,\quad \omega ^{2}=gk' \tanh (kh_{2}) , &{} \quad m=0 \\ \omega ^{2}=- g \alpha _{m} \tan (\alpha _{m}h_{2}), &{} \quad \; m=1,2,3,\ldots , \\ \end{array} \right. \end{aligned}$$

with k is the wave numbers in region I, \(k'\) is the wave number in II and III regions and the gravitational acceleration is g. Therefore the potential \(\phi\) describing the diffraction wave field which satisfy the following Helmholtz equation

$$\begin{aligned} \dfrac{\partial ^2\phi }{\partial x^2}+\dfrac{\partial ^2\phi }{\partial y^2}-\lambda _{m}^{2}\phi =0 \end{aligned}$$

in the fluid region I as shown in Fig. 1.

The analytical expression of potential due to incident wave with unit amplitude from [13]

$$\begin{aligned} \phi _{inc}=-\dfrac{ig\cosh k(z+{H})}{\omega \cosh (k{H})}\sum _{m=0}^{\infty }\mu _m J_m(kr)\cos m\theta , \end{aligned}$$

where Bessel function of the first kind is \(J_m(.)\), and \(\mu _m\) is written by

$$\begin{aligned} {\mu _m=} {\left\{ \begin{array}{ll} 1, &{} m=0\\ 2i^m, &{}m> 0 \end{array}\right. } \end{aligned}$$

3 The governing equation of the problem

The diffracted velocity potential \(\phi _d(x,y,z)\) fulfills the following governing equation and the boundary conditions

$$\begin{aligned}&\nabla ^2\phi _d=0, \nonumber \\&\quad (-{H}< z< 0,\, -\infty< x< \infty ,\, -d< y < d), \end{aligned}$$
$$\begin{aligned}&\frac{\partial \phi _d}{\partial z}-\frac{\omega ^2}{g}\phi _d=0; \,\,\,\; (z=0), \end{aligned}$$
$$\begin{aligned}&\frac{\partial \phi _d}{\partial z}= 0; \,\,\, \, (z=-{H},\, r\ge {r_2};\, z=-h_2,\, r\le {r_2}), \end{aligned}$$
$$\begin{aligned}&\frac{\partial (\phi _d+\phi _{inc})}{\partial r}= 0;\nonumber \\&\quad (-e_{1}< z< 0 ,\; r={r_1};\, -{H}< z< -h_2,\,r={r_2}), \end{aligned}$$
$$\frac{\partial \phi _d}{\partial y}= 0; \quad (y=\pm d),$$
$$\begin{aligned}&\lim _{r\rightarrow \infty } \sqrt{r} \left( \frac{\partial \phi _{d}}{\partial r}-ik\phi _{d} \right) = 0. \end{aligned}$$

3.1 Matching conditions

To proceed in order to determine the unknown coefficients of the expression of velocity potentials, we introduce the proper matching conditions to preserve continuity fluid motion along the seeming interface and cylinder’s boundaries as indicated in Fig. 1. Therefore, at \(r={r_2}\), we have

$$\begin{aligned}&\phi ^{I}_{d} = \phi ^{II}_{d} \,\,\,\, \,\,(-h_2\le z \le 0), \end{aligned}$$
$$\begin{aligned}&{\frac{\partial \phi ^{I}_{d}}{\partial r}=} {\left\{ \begin{array}{ll} -\frac{\partial \phi _{inc}}{\partial r} \,\,\,\,\, &{}(-{H}\le z\le -h_2 ),\\ \frac{\partial \phi ^{II}_{d}}{\partial r} \,\,\,\, &{}(-h_2\le z\le 0 ). \end{array}\right. } \end{aligned}$$

At \(r={r_1}\), we have

$$\begin{aligned}&\phi ^{II}_{d} = \phi ^{III}_{d} \,\,\,\, \,\,(-h_2\le z \le -e_1), \end{aligned}$$
$$\begin{aligned}&{\frac{\partial \phi ^{II}_{d}}{\partial r}=} {\left\{ \begin{array}{ll} \frac{\partial \phi ^{III}_{d}}{\partial r} \,\,\,\, &{}(-h_2\le z\le -e_1),\\ -\frac{\partial \phi _{inc}}{\partial r} \,\,\,\,\, &{}(-e_1\le z\le 0). \end{array}\right. } \end{aligned}$$

4 Solution to the problem

To find the analytical expression for the velocity potential in region I, we apply the channel multipoles method given by Linton and Evans [9] and to find the velocity potential for the regions II and III, we use separation of variables method. Hence, the solution of the boundary-value problem for different regions based on the result of Linton and Evans [9] and Wu et al. [17] are given by

$$\begin{aligned} \phi _d^{I}&=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _{q=0}^{\infty }f_m(z) A_{q,m}\left[ U_{n}(\lambda _m r)\delta _{qn}\right. \nonumber \\&\quad \left. +E(n,q;m) V_{n}(\lambda _m r)\right] \cos n\theta , \end{aligned}$$
$$\begin{aligned} \phi _d^{II}&=-\phi _{inc}+\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\left[ B_{n,m}S_n(\alpha _m r)\right. \nonumber \\&\quad \left. +C_{n,m}T_n(\alpha _mr)\right] g_m(z) \cos n\theta , \end{aligned}$$
$$\begin{aligned} \phi _d^{III}&=-\phi _{inc}+\sum _{n=0}^{\infty }\left[ D_{n,0}J_n(k'r)g_0(z)\right. \nonumber \\&\quad \left. +\sum _{m=1}^{\infty }D_{n,m}I_n(\alpha _mr)g_m(z)\right] \cos n\theta , \end{aligned}$$

where the eigenvalues \(\lambda _m\) and \(\alpha _m\) are given by Eqs. (7), (8) and \(A_{n,m}\), \(B_{n,m}\), \(C_{n,m}\) and \(D_{n,m}\) are the unknown coefficients which to be determined by applying appropriate matching conditions. The radial functions \(U_{n}(\lambda _m r)\), \(V_{n}(\lambda _m r)\), \(S_n(.)\) and \(T_n(.)\) are given by

$$\begin{aligned} U_{n}(\lambda _m r)= & {} H^{(1)}_n(kr), \nonumber \\ V_{n}(\lambda _m r)(r)= & {} J_n(kr)\,\,\,\text {for}\,\,m=0, \end{aligned}$$
$$\begin{aligned} U_{n}(\lambda _m r)= & {} K_n(\lambda _mr),\nonumber \\ V_{n}(\lambda _m r)= & {} I_n(\lambda _mr)\,\,\text {for}\,\, m=1,2,3,\ldots , \end{aligned}$$
$$\begin{aligned} S_{n}(\alpha _mr)= & {} H_n^{(1)}(k'r),\,\,\,\text {for}\,\,m=0, \end{aligned}$$
$$\begin{aligned} S_{n}(\alpha _mr)= & {} K_n(\alpha _mr),\,\,\text {for}\, \,m= 1,2,\ldots , \end{aligned}$$
$$\begin{aligned} T_{n}(\alpha _mr)= & {} H_n^{(2)}(k'r),\,\,\,\text {for}\,\,m=0, \end{aligned}$$
$$\begin{aligned} T_n(\alpha _mr)= & {} I_n(\alpha _mr),\,\,\text {for}\,\, m= 1,2,\ldots , \end{aligned}$$

where \(H^{(1)}_n(.)\)and \(H^{(2)}_n(.)\) are the Hankel functions of first kind and second kind of order n, respectively and the modified Bessel functions of first and second kind of order n are \(I_n(.)\) and \(K_n(.)\), respectively. Since based on the result of Linton and Evans [9] to construct the channel multipoles for the region I, we have the following functions

$$\begin{aligned} \zeta (z)= & {} {\left\{ \begin{array}{ll} -i{(1-z)}^{1/2}, &{} z\le 1\\ {(z^2-1)}^{1/2}, &{} z> 1, \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {a_{2n}(z)}= & {} {\left\{ \begin{array}{ll} \cos (2n\sin ^{-1}z), &{} z\le 1\\ {(-1)}^n\cosh (2n\cosh ^{-1}z), &{} z> 1, \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {a_{2n+1}(z)}= & {} {\left\{ \begin{array}{ll} \cos {(2n+1)\sin ^{-1}z}, &{} z\le 1\\ {i(-1)}^n\sinh [(2n+1)\cosh ^{-1}z], &{} z> 1, \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {b_{2n+1}(z)}= & {} {\left\{ \begin{array}{ll} \sin [(2n+1)\sin ^{-1}z], &{} z\le 1\\ {(-1)}^n\cosh [(2n+1)\cosh ^{-1}z], &{} z> 1. \end{array}\right. } \end{aligned}$$

Therefore the parameter E(., .; .) appeared in Eq. (22) is given by

$$\begin{aligned}&E(2p,2n;m)= {\left\{ \begin{aligned}& -\frac{2i\mu _p}{\pi }\int _{0}^{\infty }\frac{e^{-k\zeta d}a_{2p}(z)a_{2p}(z)}{\zeta \sinh k\zeta d}dz+\frac{\mu _p}{kd}\mathop {\sum }\nolimits _{j=0}^{l}\mu _jz_j^{-1}a_{2p}(z_j)a_{2n}(z_j),\,\,\,\,m=0\\& \mu _p \int _{1}^{\infty }\dfrac{e^{-\lambda _m dz}a_{2p}(z)a_{2n}(z)}{\zeta \sinh \lambda _m dz}dz,\,\,\, m\ge 1 \end{aligned}\right. } \end{aligned}$$
$$\begin{aligned}&E(2p+1,2n+1;m)= {\left\{ \begin{aligned}& -\dfrac{4i}{\pi }\int _{0}^{\infty }\dfrac{e^{-k\zeta d}b_{2p+1}(z)b_{2n+1}(z)}{\zeta \sinh k\zeta d}dz+\dfrac{2}{kd}\mathop {\sum }\nolimits _{j=0}^{l}\mu _jz_j^{-1}b_{2p+1}(z_j)b_{2n+1}(z_j),\,\,\,\,m=0\\& 2 \int _{1}^{\infty }\dfrac{e^{-\lambda _m dz}a_{2p+1}(z)a_{2n+1}(z)}{\zeta \sinh \lambda _m dz}dz,\,\,\, m\ge 1 \end{aligned}\right. } \end{aligned}$$


$$\begin{aligned} E(2p,2n+1;m)=E(2p+1,2n;m)=0\,\, \text {for all}\, \, m \end{aligned}$$

where the integral for \(m=0\) is taken to be a principal value of integral for all the singularities which satisfies the Helmholtz equation (9) and also integrand considered as a function of complex variable z, has simple poles at \(k\zeta d=\pm j \pi i,\, j=0,1,2,\ldots ,\) i.e. at \(z=\pm z_j\) we have

$$\begin{aligned} z_j&={(1-(j\pi /kd)^2)}^{1/2},\,\, j=0,1,2,\ldots ,l\\ z_j&= i{((j\pi /kd)^2-1)}^{1/2},\,\, j\ge l+1 \end{aligned}$$

where \(l\pi<kd<(l+1)\pi\).

5 Wave force

The dynamic fluid pressure in terms of velocity potential can be derived from Bernoulli’s equation of continuity which is given by

$$\begin{aligned} P(r,\,\theta ,\, z,\, t)= & {} -\rho \dfrac{\partial \varPhi (r,\,\theta ,\, z,\, t)}{\partial t}, \end{aligned}$$

where \(\rho\) is the uniform fluid density. Since wave exciting forces are commonly corresponding to the consolidated activity of incident potential \(\phi _{inc}\) and diffraction’s potential \(\phi _d\). Suppose \(F_i\) and \(F_d\) are the horizontal force due to the incoming and diffraction potentials, respectively. Hence the total force denoted by \(F_s\) and given by

$$\begin{aligned} F_{s}= & {} F_{i}+F_{d}. \end{aligned}$$

Now the horizontal force component \(F_{i1}\) corresponding to incident potential and \(F_{d2}\) due to diffracted potential acting on the upper cylinder along the x-direction are obtained as

$$\begin{aligned} F_{i1}= & {} i\rho \omega \int _{0}^{2\pi }\int _{-e_1}^{0}\phi _{inc}({r_1},\,\theta ,\,z){r_1}\cos \theta d\theta dz, \end{aligned}$$
$$\begin{aligned} \text {or}\,\,\, F_{i1}= & {} -\dfrac{2\pi i\rho g{r_1}J_1(k{r_1})}{\cosh (k{H})}\dfrac{\sinh (k{H})-\sinh [k({H}-e_1)]}{k} \end{aligned}$$


$$\begin{aligned} F_{d2}= & {} i\rho \omega \int _{0}^{2\pi }\int _{-e_1}^{0}\phi _{d}^{II}({r_1},\,\theta ,\,z) {r_1}\cos \theta d\theta dz, \end{aligned}$$
$$\begin{aligned}&\text {or}\,\,\, F_{d2}=\dfrac{2\pi i\rho g{r_1}J_1(k{r_1})}{\cosh (k{H})}\dfrac{\sinh (k{H})-\sinh [k({H}-e_1)]}{k}\nonumber \\&\quad -i\rho \pi \omega {r_1}\sum _{m=0}^{\infty }N_{m}^{-1/2} \times \nonumber \\&\left[ B_{1,m}S_1(\alpha _m {r_1})+C_{1,m}T_1(\alpha _m {r_1}) \right] \dfrac{\sin (\alpha _m h_2)-\sin (\alpha _m e_2)}{\alpha _m}, \end{aligned}$$

where \(e_2=h_2-e_1\) i.e. gap between the cylinders as shown in Fig. 1. Now by using Eqs. (38), (40) and (42), we can derived the total horizontal force \(F_{s1}\) exerted by the upper cylinder and it can be given by

$$\begin{aligned} F_{s1}&=-i\rho \pi \omega r_1\sum _{m=0}^{\infty }N_{m}^{-1/2}\left[ B_{1,m}S_1(\alpha _m {r_1})\right. \nonumber \\&\quad \left. +C_{1,m}T_1(\alpha _m {r_1}) \right] \dfrac{\sin (\alpha _m h_2)-\sin (\alpha _m e_2)}{\alpha _m}. \end{aligned}$$

The non-dimensional horizontal force exerted by the floating cylinder can be obtained divide by \({u_0}=\rho g \pi {{r_1}}^2\) to \(F_{s1}\) and it is given by

$$\begin{aligned} F_{s1}/{u_0}&=\dfrac{-i\omega }{g{r_1}}\sum _{m=0}^{\infty }N_{m}^{-1/2}\left[ B_{1,m}S_1(\alpha _m {r_1})\right. \nonumber \\&\quad \left. +C_{1,m}T_1(\alpha _m {r_1}) \right] \dfrac{\sin (\alpha _m h_2)-\sin (\alpha _m e_2)}{\alpha _m}. \end{aligned}$$

The exciting force \(F_{d3}\) acting on the inner wall of the upper cylinder due to the diffracted potential \(\phi _{d}^{III}\) in region III can be expressed as

$$\begin{aligned}&F_{d3}=\dfrac{2\pi i\rho gr_1 J_1(kr_1)}{\cosh (kH)}\dfrac{\sinh (kH)-\sinh [k(H-e_1)]}{k}\nonumber \\&\quad -i\rho \pi \omega r_1\sum _{m=0}^{\infty }N_{m}^{-1/2} \nonumber \\&\quad \times D_{1,m}I_{1}(\alpha _m r_1)\dfrac{\sin (\alpha _m h_2)-\sin (\alpha _m e_2)}{\alpha _m}. \end{aligned}$$

Similarly, we derived the expression for the total horizontal force \(F_{s2}\) exerted by the bottom-mounted cylinder due to diffracted velocity potential \(\phi _{d}^{I}\) and incident velocity potential \(\phi _{inc}\) in region I and it can written as

$$\begin{aligned} F_{s2}&=-i\rho \pi \omega {r_2}\sum _{m=0}^{\infty }\sum _{q=0}^{\infty }A_{q,m}M_{m}^{-1/2}\left[ U_{1,m}(\lambda _m {r_2})\delta _{q,1}\right. \nonumber \\&\qquad \left. +E(1,q;m)V_{1,m}(\lambda _m {r_2})\right] \times \nonumber \\&\quad \dfrac{\sin \lambda _m({H}-h_2)}{\lambda _m}. \end{aligned}$$

In order to make it non-dimensionalized, we divide by \({u_1}=\rho g \pi {r_{2}^{2}}\) to \(F_{s2}\), which gives

$$\begin{aligned}&F_{s2}/{u_1}=\dfrac{-i \omega }{g {r_2}}\sum _{m=0}^{\infty }\sum _{q=0}^{\infty }A_{q,m}M_{m}^{-1/2}\left[ U_{1,m}(\lambda _m {r_2})\delta _{q,1}\right. \nonumber \\&\left. \quad +E(1,q;m)V_{1,m}(\lambda _m {r_2})\right] \nonumber \\&\qquad \times \dfrac{\sin \lambda _m({H}-h_2)}{\lambda _m}. \end{aligned}$$

6 Numerical results and conclusion

From the above discussions, we have seen that to get horizontal exciting forces, we have to evaluate the associated unknown coefficients appearing in the expressions of potentials given by the Eq. (2224) with the help of matching conditions (1821). As every expression of potential is found to be a series with infinite terms, therefore, calculate the corresponding values, we must be truncated appropriately after some terms. For our convenience, let us truncate all the infinite series into a limited number of terms by assuming \(M=30\). Proceeding in this way, we arrived at a system of simultaneous linear equations with aforesaid unknown coefficients. Then the solutions of these unknown coefficients corresponding to above system of linear equations are obtained by using MATLAB programming. Once found the unknown coefficients, then the velocity potentials allow to find the exciting forces exerted by the cylinder. We discussed the influence of the various parameters like radius, draft of cylinder (submerged part), the gap between the cylinders and mainly width of the channel walls on the exciting forces. Let us consider the values of the parameters throughout our calculations as \({H}=3\) m, \(g=9.807\) m/s\(^2\) and \(e_2=0.25{H}\). For fixed \({r_1}=0.2{H},\) we proceed to study the effect on the horizontal exciting force of the floating cylinder due to the radius of the lower cylinder. In Fig. 2, we plot \(F_{s1}/{u_0}\) against \(k{r_1}\) taking distinct values of radius to the bottom-mounted cylinder say \({r_2}=0.2{H},\, 0.3{H},\,0.4{H},\,0.5{H},\,0.6{H}\). From Fig. 2, we saw that no oscillation happens for those value of \({r_2}\) when \({r_2}>\dfrac{1}{2}d\). Also we observed oscillation for higher value of \({r_2}\), e.g. \({r_2}=0.2{H}\) and 0.3H in which case \({r_2}\le \dfrac{1}{2}d\). It is likewise seen that the oscillation, at whatever point they happen, just at lower values of k. Again when \(r_2\le \dfrac{1}{2}d\), it is seen that peak value of exciting force occurs at more than one points. For all values of \({r_2}\), the exciting force attains its most extreme value only at lower range of k. For higher value of k, the force relentlessly lesser achieving right around zero. It is to be noted that when the radius of lower cylinder increases, the force exerted by cylinder is decreases. Next we investigate the effect of width of the channel walls on the exciting forces on the upper and lower cylinders. Here we fixed the parameters \(r_2,\,r_1\) and \(e_1\) by choosing \({r_2}=0.4{H},\,{r_1}=0.2{H}\) and \(e_1=0.1{H}\). In Figs. 3 and 4, we plot \(F_{s1}/{u_0}\) and \(F_{s2}/{u_1}\) against \(k{r_2}\), respectively, for different values of width of channel walls, \(d=0.5{H},\,0.6{H},\,0.7{H}\). From Fig. 3, we observed that the values of the exciting force increases with width of the channel walls decreases and Fig. 4 show that similar effect on exciting force on the upper cylinder. Remarkably both the exciting forces attains maximum value only for lower value of wave number k and forces are almost vanishing for higher value of k.

Fig. 2
figure 2

Variation of dimensionless horizontal exciting force \(F_{s1}/u_0\) versus dimensionless wave number \(kr_1\) for different radii of the lower cylinder \({r_2}\) with \(e_1=0.1{H}\), \({r_1}=0.2{H}\), \(d=0.8{H}\)

Fig. 3
figure 3

Variation of dimensionless horizontal exciting force \(F_{s1}/u_0\) versus dimensionless wave number \(kr_2\) for different width of the channel walls d with \({r_1}=0.2{H}\), \({r_2}=0.4{H}\), \({e_1}=0.1{H}\)

Fig. 4
figure 4

Variation of dimensionless horizontal exciting force \(F_{s2}/u_1\) versus dimensionless wave number \(kr_2\) for different width of the channel walls d with \({r_1}=0.2{H}\), \({r_2}=0.4{H}\), \({e_1}=0.1{H}\)

Now we determine the variation of exciting forces on the cylinders for different radius of the hollow cylinder. When we compute the exciting forces for different \(r_1\), we fixed the value of \({r_2},\,e_1\) and d by taking \({r_2}=0.4{H},\,e_1=0.1{H}\) and \(d=0.6{H}\). In Fig. 5, we plot \(F_{s2}/{u_1}\) against \(k{r_2}\) for distinct values of radius to the hollow cylinder, \({r_1}=0.2{H},\,0.3{H},\,0.4{H}\). Figure 5 shows that the forces increases with the decreasing radius \(r_1\). For \(r_1=0.2H\), the dimensionless force \(F_{s2}/u_1\) attains the maximum values equal to 0.251901. The highest values of the forces occurs only at values of lower range of k.

Fig. 5
figure 5

Variation of dimensionless horizontal exciting force \(F_{s2}/u_1\) versus dimensionless wave number \(kr_2\) for different radius of the hollow cylinder \({r_1}\) with \(e_1=0.1{H}\), \({r_2}=0.4{H}\), \(d=0.6{H}\)

Again for fixed \({r_2}\) by taking \({r_2}=0.4{H}\), now proceed to investigate how the exciting forces on the upper and lower cylinders are affected by the gap between the cylinders. In Figs. 6 and 7, we plot \(F_{s1}/{u_0}\) and \(F_{s2}/{u_1}\) against \(k{r_2}\), respectively, for different values of \(e_{2}\), say, \(e_2=0.22{H},\,0.20{H},\,0.15{H}\), i.e. for \(e_1=0.03{H},\,0.05{H},\,0.1{H}\), respectively. Figure 6 clearly shows that as the \(e_2\) decreases, the force increases only at lower values of k, i.e. the larger value of exciting force happen for \(e_2=0.15{H}\), i.e. \(e_1=0.1{H}\) and Fig. 7 shows that deviation of exciting force is very negligible for different gap between the cylinders.

Fig. 6
figure 6

Variation of dimensionless horizontal exciting force \(F_{s1}/u_0\) versus dimensionless wave number \(kr_2\) for different gap (\(e_2=h_2-e_1,\) for a fixed \(h_2\)) between the cylinders with \({r_1}=0.2{H}\), \({r_2}=0.4{H}\), \(d=0.6{H}\)

Fig. 7
figure 7

Variation of dimensionless horizontal exciting force \(F_{s2}/u_1\) versus dimensionless wave number \(kr_2\) for different gap (\(e_2=h_2-e_1,\) for a fixed \(h_2\)) between the cylinders with \({r_1}=0.2{H}\), \({r_2}=0.4{H}\), \(d=0.6{H}\)

Fig. 8
figure 8

Comparison of exciting force \(F_{s2}/u_1\) with result of Yeung and Sphaier [19] with \(r_2/H=0.1\), \(d/H=0.2\) and \(h_2/H=0.1\)

Table 1 Comparison of exciting force \(F_{s2}/u_1\) with numerical results of Yeung and Sphaier [19] along with the parameters \(r_2/H=0.1\), \(d/H=0.2\) and \(h_2/H=0.1\)

Now it is reasonable to compare our results with the results of Yeung and Sphaier [19] for the modulus of the non-dimensional horizontal exciting force. All the parameters considered in our problem are reconsidered according to the ones from Yeung and Sphaier’s work in order to have the same physical problem. So we consider the value of the draft \(e_1=0.0001\) in order to have a single structure as in Yeung and Sphaier [19]. The values of the other parameters are considered as: \(r_2/H=0.1\), \(d/H=0.2\) and \(h_2/H=0.1\) and draft of lower cylinder \(e_3=H-h_2\). In the Yeung and Sphaier [19] the parameters adw represent radius, a draft of the cylinder and the width of the channel wall, respectively. The comparison of both the results is shown in Fig. 8 and Fig. 8 shows good agreement between the results of the present work and Yeung and Sphaier [19]. Also Table 1 shows the numerical comparison between the results of present work and Yeung and Sphaier [19]. It shows good numerical comparison between both the results.