1 Introduction

Undertaking projects,such as the construction of underwater tubular bridges across several fjords and straits as a viable mean of improving/increasing modes of transportation(Friis et al.1991),construction of floating bridges or floating airports(Maniar and Newman 1997),require the investigation of various hydrodynamic phenomena that may arise.One such phenomenon is the occurrence of trapped waves:such waves have modes which are the finite-energy solution of the unforced problem.

在铣床上加工多孔专用夹具采用调整垫和紧固螺杆配合以及定位螺丝的配合调整工件的位置以达到工件加工的要求。

Trapped waves are of considerable importance in providing examples of discrete wave frequencies in the presence of a continuous spectrum.They are the non-trivial solutions of the homogenous boundary value problem and they represent fluid oscillations that are essentially confined to the vicinity of a structure.Their existence indicates that large-amplitude motions of the fluid and structure(s)are possible when the system is forced at a frequency close to that of the trapped mode.For a homogenous fluid,the topic of uniqueness and the existence of trapped modes have received a plenty of attention over several decades.A brief literature survey in this regard can be found in Kuznetsov et al.(1998,2002).Harter et al.(2007,2008)incorporated the effect of surface tension on trapped modes.They showed that its exclusion from the problem is not always justifiable since its inclusion in a particular submerged body problem changes the topological nature of the stream line pattern.

Since straits and fjords rarely contain a fluid of constant density,it seems that a more appropriate model for studies of the existence of trapped waves should perhaps include more than one fluid layer because the number and frequencies of such waves,if they exist in multi-layer cases,may depend on the number of such fluid layers and their densities.Since a two-layer fluid is the simplest representation of a multi-layer fluid,diffraction and radiation problems in a two-layer fluid have been successfully attempted to a reasonable extent but not much work has been accomplished with regard to the existence of trapped waves.The number of investigations carried out for trapped waves in three-layer fluids is even less.

Of late,there has been a significant progress on study of trapped waves in a two-layer fluid of finite and infinite depths.The first result about trapped modes in a two-layer fluid was obtained by Kuznetsov(1993)where he investigated the existence of trapped modes above a submerged cylinder in the lower layer of infinite depth.Using the density difference as a small parameter in a formal perturbation analysis and reducing the equations to a problem in the lower layer,he studied the existence of trapped modes on the free surface as well as at the interface between the two liquid layers.Later,Linton and Cadby(2003)computed the trapped mode frequencies for a horizontal circular cylinder submerged either in the upper layer or in the lower layer.They also considered the case of a pair of submerged identical circular cylinders in the lower layer and predicted the existence of trapped modes embedded in the continuous spectrum.Kuznetsov et al.(2003)provided examples of bodies which supported trapped modes in a two-layer fluid.The general sufficient condition for wave trapping in a two-layer fluid was derived by Nazarov and Videman(2009)in which it was shown in particular that a trapped mode always exists when the submerged body intersects neither the free surface nor the interface.Cal et al.(2012)examined the existence of trapped modesalong periodic structures in a two-layer fluid of finite depth.Nazarov et al.(2013)discussed the asymptotic behavior of trapped modes in two layer fluids,where density ratio limits were studied.The existence of trapped waves above a submerged horizontal circular cylinder placed in either of the layers of a two-layer fluid of finite depth with different upper and lower boundary conditions and inclusion of surface tension was studied under different set-up and conditions by Saha and Bora(2013,2014,2015a,b).Trapped mode frequencies were numerically computed by finding the zeros of a suitably truncated determinant.Wave trapping by porous structures in a two-layer fluid has also been studied by Behera et al.(2013).

However,not much progress has been made on analyzing the trapped waves in a multi-layer fluid.The sharp interface between the layers in all the above two-layer models is basically a crude representation of a smooth pycnocline that exists between fresh and salt water.However,a better model would perhaps involve replacing the sharp interface in the two-layer model by a layer of finite width in which the density either varies linearly between the upper and lower values or remains constant representing some sort of mean density of the middle layer.Sturova(1999)considered the same and solved the diffraction and radiation problem when a circular cylinder was located beneath the middle layer in which the density was constant.Even though this is a better model but one can approximately be the same by considering different but constant densities in each of the three layers.Such sharp density gradients can actually be generated in an ocean due to gravitational settling of sediments carried by fluids.This model can also replicate a fjord into which fresh river water flow s over oceanic water,which is more saline and consequently heavier.Taylor(1931)analyzed the linear stability of a three-layer fluid.Craik and Adam(1979)considered a three-layer fluid and studied the resonance in wave interaction when each layer had a uniform current speed.Chen and Forbes(2008)studied the steady periodic wave in a three-layer fluid due to the shear force in middle layer by assuming a uniform current speed in each layer.Mondal and Sahoo(2014)studied wave structure interaction in a three layer fluid of finite and infinite depth with an elastic plate covering the free surface.However,to the best of the know ledge of the authors,not much work has been done on trapped mode in a three-layer fluid.Chakrabarti et al.(2005)showed the existence of trapped mode by a submerged cylinder placed in the bottom layer of infinite depth by using perturbation method.They showed that this problem reduced to the ones involving a homogenous fluid of infinite depth and the two layer fluid(Kuznetsov 1993)in the limiting cases.

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Though in study of Chakrabarti et al.(2005),the authors showed the existence of trapped modes in a three-layer fluid,they did not provide any numerical justification as is available for a two-layer case.This motivates us to consider the same set-up and investigate the existence of trapped modes in a three-layer fluid by providing numerical evidence.In this article,a three-layer fluid is considered in which each layer has a distinct constant density.The upper and the middle layers are considered to be of finite depth while the lower layer is of infinite depth.The objective of the present work is to investigate the existence of trapped modes and the effect of the width of the middle layer when a submerged horizontal circular cylinder is placed in either the lower or the upper layer of the three-layer fluid.Finding of trapped modes is based on the multipole expansion method adopted by Ursell(1987)in which the singular solutions of the modified Helmholtz equation were modified to include all the prescribed boundary conditions.

The generalized problem considered here contains two parameters ρ and ρ∗ representing the ratios of the densities of the adjacent layers of the three-layer fluid under consideration.For the hydrodynamic set-up considered here,two different trapped waves developed:at the free surface corresponding to the lowest wave number and on the internal interfaces corresponding to the other two wave numbers.Two-layer fluid results can be recovered only by taking either ρ or ρ∗ to be zero depending on in which layer the cylinder is placed.But the same cannot be recovered when the limits ρ and ρ∗ tend to 1.Moreover,single-layer result is also not a possible outcome with the double limit of ρ and ρ∗ tending to 1.The effects of the width of the middle layer and the submergence depth on the dispersion curves and the trapped modes are observed.

2 Mathematical Formulation of the Problem

Under the usual assumptions of linear water wave theory,the proposed problem is considered in three dimensional Cartesian coordinate system with xy plane in the horizontal direction and the z-axis in the vertically upward direction.We consider a three-layer incompressible fluid in which the lower layer is of infinite depth.The topmost layer WI contains fresh water of density ρ1 and has depth h-d with z=h as the mean position of the free surface.The middle layer WII consists of saltwater of constant density ρ2 and has depth d ＞ 0 with z=d as the mean position of the interface between the middle and top layers.The bottom(lowermost)layer WIII contains muddy water of constant density ρ3 and is of infinite depth with z=0 as the mean position of the interface between the bottom and the middle layers(Fig.1).It is physically more acceptable if we consider the lower layer to be of finite depth.We had considered the same in Saha and Bora(2013,2014)and observed that its inclusion would not affect the nature of the trapped modes too much but rather increase the complexity of the problem.Further,in a three-layer problem,we are presently interested to work on the model of Chakrabarti et al.(2005).Therefore,we consider the lower layer to be of infinite depth.The effect due to surface tension at both the interfaces is neglected.

Let ΦI(x,y,z,t), ΦII(x,y,z,t),and ΦIII(x,y,z,t)be,respectively,the time dependent velocity potentials corresponding to the irrotational motion of the fresh,salty and muddy water.Then for oblique waves,they can be written in the form

where φI(x,z),φII(x,z),and φIII(x,z)are,respectively,complex-valued potential functions for the fresh,salty,and muddy water,and Re denotes the real part.Both the radian frequency ω and the wave number l along the y-direction are taken to be real and positive so that the solution stays bounded for all y and t.

The governing equation for the boundary value problems involving these spatial potentials φI(x,z),φII(x,z),and φIII(x,z)is the modified Helmholtz equation

Fig.1 Cross-sectional view of a three-layer fluid of infinite depth

Denoting the ratio of the densities of the fresh and salty water ρ1/ρ2(＜1)by ρ and that of the densities of the salty and muddy water ρ2/ρ3(＜1)by ρ∗,the linearized boundary conditions at the free surface and both the interfaces are

where K=ω2/g with g being the acceleration due to gravity.Equation(1)is the combined linearized dynamic and kinematic boundary condition on the free surface.Boundary conditions(2)and(3),respectively,represent the continuity of normal velocity and pressure at the upper interface z=d.Similarly,boundary conditions(4)and(5)arise due to the same reason at the lower interface z=0.

Since the lower layer is of infinite depth,therefore,the following limiting values hold:

Within this framework,we discuss the nature of dispersion relation and the condition on the roots of dispersion relation for which trapped modes exist.Hence we consider progressive waves or incident waves which take the form(up to an arbitrary multiplicative constant)

where

and we have either u=K or u satisfying the dispersion relation

w ith

For a fixed geometrical configuration and fixed values of both density ratios,Eq.(10)has exactly two positive real roots u1 and u2(u1＜u2,say)corresponding to a value of K.Hence,just as in Mondal and Sahoo(2014),there exist three wave numbers:u=K at the free surface;u=u1 and u=u2 at the interfaces.Detail discussion on these wave numbers can be found in Mondal and Sahoo(2014).If ρ=0,then either u-.,if the density ratio ρ equals zero,then the dispersion relation reduces to that of a two-layer free surface problem of infinite extent with d as the depth of upper layer in the former case or to a single-layer free surface problem with h-d as finitedepth in the latter case.If ρ∗=0,then

i.e.,if the density ratio ρ∗ equals zero,then the dispersion relation reduces to that of a two-layer free surface problem of finite depth with d as the depth of the lower layer and h-d as the depth of the upper layer.Similar dispersion relation has been derived in Linton and Cadby(2002).

For the existence of trapped modes,the following are required to be valid:

and hence,l is restricted to be in the range l＞u2＞ u1＞K which ensures that no wave propagation to infinity takes place at both the interfaces or near the free surface.Hence,l is the cut-off value for the trapped modes to exist and the values of the frequency corresponding to the wave numbers u1,u2,and K for the cut-off value l are the cut-off frequencies for the existence of trapped modes.A detailed discussion on the cut-off frequencies can be found in Mohapatra and Sahoo(2014)and Mohapatra and Soares(2016).

3 Solutions by Multipoles

A horizontal circular cylinder of radius a having its axis along z=f,|f|＞a,and its generator running parallel to the y-axis is placed in either the top layer or the lower layer of a three-layer fluid.For the cylinder to be totally submerged in the lower layer,we need f＜0.For that same cylinder to be totally submerged in the upper layer,we need f＞0 along with the constraintso that it does not touch the free surface as well as the interface between the upper and middle layers.Considering the origin of the rectangular Cartesian coordinates at the mean position of the axis of the cylinder,polar coordinates(r,θ)are defined as

3.1 Cylinder Submerged in the Bottom Layer

Symmetric multipoles ,about the line x=0 are defined,as detailed in Linton and Cadby(2002),by

where v=l cosh u and Kn(lr)are modified Bessel functions of the second kind of order n.

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With the help of the boundary conditions at the interfaces and the free surface,the coefficients AL(v),BL(v),CL(v),DL(v),and EL(v)appearing in Eqs.(12-14)are obtained as

where

It is to be noted that G(v)=0 is the dispersion relation in the variable v which is the same as in Eq.(10)and consequently v=u1 and v=u2 are simple zeros of G(v).Therefore,all the integrals in Eqs.(12-14)have simple poles at v=u1,v=u2,and v=K.But by using the trapped mode condition(11)and noting that v＞l,we get v＞l＞u2＞u1＞K which implies that there will be no singularities of the integrand on the real axis.

The total velocity potential can now be written as(Linton and Cadby 2003)

with

where Im(lr)are modified Bessel functions of the first kind of order mand

with∈0=1,∈n=2,n≥1

Applying the body boundary condition, on r=a,we obtain an infinite system of linear equations in the unknowns αn which is

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where′denotes differentiation with respect to r.

where

3.1.1 Results and Discussion

Figures 2,3,4,and 5 show the results obtained for trapped modes above a horizontal circular cylinder placed in the lowest layer of a three-layer fluid.In all the figures,when trapped modes are plotted for different depths d/a of the middle layer,the submergence depth f/a of the cylinder is taken as-1.01 which indicates that the cylinder is very close to the interface between the lower and middle layers.Itis already discussed in Nazarov and Videman(2009)that trapped mode would exist when the submerged body is located near the free surface or the interface.

第1天，将召开一个首次会议，受检方的主要人员参加。主检查员会给受检方一封认证通知函，告之检查的依据和检查员名单。之后，会要求受检方出具一个书面函件，确认本次认证的范围。同时，检查员出具另一个通知，要求受检方准备好所列目录中的资料，以便开展检查。

In all the figures,the curves with continuous lines give the first modes and the ones with dashed lines give the second modes.In Fig.6,the trapped mode wave numbers are shown against the density ratioρ∗when la=2,h/a,=3.1,f/a,=2.09,andρ=0.4.We consider the case when the cylinder is nearer to the free surface,i.e.,when(h-f)/a.The results are presented for four different values of the depth of the middle layer:d/a=0.7,0.8,0.9,and 1.0.Corresponding to each value of d/a,there are two curves showing the first and second modes(thin and dashed line).For the wave number Ka,with an increase in ρ∗,the first mode remains constant and the second mode decreases until it appears to nearly cross the first mode.With a further increase in ρ∗,the second mode ceases to exist and the first mode decreases to zero.Same behavior is observed for the case of the wave number u1a.But for the wave number u2a,with increase in ρ∗,both the modes increase to la=2,in the limit ρ∗ → 1,both the first and second modes for the wave number u2a terminate,which is different as compared to the two-layer fluid result by Linton and Cadby(2003)in which the second mode terminated but first mode existed with some finite value.Therefore,the result of infinite depth two-layer fluid can not be recovered in the limit ρ∗→ 1.But as expected,the result for the finite depth two-layer fluid with density ratio 0.4 when the cylinder is placed in the upper layer is recovered when ρ∗ =0(with the lower interface now playing the role of the rigid horizontal bottom).The total depth of the fluid is 2.1 and the submergence depth of the cylinder is 1.01,i.e.,nearer to the upper surface.These results are similar to the ones in Fig.5 in Linton and Cadby(2003).

In Fig.3,trapped mode wave numbers are plotted against the density ratio ρ∗ with ρ taken to be zero.With an increase in ρ∗,Ka decreases to zero in the same pattern as shown in Linton and Cad by(2003).For the first mode,as the depth of the middle layer increases,wave number u1a increases and wave number u2a decreases,and then they interchange their properties at near crossing points.Further,as ρ∗ → 1,u2a tends to some finite limit corresponding to each value of d/a,but u1a→0 and Ka→0.For the second mode also,the same features are observed as can be seen in Fig.3c.Trapped mode wave numbers for the case of the cylinder placed in the lower layer of an infinite depth two-layer fluid can be recovered by taking ρ=0(since the cylinder is placed in the lower layer).Before the near crossing point,for each depth d/a,of the middle layer,u1a will correspond to the two-layer fluid results and after that u2a will continue with the same value for both the first and second modes.

Fig.2 Dispersion curves for a cylinder of radius a placed in the lower layer for different depths d/a of the middle layer;f/a=-1.01,h/a=3,ρ=0.4,ρ∗=0.5

Fig.3 Trapped mode wave numbers plotted against ρ∗ for a cylinder of radius a in the lower fluid layer for different depths d/a of the middle layer;f/a=-1.01,h/a=3,ρ=0,la=2

In Fig.4,ρ=0.4 is considered and in this case second mode or the higher mode for the three wave numbers are observed only for values of ρ∗ ＞ 0.3.When ρ =0(Fig.3),it was observed that the first mode increased steadily to some fixed value as ρ∗ → 1 but in this case,it is observed that though both modes corresponding to the wave numbers u1a and u2a tend to some fixed values as ρ∗ → 1,they do not follow a fixed pattern as was observed in Fig.3.It is seen that asρ∗→1,the wave numbers Ka and u1a for both the first and second modes tend to 0;the wave number u2a tends to some finite limit for the first mode,and for the second mode it tends to la=2.

Thus,for the second mode,the rate of decay of the exponential term expdecreases as ρ∗ comes closer to unity and hence in the limit there exists no trapped mode.But the first mode will always exist for some finite value of u2a in the limit of ρ∗tending to 1.Figure 5 shows the variation of the first mode for all the wave numbers Ka,u1a,and u2a with varying submergence depth.W ith an increase in f/a,trapped mode wave numbers increase to la=2.

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3.2 Cylinder Submerged in the Upper Layer

For the case of cylinder placed in the upper layer,the velocity potential can be expanded exactly in the same manner as was done for the case with the cylinder in the lower layer.Now,the multipoles can be written as

该题学生的易错点为：1.题目中有两个质量、学生代数据时、容易张冠李戴，把水的质量和干木柴的质量混淆。2.对题目给的隐含条件没注意、有的学生可能得出水的末温是120益。我们都知道，水沸腾时，吸收热量，但温度保持不变。标准大气压下，水的最高温度为100益、不可能为120益。

For a fixed geometrical configuration and fixed density ratios,the problem of finding the trapped mode frequencies is completely specified by the two non-dimensional parameters Ka and la.For a fixed value of la,the parameter Ka is varied in such a way that the roots of the dispersion equation never cross la.With this variation of Ka,we locate the zeros of the truncated determinant.Then corresponding to those specific values of Ka,we can determine two wave numbers u1a and u2a by using dispersion relation(10).For the numerical evaluation of the zeros of the determinant,we truncate the system to a 32×32 system and the results presented in Section 3.1.1 are obtained correct up to three decimal places.

By applying the body boundary condition,we obtain a system of equations for βn identical to Eq.(20):

where G(v)is the same as in Eq.(16).Here,also,due to the trapped mode condition,there will be no singularities on the real axis.

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虽然疫苗是促进个体健康和减少疾病负担最有效的公共卫生措施之一，但没有任何一种疫苗、任何一种预防接种体系是绝对安全的。免疫规划作为一项复杂的系统工程，其工作的优劣受到社会各方面的影响，尤其与儿童家长的认知密切相关。文献[4-5]及课题组前期对家长疫苗知识获取途径的调查显示，专业机构医护人员是家长最希望和最信任的疫苗信息获取途径。国外也有研究表明，基层医务人员作为儿童接种疫苗的主要和可信赖的信息来源[6]，其对免疫接种的KAP对所在社区居民的免疫接种KAP会产生极大的影响。因此，了解其疫苗及安全接种的KAP，对提高儿童监护人预防接种的主动接受意识具有重要意义。

Here also,as in the previous case,by varying the frequencies Ka and fixing the other parameters,the zeros of the truncated determinant are conveniently located.It is already known that the values of those frequencies correspond to the trapped modes.The results presented in Section 3.2.1 are obtained correct up to three decimal places where a 32×32 system is used after truncating the system arising from Eq.(22).However,it is noticed that the first modes can be located even by considering a system of lesser order,say a 10×10 system.

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Limit as ρ∗→ 1:The dispersion relation(10)can also be written as

Figure 2 shows the trapped mode wave number u2a for density ratios ρ =0.4 and ρ∗ =0.5 which clearly implies that the fluid consists of three layers.The different curves correspond to the four different depths of the middle layer:d/a=0.5,1.0,1.5,2.0,and to each value of d/a there corresponds two curves.As discussed in the previous section,the existence of trapped modes requires the wave numbers to be less than the cut-off value,i.e.,u2＜l so that l=u2(the thin dashed line)gives an upper bound for these curves.For a fixed la within the range 0 to 4,there exist two values of the wave number Ka for which trapped mode exists(determinant of the truncated system vanishes).Using Eq.(10),the values of the other two wave numbers u1a and u2a can be found for the corresponding values of Ka.In this figure,we consider the dispersion curves for the wave number u2a.Interestingly,these curves are similar to those obtained for a two-layer fluid in Linton and Cadby(2003)even though a three-layer fluid is considered in the present case.This clearly indicates that the presence of the upper layer has no effect on the dispersion curves.It is observed that with an increase in the depth of the middle layer,these trapped mode frequencies decrease.It is also observed that the wave number decreases as ladecreases and converges in the lower range of la.

In Fig.7,we assign a non-zero value to the density ratio ρ∗by considering ρ∗ =0.5 and plot trapped mode wave numbers against the density ratio ρ.When f/a=2.09,i.e.,the cylinder is near the free surface,then there exist two trapped modes for the cases of all the three wave numbers.But if we move away from the free surface towards the interface,then the second mode terminates and only the first mode exists for all the cases.In the limit ρ → 1,Ka,u1a tend to zero and u2a tends to some finite non-zero limit.

Figure 8 shows the plots of trapped wave number against the density ratio ρ for the fixed values la=2,ρ∗ =0.5.In this case,h/a=3.1 and f/a=2.09 are considered such that the cylinder is very much nearer to the free surface.It is observed that near the free surface there exist s two modes and we want to study the behavior of these modes with change in the depth of the middle layer.For all values of depth of the middle layer,the second trapped mode for all the wave numbers start from ρ=0.2.Then as ρ increases,the curve behaves in a similar pattern as was in Fig.6,for the wave numbers Ka and u1a.But for the wave number u2a,with an increase in ρ,both modes come closer to each other at near crossing points.With further increase in ρ,the second mode terminates but the first mode tends to some finite limit.

3.3 Limiting Values of the Density Ratios

Limit asρ→1:The dispersion relation(10)can be written as:

Considering the limit ρ→1,we have σ→∞.Taking limit σ→∞on both sides of Eq.(23),we get

This gives the dispersion relation for a two-layer fluid with the lower layer of infinite depth and upper layer is of depth h.For a fixed value of Ka,the wave number ka=u1a can be determined from this relation.From Figs.7 and 8,we observe that in the limit ρ →1,the trapped mode frequency Ka and wave number u1a tend to 0.

The observation on trapped mode in a two-layer fluid of infinite depth made by Linton and Cadby(2003)is that for the density ratio 0.5,there exists one trapped mode(Fig.5;Linton and Cadby 2003).For that trapped mode,Ka has an approximate value of 0.5 and ka takes an approximate value of 1.40.Hence,results on trapped mode given in Linton and Cadby(2003)is not recovered from the result in a three-layer fluid in the limit ρ→1.

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3.2.1 Results and Discussion

Similarly,as before,taking the limit ρ∗→1,we get

Corresponding to a value of frequency Ka,we get wave number ka=u2a from this dispersion relation.For a cylinder in either fluid layer,the wave number u2a tends to some value(Figs.5 and 6)which is a trapped mode but certainly does not correspond to one of a two-layer fluid because Ka→0 in the limit.But for a density ratio 0.4 of a two-layer fluid of infinite depth,the value of Ka is not equal to zero for a cylinder placed in either of the layers(Figs.2 and 5 of Linton and Cadby(2003).

Fig.4 Trapped mode wave numbers plotted against ρ∗ for a cylinder of radius a in the lower fluid layer for different depths d/a of the middle layer;f/a=-1.01,h/a=3,ρ=0.4,la=2

Fig.5 Trapped mode wave numbers plotted against ρ∗ for a cylinder of radius a in the lower fluid layer for different submergence depths f/a;d/a=2.0,h/a=3,ρ=0.4,la=2

Fig.6 Trapped mode wave numbers plotted against ρ∗ for a cylinder of radius a in the upper fluid layer for different depths d/a of the middle layer;f/a=2.09,h/a=3.1,ρ=0.4,la=2

Double limit as ρ → 1 and ρ∗ → 1:When we take this double limit in the dispersion relation,we get K/u→0 and for a fixed value of l(＞u),we get K→0.Same observation follows from all the figures.And hence in the double limit to 1,it is not possible to recover the single-layer results.To explain analytically what actually takes place,we shall consider the boundary conditions in the limit as ρ→1 andρ∗→1 and K → 0 simultaneously.We introduce small parameters ε,δ,δ′such that δ≈ δ′and define

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In this limit,boundary conditions(1-5)become

The dispersion relation will have two positive solutions u1 and u2 which must satisfy

In the absence of any structure,oblique waves propagating in the fluid take the form

where

Fig.7 Trapped mode wave numbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different submergence depths f/a;d/a=1.0,h/a=3.1,ρ∗=0.5,la=2

Fig.8 Trapped mode wave numbers plotted against ρ for a cylinder of radius a in the upper fluid layer for different depths d/a of the middle layer;f/a=2.09,h/a=3.1,ρ∗=0.5,la=2

Thus,in the double limit for a fixed l,we obtain a boundary value problem in terms of the new spectral parameter K′.A fter finding the forms of the multipoles for this problem and computing the trapped mode frequencies the results match those found in the limit of Figs.5,6,7,and 8.Hence,for a fixed l,the trapped mode problem in the double limit ρ→1 and ρ∗→1 is related to the limits of the trapped mode curves in these figures.Thus,it is not possible to recover the single layer fluid results in the double limit.

4 Conclusions

A three-layer incompressible fluid is considered with the lowermost layer being of infinite depth and the other two layers of finite depth.Under the usual assumptions of linear water wave theory and by using a multipole expansion method,the existence of trapped mode is shown when a horizontal circular cylinder is placed in either the lowermost or the uppermost layer.This generalizes the results of Linton and Cadby(2003)from a two-layer case to a three-layer one.In the case of a two-layer fluid,we have seen that trapped waves correspond to two wave numbers but we now show in the three-layer case that trapped waves correspond to three wave numbers at the free surface and the internal interfaces.But similar to the two-layer case,it is observed here also that for each set,there exists two modes and with an increase in any of the two density ratios to 1,the second mode ceases to exist but the first mode does exist in nearly all the cases.The effects of the submergence depth and the depth of the middle layer on trapped modes are also observed.

然而，目前的文献多是纯粹的飞行器或无人潜艇的路径规划问题，缺乏对信息采集结合路径规划的研究。飞行器上搭载信息采集设备不仅需要飞行器在飞行过程中规避障碍物，还需要确保信息采集设备能够采集到相应的数据信息。

By considering the density ratio ρ=0,it is shown that the corresponding results for a two-layer fluid of infinite depth can be recovered when the cylinder is placed in the lower layer.Similarly for ρ∗ =0,the results of a two-layer fluid of finite depth is recovered when the cylinder is placed in the upper layer.A ll the trapped modes discussed above occur at wave numbers below a cut-off value la,i.e.,no wave propagates to infinity for the values of the wave numbers below this cut-off value.Above this cut-off value,it is possible to formulate a scattering problem corresponding to the interaction of an obliquely incident wave with the cylinder,but below it no waves can propagate into the far field.When either of the density ratios or both tend to 1,then it is not possible to recover either the two-layer or the single-layer case as is evident from the present investigation—contrary to what was concluded in Chakrabarti et al.(2005).That is,however small the width of the pycnocline may be,it still commands influence over the trapped modes and the three-layer case cannot be considered as equivalent to the two-layer case in the limit.Sharp and smooth pycnoclines are,respectively,simulated by two-layer and three-layer fluids.In the latter,the middle layer is linearly stratified,whereas the upper and lower layers are homogeneous.The middle layer with constant density,as considered in our problem,is basically a crude representation of a smooth pycnocline.

When either of the density ratios or both tend to 1,then we are not able to recover either the two-layer or the single-layer case,perhaps due to lack of high resolution computations of these limiting cases.These issues require further investigations.

With out doubt,some investigations can be carried out for a stratified ocean with a number of layers.But from practical point of view,two-layer or three-layer models seem to be more appropriate.It would be interesting to extend our problem to investigation of trapped waves at the coast or near breakwaters.Any feasible solution in this regard will throw light on a number of queries related to various aspects of the maintenance of the coast and/or the breakwaters.It will also probably be appropriate to carry out scientific study to check if the trapped waves in our problem can be connected to Kelvin and/or Ross by waves prevalent in coastal Engineering.

Acknowledgements The authors are grateful to the esteemed reviewers and the Editor-in-Chief for giving the chance to revise the manuscript which has definitely improved to a large extent after corrections were carried out taking into consideration of the insightful comments of the reviewers.

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