1   Introduction

The study of wave propagation problems concerning fully submerged or semi-immersed structures of spherical shape within the fluid has been essential and received immense importance in the literature for their utilization as wave power devices or as a spherical hull in submerged vehicles and others. Havelock (1955) initiated the study of spheres in the fluid, who investigated the radiation by a half-immersed heaving sphere in deep water and solved the problem considering the velocity potential as a summation of wave source potentials of three-dimension and harmonic wave-free potentials taken in a linear combination.This method was used by Evans and Linton (1989) in the water of finite depth to study scattering and radiation by submerged horizontal cylindrical structure. Extending this in a similar approach, Linton (1991) dealt with the case of a submerged spherical object. The multipole expansion method has made a prime place in the study of various hydrodynamic characteristics of different geometries that are complaint to separable solutions of Laplace's equation.This method was practised initially by Ursell (1950) to study waves considering the presence of a submerged long horizontal circular cylinder in the water of infinite depth.Srokosz (1979) used the method of multipoles for the radiation problem of water waves by a sphere, considering it as a wave power absorber fully submerged in deep water under a free surface. Many researchers, including Das and Mandal (2008; 2010) have worked on problems of hydrodynamic concerns, analysing fixed rigid spherical structures. Das and Thakur (2013) analysed the problem of water wave scattering in the presence of a submerged sphere considering a thin ice-cover as an elastic plate in the water of uniform finite depth, and that in a two-layer fluid was investigated by Das and Thakur (2014), applying the method of multipoles. Such studies emphasize spherical shape devices of wave energy in oceans of polar regions. Most of the research works are concerned with completely immersed spherical objects. There are rare researchers on spheroids, a more complex geometry (cf. Wu and Taylor(1987; 1989), Chatjigeorgiou (2013), Chatjigeorgiou and Miloh (2015; 2017), and others.)

Geophysical flows of oceans and atmosphere have an essential feature of density inhomogeneity, due to which there can be significant effects in the dynamics of the flow. Density stratification in the fluid flows of nature and industrial processes are due to the differences in salinity, the concentration of several solutes, temperature, and their combination. Estuaries or fjords have fresh river water flowing over oceanic saline water. Even though almost all fluids in the earth are stratified, the effects of density strati-fication of seawater were not considered in the initial researches, and an assumption of fluids of uniform density was usual. A general theory of propagation of water waves in a density stratified fluid of two layers with a free surface was developed by Linton and McIver (1995), considering the presence of long horizontal cylindrical structures in either of the two layers. This study was motivated by the model of underwater pipe bridges across the stratified fluid of Norwegian fjords where the freshwater of around 10m depth flows over the saltwater. Extending this research, Cadby and Linton (2000) investigated the three-dimensional scattering and radiation problem in the presence of a submerged sphere in any of the two layers, using the multipole expansion method. Interesting flows of polar oceans were studied considering the more general class of problems of the density stratified two-layer fluid with ice cover or floating thin elastic plate (cf. Das (2008; 2015), Das and Mandal (2006; 2007)). In the same fluid structure, the radiation of water waves by a sphere was investigated by Das and Mandal (2010). Recently, Sahu and Das (2021) studied the hydrodynamic forces on a submerged circular cylinder in two-layer fluid with an ice-cover.

We also know that based on the layered density structure, the ocean has three horizontal depth zones, namely, the mixed layer, pycnocline, and deep layer. Severe density changes take place in pycnocline. The density gradients may occur by the gravitational settling of sediments or solar heating of the surface water and also as a result of minimal mixing forces of wind and wave action, which is more often in the summer months. In recent years, the study of stratified fluid dynamics has drawn more attention, understanding the vital effect of inhomogeneity of density in ocean engineering applications (cf. Liu et al. (2020); Wang et al. (2021) and others). In a three-layer fluid model with a free upper surface and two interfaces, the water wave propagates with three possible modes of linear water waves, each with different wavenumbers. This fact makes the model mathematically difficult to handle. Each of the three modes may correspond to the oscillations confined mainly to the upper, middle and lower layer of fluid, respectively. In interaction with the body in the wave field of stable but arbitrary density ratio, the wave energy may have a chance of transferring from one mode to another.Hence, this model is considered a more accurate realization of the two-layer fluid model. There are published research works in three-layer fluid with some fascinating results to understand wave interaction with different geometrical configurations for some particular interests. The water wave in a three-layer system containing rigid horizontal walls above the top layer and below the bottom layer of fluid was investigated by Michallet and Dias (1999). Taylor (1931) studied the linear stability of a three-layer fluid.The trapped modes of wave in such fluid in a channel with a fully immersed cylinder in the lower layer fluid was discussed by Chakrabarti et al. (2005). Also, Chen and Forbes(2008) studied steady periodic waves considering shear in the middle layer of the three-layer fluid. Problems of wave structure interaction in the three-layer fluid were discussed by Mondal and Sahoo (2014). Less work has been done in this regard. Das (2016) investigated the scattering of water waves by horizontal cylindrical structure in a three-layer fluid. Recently, oblique wave scattering in the three-layer fluid was studied by Das and Majumder (2020) using the method of multipoles expansion. Newly, Das and Sahu (2021) investigated wave radiation by a sphere in a threelayer fluid.

Solution of relevant fluid mechanics problems is paramount to understand flows of interest in oceanography and construct advanced necessary off-shore structures like submerged sphere-city, called ocean spiral. In Naval hydrodynamics, experimental and numerical tools are used to study the flow field around marine vessels. Underwater models of spherical robots are analysed for their various utilisation in ocean phenomena (cf. Amran and Isa 2020). Studies of autonomous underwater vehicles (AUVs) of spherical structure is of major use for mine exploration (cf. Fernandez et al. 2018 and others). Lately, Gu et al. (2021) used a heaving spherical wave energy converter (WEC) as a point absorber to test their proposed controller. Freshly, Samayam et al. (2021) considered an oscillating sphere close to a plane boundary to investigate direct numerical simulation (DNS) of flow induced by it. Most of their review reveals that ignoring the effects of density stratification, fluid is taken to be of uniform density. The submerged spherical structure in a three-layer fluid can be studied to inspect vital problems regarding interesting ocean hydrodynamic phenomena, making the investigation more realistic. This three-dimensional body can resemble spherical submarines, various wave energy devices, subsurface storage tanks, or fuel bladder of spherical geometric configuration in the ocean. Moreover, the spheroidal structures of vehicles fixed within the ocean are also being studied. The scattering of water waves in a three-layer fluid in the presence of a submerged sphere in any one of the three layers is examined here. The focus is on the quantity of wave energy reflected and transmitted due to the obstruction of the incident wave by the spherical structure and thus calculating the resultant vertical and horizontal forces. This information is essential for the stable and efficient construction of immersed bodies having sphere shapes in the ocean. The method of multipole expansions is employed to express the velocity potentials in spherical harmonics that describe the motion in either of the three layers. Applying the structural boundary conditions of the surface of the submerged sphere, the problem is reduced to a system of linear algebraic equations. These equations are truncated and simultaneously solved using numerical methods. Finally, the vertical and horizontal exciting forces on the sphere are obtained respectively for the heave and sway motions of the submerged body. These forces for the structure submerged in either lower, middle or upper layer are depicted graphically against the wave number in several figures, varying the submersion depth of the sphere. The curves are almost similar to those of Cadby and Linton (Cadby and Linton 2000) when the densities of the upper and the middle layer of the three-layer fluid are nearly equal, as the fluid represents only two layers.

2   Mathematical formulation

It is concerned with irrotational motion in three superposed non-viscous incompressible fluids under the action of gravity and neglecting any effect due to surface tension at the interfaces. H and h are the depths of the upper and the middle layer respectively, while the lower layer is infinitely deep. The densities of the upper, middle and lower layers are ρ₁, ρ₂ and ρ₃ (ρ₃ > ρ₂ > ρ₁) respectively. Cartesian co-ordinates are chosen such that (x, z) plane coincides with the undisturbed interface between the middle and lower layer (ML). The y- axis points vertically upwards with y = 0 as the mean position of the interface of ML, y = h(> 0) as the mean position of the interface of the upper and middle (UM) and y = H + h(> 0) as the mean position of the linearized free surface. Under the usual assumptions of linear water wave theory, a velocity potential can be defined for waves in the form

where ϕ (x, y, z) is a complex valued potential function, ω is the angular frequency.

The upper fluid, h < y < h + H, will be referred to as region Ⅰ, the middle fluid, 0 < y < h, will be referred to as region Ⅱ, while the lower fluid, y < 0, will be referred to as region Ⅲ (cf. Figure 1(a), 1(b), 1(c)).

Figure  1  Schematic diagrams of a sphere submerged in either layer of the three-layer fluid

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The potential in the upper fluid will be denoted by ϕ^Ⅰ and that in the middle and lower fluids by ϕ^Ⅱ, ϕ^Ⅲ respectively, ϕ^Ⅰ, ϕ^Ⅱ and ϕ^Ⅲ satisfied Laplace's equation (cf. Das and Sahu 2021)

Linearized boundary conditions on the interfaces and at the free surface are

where ${s_1} = \frac{{{\rho _1}}}{{{\rho _2}}} $ (< 1) and $ {s_2} = \frac{{{\rho _2}}}{{{\rho _3}}}$ (< 1)

where K = ω²/g. The boundary conditions (2) and (4) are obtained from the continuity of normal velocity at the interface between UM and ML respectively, while the conditions (3) and (5) are obtained from the continuity of pressure at the interface between UM and ML respectively.

Also, condition at large depth is

Now the total potential function can be decomposed into two parts:

where ϕ_inc is the incident wave potential function and ϕ_s is scattering potential function which must satisfy Equation (1) to (7) and also the body boundary condition

and behave as an outgoing wave far from the sphere. Without loss of generality, it can be assumed that the incident wave is from x =−∞ so that α_inc = 0.

3   Scattering by a submerged sphere

The centre of the sphere of radius a is taken as (0, f, 0) so that for f < 0 and a > f, the sphere is in the lower layer and for f > 0 and a < min (h − f, f), f < h, the sphere is in the middle layer and f > 0 and a < min(H + h − f, f), h < f < h + H, the sphere is in the upper layer. Polar co-ordinates (r, θ, α) are defined by

so that r = a denotes the surface of the sphere.

3.1   Sphere in the lower layer

A solution of Laplace's equation in the spherical polar co-ordinate system (r, θ, α) and singular at r = 0 is r^{− n − 1}P_n^m(cos θ) cos mα, n ≥ m ≥ 0, where P_n^m are associated Legendre functions. This has the integral representation, valid for y > f

where J_m are Bessel functions and R = (x² + z²)^1/2. Let the multipole potentials ϕ_n^jm cos mα, j = Ⅰ, Ⅱ, Ⅲ, m=0, 1 be the singular solutions of the Laplace's equation and satisfy (2)‒(6) and behave as outgoing waves as R → ∞ which is the radiation condition. The multipole potentials ϕ_n^Ⅰm, ϕ_n^Ⅱm, ϕ_n^Ⅲm are obtained as (cf. Das and Sahu 2021)

where

where $ {\sigma _1} = \frac{{1 + {s_1}}}{{1 - {s_1}}}, {\sigma _2} = \frac{{1 + {s_2}}}{{1 - {s_2}}}$

and

Since the equation H (k) = 0 has exactly three positive real roots K, k₁and k₂ (k₂ > k₁) (say), the path of integration is indented to pass beneath the poles of the above three integrands at k = K, k = k₁ and k = k₂.

The far-field forms of the multipoles, in the lower layer, is given by (cf. Das and Sahu 2021)

as R → ∞. Here E^K, E^k₁ and E^k₂ are the residues of E (k) at k = K, k = k₁ and k = k₂ respectively, given by

Using the result

(14) can be expressed as

where

3.1.1   Incident wave train of wavenumber K

First we consider an incident plane wave of wave num‐ ber K and amplitude A on the free surface y = h + H whose potential can be expanded in spherical plolar co-ordinates and get

where ϵ₀ = 1, ϵ_m = 2 for m ≥ 1.

For the scattering problems considered, we write

where m₁ = max (m, 1) and ϕ_n^m is given (in the lower fluid layer) by (26).

If we then apply the boundary condition (9) and use the orthogonality of the associated Legendre functions and also the functions cos mα we can derive an infinite system of equations for the sets of coefficients c_n^m, n ≥ m1 for each m ≥ 0, which is

These system can be solved by truncation.

The hydrodynamic force on the body in the ith mode of motion can be written as F_i(t) = Re { f_ie^{− iωt}}, where f_i is found by integrating the dynamic pressure times the appropriate component of the normal over the body surface. In other words,

where S_B is the body boundary and ni is the component of the inward normal to the body in the ith mode of motion.

The vertical and horizontal exciting forces on the sphere, f_K⁰ and f_K¹ can be obtained as

and

These can be simplified using (31) with s = 1 giving

and

The constants c₁⁰ appearing in (34) and c₁¹ appearing in (35) can be obtained numerically by solving the linear system (31) after truncation.

3.1.2   Incident wave train of wavenumber k_j, j=1, 2

Now, we consider the case of an incident plane wave of amplitude A on the interface y = h for k₁ and y = 0 for k₂ and the wavenumber k_j described by

The analysis is very similar to that given above for an incident wave of wavenumber K. We use the same expansion for ϕ_S as before, Equation (30), but denote the unknown coefficients by d_n^m and we obtain the infinite system of equations

for each m ≥ 0.

The expressions for the vertical and horizontal exciting forces are

and

The constants d₁⁰ appearing in (38) and d₁¹ appearing in (39) can be obtained numerically by solving the linear system (37) after truncation.

3.1.3   Numerical results

To study the numerical results the density ratios s₁ and s₂ are both taken to be 0.95. Figures 2 to 7 depict the time-independent and non-dimensional vertical and horizontal exciting forces on the sphere submerged in the lower layer, plotted against the wavenumber Ka for the incident wave of wavenumbers K, k₁ and k₂.We have chosen h/a and H/a as 2, when the sphere is submerged in the lower layer for various immersion depths f/a =− 1.1, − 1.5, − 2 and − 3, shown using four distinct curves. f/a =− 1.1 represents the immersion depth of the sphere almost close to the interface y = 0, between the lower and the middle layer. The curves corresponding to other values of f/a represent the sphere submerged deeper below the interface. It is noted that the forces in all of these figures increase with the increase in Ka and attain a maximum after which they decrease with further increase in Ka. Naturally, the forces increase as the submersion depth of the sphere decreases when the surface of the sphere comes nearer to the interface y = 0 (f/a =− 1.1) and consequently, to the free surface. The forces have similar behaviour as those of Cadby and Linton (2000), when the sphere is submerged in the lower layer, though here, in three-layer fluid the forces are higher and even the increase in forces becomes larger with the increase in submersion depths. Again, the range of forces is more here than that of Cadby and Linton (2000), where the fluid was considered of two layers.

Figure  2  Vertical forces f_K⁰ plotted against Ka in lower layer

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Figure  3  Horizontal forces f_K¹ plotted against Ka in lower layer

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Figure  4  Vertical forces $ \mathit{f}_{{\mathit{k}_{\rm{1}}}}^{\rm{0}}$ plotted against Ka in lower layer

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Figure  5  Horizontal forces $ \mathit{f}_{{\mathit{k}_{\rm{1}}}}^1$ plotted against Ka in lower layer

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Figure  6  Vertical forces $ \mathit{f}_{{\mathit{k}_2}}^0$ plotted against Ka in lower layer

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Figure  7  Horizontal forces $ \mathit{f}_{{\mathit{k}_2}}^1$ plotted against Ka in lower layer

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Also, the Tables 1 and 2 corresponding to heaving and swaying spheres show the values of vertical and horizontal forces in the two-layer fluid (paper of Cadby and Linton (2000)) and the present paper of three-layer fluid. For all data we consider s₁ = 0.99, depth of the upper layer in two-layer fluid being 4, h/a = 2 and H/a = 2 in three layer fluid and f/a =− 2 for both the cases. Thus it may be noted that for s₁ = 0.99, the density ratio of the upper and middle layer, then the density of the upper and middle layer are almost same and we see that the three-layer fluid becomes two-layer fluid. For this case it is observed that from the Tables 1 and 2 the values of the vertical and hori‐zontal forces almost coincide with the corresponding values for a two-layer fluid.

The vertical and horizontal exciting forces for an incident wave of wavenumber K are shown in Figures 2 and 3. They are very similar. Figures 4 and 5 depict the vertical and horizontal exciting forces respectively, for the incident wave of wavenumber k₁ and Figures 6 and 7, depict the same respectively, for the incident wave of wavenumber k₂, where in both the cases, vertical exciting forces are slightly greater than horizontal exciting forces. For Figures 4 and 5, the forces are much smaller than those of Figures 2 and 3, and the forces of Figures 6 and 7 are smaller than those of the previous figures.

3.2   Sphere in the middle layer

For the problem involving a sphere in the middle layer, one needs to construct the multipoles which are singular at y = f > 0. Suitable multipoles are obtained as (cf. Das and Sahu 2021)

where

and the path of integration is indented to pass beneath the poles of the above three integrands at k = K, k = k₁ and k = k₂.

The polar expansions of the multipoles, similar to the case when sphere is in the lower fluid, are

where

3.2.1   Incident wave train of wavenumber K

An incident wave of wavenumber K on the free surface has the same form in the middle layer as in the lower layer given by (28). The total Potential ϕ_S can be expanded using (30), but it now uses the multipole expansions developed for the middle layer, (48). Thus the coefficients c_n^msatisfy the infinite system of equations

and the non-dimensional vertical and horizontal forces for a sphere in the middle layer fluid through the equations

and

3.2.2   Incident wave train of wavenumber k_j, j=1, 2

For this problem ϕ_inc^Ⅱ is given, in the middle fluid, by−$ \frac{{igAK}}{{\omega {k_j}}}g_2^j\left( y \right){e^{i{k_j}\cos a}}, j = 1, 2, $ where

The polar expansion of ϕ_inc^Ⅱ is given by

where ϵ₀= 1, ϵ_m= 2 for m ≥ 1, where

For each m ≥ 0 the coefficients d_n^m, in the expansion ϕ_S satisfy the infinite system of equations

for each m ≥ 0.

Also, the expressions for the non-dimensional vertical and horizontal exciting forces are

and

The constants d₁⁰ appearing in (56) and d₁¹ appearing in (57) can be obtained numerically by solving the linear system (55) after truncation. Here the linear system (55) is truncated up to five terms. This provides an accuracy up to five decimal places, because if the system is truncated up to five or six terms, there is practically no change in the numerical results.

3.2.3   Numerical results

When the sphere is submerged in the middle layer, the vertical and horizontal exciting forces are represented with four curves corresponding to the various submersion depths of the sphere f/a = 1.1, 1.7, 2.3 and 2.9 and the density ratios s₁, s₂ are both taken to be 0.95. Here we have chosen, h/a and H/a both as 4. In all the figures, the forces are higher when the surface of the sphere is closer to either interface y = 0 or = h (f/a = 1.1, 2.9). Figures 8 and 9 portray that the vertical and horizontal exciting forces associated with the incident wave of wavenumber K are of similar characteristics. Figures 10 and 11 depict the vertical and horizontal exciting forces respectively, for the incident wave of wavenumber k₁. The maximum of vertical exciting forces occurs at higher values of Ka than that of horizontal exciting forces. Again, in this case, the maximums for vertical exciting forces are higher than those of horizontal exciting forces. This same nature is also noted for the vertical and horizontal exciting forces associated with the incident wave of wavenumber k₂, as shown in Figures 12 and 13.

Figure  8  Vertical forces f_K⁰ plotted against Ka in middle layer

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Figure  9  Horizontal forces f_K¹ plotted against Ka in middle layer

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Figure  10  Vertical forces $ f_{{k_1}}^0$ plotted against Ka in middle layer

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Figure  11  Horizontal forces $ f_{{k_1}}^1$plotted against Ka in middle layer

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Figure  12  Vertical forces $ f_{{k_2}}^0$ plotted against Ka in middle layer

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Figure  13  Horizontal forces $ f_{{k_2}}^1$ plotted against Ka in middle layer

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For the sphere submerged in the middle layer, all the forces increase with the increase in Ka, and after attaining the maximum, they decrease with further increase in Ka, but only some horizontal forces corresponding to particular submersion depths of the sphere, associated with the incident wave of wavenumber k₁, become zero which again increase and after reaching a local maximum, they decrease as Ka further increases. The forces for the sphere near to the interface y = 0, between the lower and middle layer (f/a = 1.1), is the highest for the incident wave of wavenumber k₁ (Figures 10, 11), whereas those for the sphere near to the interface y = h, between the upper and the middle layer (f/a = 2.9), is the highest for the incident wave of wavenumber k₂ (Figures 12, 13) as it is associated with this interface. From Figures 12 and 13, we also observe that for the incident wave of wavenumber k₂, the maximum of vertical and horizontal exciting forces occurs for larger waves (smaller wavenumbers) compared to those for the incident wave of wavenumber K (Figures 8, 9), but compared to those for the incident wave of wavenumber k₁ (Figures 10, 11), the maximum of vertical exciting forces for the incident wave of wavenumber k₂ occur for slightly smaller values of Ka and those for horizontal exciting forces occur for slightly larger values of Ka.

Also, the tables 3 and 4 corresponding to heaving and swaying spheres show the values of vertical and horizontal forces in the two-layer fluid (paper of Cadby and Linton2000) and the present paper of three-layer fluid. For all data we consider s₁ = 0.99, depth of the upper layer in two-layer fluid being 6, h /a = 3 and H /a = 3 in three-layer fluid and f /a = 1.7 for both the cases. Thus, it may be noted that for s₁ = 0.99, the density ratio of the upper and middle layer, then the density of the upper and the middle layer are almost same and we see that the three-layer fluid becomes two-layer fluid. For this case it is observed that from the tables 3 and 4 the values of the vertical and horizontal forces almost coincide with the corresponding values for a two-layer fluid.

3.3   Sphere in the upper layer

For the problem involving a sphere in the upper layer, one needs to construct the multipoles which are singular at y = f > 0. Suitable multipoles are obtained as (cf. Das and Sahu 2021)

where

and the path of integration is indented to pass beneath the poles of the above three integrands at k = K, k = k₁ and k = k₂.

The polar expansions of the multipoles, similar to the case when sphere is in the lower fluid, are

where

3.3.1   Incident wave train of wavenumber K

An incident wave of wavenumber K on the free surface has the same form in the upper layer as in the lower layer given by (28). The total Potential ϕ_S can be expanded using (30), but it now uses the multipole expansions developed for the upper layer, (66). Thus the coefficients c_n^m satisfy the infinite system of equations

and the non-dimensional vertical and horizontal forces for a sphere in the upper layer fluid through the equations

and

3.3.2   Incident wave train of wavenumber k_j, j=1, 2

For this problem ϕ_inc^Ⅰ is given, in the upper fluid, by $ -\frac{i g A K}{\omega k_{j}} g_{1}^{j}(y) e^{i k_{j} \cos \alpha}, j=1, 2$ where

For this problem ϕ_inc^Ⅰ is given, in the upper fluid, by

where ϵ₀= 1, ϵ_m= 2 for m ≥ 1, where

For each m ≥ 0 the coefficients d_n^m, in the expansion ϕ_S satisfy the infinite system of equations

for each m ≥ 0.

Also, the expressions for the non-dimensional vertical and horizontal exciting forces are

and

The constants d₁⁰ appearing in (74) and d₁¹ appearing in (75) can be obtained numerically by solving the linear system (73) after truncation. Here the linear system (73) is truncated up to five terms. This provides an accuracy up to five decimal places, because if the system is truncated up to five or six terms, there is practically no change in the numerical results.

3.3.3   Numerical results

The vertical and horizontal exciting forces on the sphere submerged in the upper layer of the three-layer fluid with the submersion depths f/a = 5.1, 5.7, 6 and 6.8, are depicted by four curves for each f/a. To analyze this case also, we have chosen h/a and H/a both as 4. It is observed that the maximum for vertical and horizontal exciting forces are similar for an incident wave of a particular wavenumber.

The vertical and horizontal exciting forces for the incident wave of wavenumber K, k₁ and k₂ are shown by Figures 14, 15; 16, 17; and 18, 19 respectively. For the incident wave of wavenumber K, the forces are higher when the sphere is near to the free surface (f/a = 6.8). However, for the incident waves of wavenumbers k₁ and k₂, the forces increase with the decrease in f/a. Hence, the forces are high when the surface of the submerged sphere comes closer to the interface y = h (f/a = 5.1). Both the vertical and horizontal exciting forces are higher for the incident wave of wavenumber K (Figures 14, 15) than those for the incident wave of wavenumbers k₁ (Figures 16, 17) and k₂ (Figures 18, 19). For the incident wave of wavenumbers k₁ and k₂, the vertical exciting forces may attain the maximum at a slightly smaller value of Ka than those of horizontal exciting forces.

Figure  14  Vertical forces f_K⁰ plotted against Ka in upper layer

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Figure  15  Horizontal forces f_K¹ plotted against Ka in upper layer

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Figure  16  Vertical forces $ f_{{k_1}}^0$ plotted against Ka in upper layer

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Figure  17  Horizontal forces $ f_{{k_1}}^1$ plotted against Ka in upper layer

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Figure  18  Vertical forces $ f_{{k_2}}^0$ plotted against Ka in upper laye

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Figure  19  Horizontal forces $ f_{{k_2}}^1$ plotted against Ka in upper layer

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Also, the Tables 5 and 6 corresponding to heaving and swaying spheres show the values of vertical and horizontal forces in the two-layer fluid (paper of Cadby and Linton 2000) and the present paper of three-layer fluid. For all data we consider s₁ = 0.99, depth of the upper layer in two layer fluid being 6, h /a = 3 and H /a = 3 in three-layer fluid and f /a = 5.7 for both the cases. Thus, it may be noted that for s₁ = 0.99, the density ratio of the upper and middle layer, then the density of the upper and the middle layer are almost same and we see that the three-layer fluid becomes two-layer fluid. For this case it is observed that from the Tables 5 and 6 the values of the vertical and horizontal forces almost coincide with the corresponding values for a two-layer fluid.

In all three cases, it is noted that the maximum of the forces for the incident wave of wavenumbers k₁ and k₂, occur for larger waves (smaller wavenumbers) than those for the incident wave of wavenumber K. Thus, it is observed that when the densities of the upper and middle layer are taken as almost the same, that is when s₁ is almost equal to 1, the three-layer fluid behaves similar to two-layer fluid with free surface and both the vertical and horizontal exciting forces are somewhat similar to the vertical and horizontal exciting forces as represented by Cadby and Linton (2000) for two-layer fluid. Here due to the presence of more layer of the fluids, the curves for both vertical and horizontal forces are somewhat different from the curves for the same in the upper layer cases in the two-layer fluid given by Cadby and Linton (2000). They are oscillatory in nature. This may be attributed to interaction of the boundary of the sphere, free surface and interfaces between upper and middle layer as well as middle and lower layer.

4   Conclusion

We have examined the interaction between the incident waves with the sphere submerged in either layer of a threelayer fluid. The middle layer is of finite depth and is bounded above by an upper layer of finite depth with free surface and the lower layer extends infinitely downwards. In such a situation propagating waves can exist at three different wavenumbers for any given frequency. The method of multipoles expansion is used to solve the scattering problems for the sphere situated entirely within either of the layer of three-layer fluid. Numerical results for the vertical and horizontal forces for the sphere are obtained. The hydrodynamic forces are depicted graphically against the wavenumber as a number of figures when the sphere is submerged in either of the layers. When the density ratio of the upper and middle layer is made to approximately one, curves for vertical and horizontal forces almost coincide with the corresponding curves for the case of a twolayer fluid with a free surface. This means that in the limit, the density ratio of the upper and middle layer goes to approximately one, the solution agrees with the solution for the case of a two-layer fluid with a free surface.