Minakshi Ghosh,Manomita Sahu and Dilip Das

Received:21 September 2021/Accepted:04 January 2022

?Harbin Engineering University and Springer-Verlag GmbH Germany,part of Springer Nature 2022

Abstract Using linear water wave theory, three-dimensional problems concerning the interaction of waves with spherical structures in a fluid which contains a three-layer fluid consisting of a layer of finite depth bounded above by freshwater of finite depth with free surface and below by an infinite layer of water of greater density are considered. In such a situation timeharmonic waves with a given frequency can propagate with three wavenumbers.The sphere is submerged in either of the three layers.Each problem is reduced to an infinite system of linear equations by employing the method of multipoles and the system of equations is solved numerically by standard technique. The hydrodynamic forces (vertical and horizontal forces) are obtained and depicted graphically against the wavenumber. 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 two-layer 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.

Keywords Three-layer fluid; Wave scattering; Submerged sphere; Hydrodynamic forces; Vertical and horizontal forces;Linear water wave theory;Density-stratified three-layer fluid;Submerged spherical structure;Underwater sphere

1 Introduction

The study of wave propagation problems concerning ful‐ly submerged or semi-immersed structures of spherical shape within the fluid has been essential and received im‐mense importance in the literature for their utilization as wave power devices or as a spherical hull in submerged ve‐hicles 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 summa‐tion of wave source potentials of three-dimension and har‐monic 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 hy‐drodynamic 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 radi‐ation 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 hydro‐dynamic concerns, analysing fixed rigid spherical struc‐tures.Das and Thakur(2013)analysed the problem of wa‐ter 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 meth‐od 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 im‐mersed 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 indus‐trial 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 re‐searches, 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 sur‐face was developed by Linton and McIver (1995), consid‐ering 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 re‐search, Cadby and Linton (2000) investigated the three-di‐mensional 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 struc‐ture, the ocean has three horizontal depth zones, namely,the mixed layer, pycnocline, and deep layer. Severe densi‐ty changes take place in pycnocline.The density gradients may occur by the gravitational settling of sediments or so‐lar heating of the surface water and also as a result of mini‐mal 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, under‐standing 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, re‐spectively. 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 realiza‐tion of the two-layer fluid model. There are published re‐search works in three-layer fluid with some fascinating re‐sults to understand wave interaction with different geomet‐rical configurations for some particular interests. The wa‐ter wave in a three-layer system containing rigid horizon‐tal walls above the top layer and below the bottom layer of fluid was investigated by Michallet and Dias (1999). Tay‐lor(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 dis‐cussed 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 para‐mount 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. Un‐derwater 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 ex‐ploration (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 pro‐posed controller. Freshly, Samayam et al. (2021) consid‐ered an oscillating sphere close to a plane boundary to in‐vestigate direct numerical simulation (DNS) of flow in‐duced by it.Most of their review reveals that ignoring the effects of density stratification,fluid is taken to be of uni‐form 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-dimen‐sional 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 scatter‐ing of water waves in a three-layer fluid in the presence of a submerged sphere in any one of the three layers is ex‐amined here.The focus is on the quantity of wave energy reflected and transmitted due to the obstruction of the in‐cident wave by the spherical structure and thus calculat‐ing the resultant vertical and horizontal forces.This infor‐mation is essential for the stable and efficient construc‐tion 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.Ap‐plying the structural boundary conditions of the surface of the submerged sphere,the problem is reduced to a sys‐tem of linear algebraic equations. These equations are truncated and simultaneously solved using numerical methods.Finally,the vertical and horizontal exciting forc‐es 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 up‐per layer are depicted graphically against the wave num‐ber in several figures, varying the submersion depth of the sphere.The curves are almost similar to those of Cad‐by and Linton (Cadby and Linton 2000) when the densi‐ties 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 super‐posed non-viscous incompressible fluids under the action of gravity and neglecting any effect due to surface tension at the interfaces.Handhare the depths of the upper and the middle layer respectively,while the lower layer is infi‐nitely deep. The densities of the upper, middle and lower layers areρ1,ρ2andρ3(ρ3>ρ2>ρ1)respectively. Carte‐sian co-ordinates are chosen such that (x,z) plane coin‐cides with the undisturbed interface between the middle and lower layer (ML). They- axis points vertically up‐wards withy= 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) andy=H+h( > 0) as the mean position of the linearized free surface.Under the usu‐al assumptions of linear water wave theory, a velocity po‐tential can be defined for waves in the form where?(x,y,z) is a complex valued potential function,ωis the angular frequency.

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

The potential in the upper fluid will be denoted by?Iand that in the middle and lower fluids by?II,?IIIrespec‐tively,?I,?IIand?IIIsatisfied Laplace's equation (cf. Das and Sahu 2021)

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

whereK=ω2g. The boundary conditions (2) and (4) are obtained from the continuity of normal velocity at the in‐terface between UM and ML respectively,while the condi‐tions (3) and (5) are obtained from the continuity of pres‐sure at the interface between UM and ML respectively.

Also,condition at large depth is

Now the total potential function can be decomposed in‐to two parts:

where?incis the incident wave potential function and?sis 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.With‐out loss of generality, it can be assumed that the incident wave is fromx=?∞so thatαinc= 0.

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

3 Scattering by a submerged sphere

The centre of the sphere of radiusais taken as(0,f,0)so that forf＜ 0 anda>f, the sphere is in the lower layer and forf> 0 anda＜ min(h?f,f),f＜h, the sphere is in the middle layer andf> 0 anda＜ 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 thatr=adenotes 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 atr= 0 isr?n?1P(cosθ)cosmα,n≥m≥0, wherePare associ‐ated Legendre functions. This has the integral representa‐tion,valid fory>f

whereJmare Bessel functions andR=(x2+z2)12. Let the multipole potentials?cosmα,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 asR→∞which is the radiation condition. The multipole potentials ?Inm ,??, ??are obtained as(cf.Das and Sahu 2021)

where

Since the equationH(k)= 0 has exactly three positive real rootsK,k1andk2(k2>k1) (say), the path of integra‐tion is indented to pass beneath the poles of the above three integrands atk=K,k=k1andk=k2.

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

asR→∞. HereEK,Ek1andEk2are the residues ofE(k)atk=K,k=k1andk=k2respectively,given by

Using the result

(14)can be expressed as

where

3.1.1 Incident wave train of wavenumberK

First we consider an incident plane wave of wave num‐berKand amplitudeAon the free surfacey=h+Hwhose potential can be expanded in spherical plolar co-or‐dinates and get

where?0= 1,?m= 2 form≥1.

For the scattering problems considered,we write

wherem1= max(m,1) and?mnis 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 al‐so the functions cosmαwe can derive an infinite system of equations for the sets of coefficientscmn,n≥m1for eachm≥0,which is

These system can be solved by truncation.

The hydrodynamic force on the body in theith mode of motion can be written asFi(t)= Re{fie?iωt}, wherefiis found by integrating the dynamic pressure times the appro‐priate component of the normal over the body surface. In other words,

whereSBis thebodyboundaryandniisthecomponent of theinward normaltothebody intheith modeofmotion.

Theverticaland horizontal exciting forces on the sphere,andcan beobtained as

and

These can be simplified using(31)withs= 1 giving

and

The constantscappearing in (34) andcappearing in(35)can be obtained numerically by solving the linear sys‐tem(31)after truncation.

3.1.2 Incident wave train of wavenumberkj,j=1,2

Now, we consider the case of an incident plane wave of amplitudeAon the interfacey=hfork1andy= 0 fork2and the wavenumberkjdescribed by

The analysis is very similar to that given above for an incident wave of wavenumberK. We use the same expan‐sionfor?Sasbefore,Equation (30), butdenotethe un‐known coefficients bydand weobtain the infinite sys‐tem of equations

for eachm≥0.

The expressions for the vertical and horizontal exciting forces are

and

The constantsdappearing in (38) anddappearing in(39)can be obtained numerically by solving the linear sys‐tem(37)after truncation.

3.1.3 Numerical results

To study the numerical results the density ratioss1ands2are both taken to be 0.95. Figures 2 to 7 depict the time-independent and non-dimensional vertical and hori‐zontal exciting forces on the sphere submerged in the lower layer, plotted against the wavenumberKafor the incident wave of wavenumbersK,k1andk2.We have cho‐senh/aandH/aas 2,when the sphere is submerged in the lower layer for various immersion depthsf/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 interfacey= 0, between the lower and the middle layer.The curves corresponding to other values off/arepresent the sphere submerged deeper below the in‐terface.It is noted that the forces in all of these figures in‐crease with the increase inKaand attain a maximum af‐ter which they decrease with further increase inKa.Natu‐rally, the forces increase as the submersion depth of the sphere decreases when the surface of the sphere comes nearer to the interfacey= 0(f/a=?1.1) and consequent‐ly, 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 Cad‐by and Linton (2000), where the fluid was considered of two layers.

Figure 2 Vertical forces f plotted against Ka in lower layer

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 considers1= 0.99, depth of the upper layer in two-layer fluid being 4,h/a= 2 andH/a= 2 in three layer fluid andf/a=?2 for both the cases.Thus it may be noted that fors1= 0.99, the density ratio of the upper and mid‐dle layer, then the density of the upper and middle layer are almost same and we see that the three-layer fluid be‐comes 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 val‐ues for a two-layer fluid.

Figure 3 Horizontal forcesplotted against Ka in lower layer

Figure 4 Vertical forcesplotted against Ka in lower layer

Figure 5 Horizontal forcesplotted against Ka in lower layer

Figure 6 Vertical forces plotted against Ka in lower layer

Figure 7 Horizontal forcesplotted against Ka in lower layer

Table 1 Vertical exciting forces for the sphere in lower layer fluid

Table 2 Horizontal exciting forces for the sphere in lower layer fluid

The vertical and horizontal exciting forces for an inci‐dent wave of wavenumberKare 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 wavenumberk1and Figures 6 and 7, depict the same respectively, for the incident wave of wavenumberk2, 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 aty=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 atk=K,k=k1andk=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 wavenumberK

An incident wave of wavenumberKon the free surface has the same form in the middle layer as in the lower layer given by (28). The total Potential?Scan be expanded us‐ing (30), but it now uses the multipole expansions devel‐oped for the middle layer, (48). Thus the coefficientscsatisfy 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 wavenumberkj,j=1,2

For eachm≥0 the coefficientsd, in the expansion?Ssatisfy the infinite system of equations

for eachm≥0.

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

and

The constantsd01appearing in (56) andd11appearing in(57)can be obtained numerically by solving the linear sys‐tem (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 spheref/a= 1.1,1.7,2.3 and 2.9 and the den‐sity ratioss1,s2are both taken to be 0.95. Here we have chosen,h/aandH/aboth as 4.In all the figures,the forces are higher when the surface of the sphere is closer to either interfacey= 0 or =h(f/a= 1.1,2.9). Figures 8 and 9 portray that the vertical and horizontal exciting forces as‐sociated with the incident wave of wavenumberKare of similar characteristics. Figures 10 and 11 depict the verti‐cal and horizontal exciting forces respectively,for the inci‐dent wave of wavenumberk1. The maximum of vertical exciting forces occurs at higher values ofKathan that of horizontal exciting forces. Again, in this case, the maxi‐mums 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 wavenumberk2, as shown in Figures 12 and 13.

Figure 8 Vertical forcesplotted against Ka in middle layer

Figure 9 Horizontal forcesplotted against Ka in middle layer

Figure 10 Vertical forcesplotted against Ka in middle layer

Figure 11 Horizontal forcesplotted against Ka in middle layer

Figure 12 Vertical forcesplotted against Ka in middle layer

Figure 13 Horizontal forcesplotted against Ka in middle layer

For the sphere submerged in the middle layer, all the forces increase with the increase inKa, and after attaining the maximum, they decrease with further increase inKa,but only some horizontal forces corresponding to particu‐lar submersion depths of the sphere,associated with the in‐cident wave of wavenumberk1, become zero which again increase and after reaching a local maximum, they de‐crease asKafurther increases. The forces for the sphere near to the interfacey= 0, between the lower and middle layer (f/a= 1.1), is the highest for the incident wave of wavenumberk1(Figures 10, 11), whereas those for the sphere near to the interfacey=h, between the upper and the middle layer(f/a= 2.9),is the highest for the incident wave of wavenumberk2(Figures 12,13)as it is associated with this interface. From Figures 12 and 13, we also ob‐serve that for the incident wave of wavenumberk2, the maximum of vertical and horizontal exciting forces occurs for larger waves (smaller wavenumbers) compared to those for the incident wave of wavenumberK(Figures 8,9),but compared to those for the incident wave of wavenumberk1(Figures 10, 11), the maximum of vertical exciting forces for the incident wave of wavenumberk2occur for slightly smaller values ofKaand those for horizontal exciting forc‐es occur for slightly larger values ofKa.

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 Linton 2000) and the present paper of three-layer fluid. For all data we considers1= 0.99, depth of the upper layer in two-layer fluid being 6,h/a= 3 andH/a= 3 in three-layer fluid andf/a= 1.7 for both the cases. Thus, it may be noted that fors1= 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 coin‐cide with the corresponding values for a two-layer fluid.

Table 3 Vertical exciting forces for the sphere in middle layer fluid

Table 4 Horizontal exciting forces for the sphere in middle 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 aty=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 atk=K,k=k1andk=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 wavenumberK

An incident wave of wavenumberKon the free surface has the same form in the upper layer as in the lower layer given by (28). The total Potential?Scan be expanded us‐ing (30), but it now uses the multipole expansions devel‐oped for the upper layer,(66).Thus the coefficientscsat‐isfy 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 wavenumberkj,j=1,2

For this problem?is given, in the upper fluid, byj= 1,2 where

The polar expansion of?is given by

For eachm≥0 the coefficientsd, in the expansion?Ssatisfy the infinite system of equations

for eachm≥0.

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

and

The constantsdappearing in (74) anddappearing in(75)can be obtained numerically by solving the linear sys‐tem (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 depthsf/a= 5.1,5.7,6 and 6.8, are depict‐ed by four curves for eachf/a.To analyze this case also,we have chosenh/aandH/aboth 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 inci‐dent wave of wavenumberK,k1andk2are shown by Figures 14, 15; 16, 17; and 18, 19 respectively. For the in‐cident wave of wavenumberK, the forces are higher when the sphere is near to the free surface(f/a= 6.8). Howev‐er, for the incident waves of wavenumbersk1andk2, the forces increase with the decrease inf/a. Hence, the forces are high when the surface of the submerged sphere comes closer to the interfacey=h(f/a= 5.1). Both the vertical and horizontal exciting forces are higher for the incident wave of wavenumberK(Figures 14,15)than those for the incident wave of wavenumbersk1(Figures 16, 17) andk2(Figures 18,19).For the incident wave of wavenumbersk1andk2, the vertical exciting forces may attain the maxi‐mum at a slightly smaller value ofKathan those of hori‐zontal exciting forces.

Figure 14 Vertical forcesplotted against Ka in upper layer

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 da‐ta we considers1= 0.99, depth of the upper layer in twolayer fluid being 6,h/a= 3 andH/a= 3 in three-layer flu‐id andf/a= 5.7 for both the cases. Thus, it may be noted that fors1= 0.99, the density ratio of the upper and mid‐dle layer,then the density of the upper and the middle lay‐er are almost same and we see that the three-layer fluid be‐comes two-layer fluid. For this case it is observed that from the Tables 5 and 6 the values of the vertical and hori‐zontal forces almost coincide with the corresponding val‐ues for a two-layer fluid.

Figure 15 Horizontal forcesplotted against Ka in upper layer

Figure 16 Vertical forcesplotted against Ka in upper layer

Figure 17 Horizontal forcesplotted against Ka in upper layer

Figure 18 Vertical forcesplotted against Ka in upper layer

Figure 19 Horizontal forcesplotted against Ka in upper layer

Table 5 Vertical exciting forces for the sphere in upper layer fluid

Table 6 Horizontal exciting forces for the sphere in upper layer fluid

In all three cases, it is noted that the maximum of the forces for the incident wave of wavenumbersk1andk2,oc‐cur for larger waves(smaller wavenumbers)than those for the incident wave of wavenumberK. Thus, it is observed that when the densities of the upper and middle layer are taken as almost the same,that is whens1is almost equal to 1, the three-layer fluid behaves similar to two-layer fluid with free surface and both the vertical and horizontal excit‐ing forces are somewhat similar to the vertical and hori‐zontal 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 be‐tween 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 dif‐ferent 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 ver‐tical 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 coin‐cide 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 ap‐proximately one, the solution agrees with the solution for the case of a two-layer fluid with a free surface.