TWO-DIMENSIONAL SOURCE POTENTIALS IN A TWO-FLUID MEDIUM FOR THE MODIFIED HELMHOLTZ ’ S EQUATION

Velocity potentials describing the irrotational infinitesimal motion of two superposed inviscid and incompressible fluids under gravity with a horizontal plane of mean surface of separation, are derived due to a vertical line source present in either of the fluids, whose strength, besides being harmonic in time, varies sinusiodal(y along its length. The technique of deriving the potentials here is an extension of the technique used for the case of only time harmonic vertical line source. The present case is concerned with the two-dimensional modified Helmholtz's equation while the previous is concerned with the two-dimenslonal Laplace's equation.

etc.) or vertical plane barriers (el.Mandal and Goswaml [II], [12] [13]), half- immersed or fully submerged infinitely long circular cylinder (cf.Mandal and Goswami [14], Levine [15], by exploiting the geometry of the obstacles, the velocity potential can be as:;umed to have a harmonic variation in the lateral (z) direction, same as the incident wave field.
Thus the potential function satisfies a two-dimensional reduced Helmholtz's equation.Hence the problems are essentially boundary value problems (BVP) involving the He[mholtz's equation, and the construction of a two-dimenslonal source potential (the Green's function) is necessary to reduce the BVP's to equivalent integral ,equations.Both for infinite and finite constant depth of fluid, this source potential can be constructed by the method of Fourier transform (in x)(cf.Heins [9], Levlne [15], Miles [16] etc.) or by the method of separation of variables (cf. Rhodes-Roblnson [17] where the effect of surface tension of FS is included), thereby obtaining a linear combination of potentials due to the source in an unbounded fluid together with an 'image' potential in the FS boundary condition.
In the present paper we consider a two-fluid medium and derive velocity potential due to a vertical llne source present in either of the fluids whose strength varies harmonically with time and also with the co-ordlnate measured along its length.This is the same as deriving the source potentials in a two-fluld medium for the reduced two dimensional Helmholtz's equation.The corresponding problem for the two-dimensional Laplace's equation was considered in [4].When the strength of the llne source is made independent of the co-ordlnate along its length, known results for a two-fluid medium are recovered.
When the density of the upper fluid is made zero, the results derived here reduce to corresponding known results for an on-fluld medium.

STATEMENT OF THE PROBLEM.
We consider a two-fluid medium, both the fluids being incompressible and inviscid.
The mean SS is horizontal and taken as the xz-plane y-axis pointing vertically downwards.A line source is assumed to be present in either of the fluids and the y-axis is chosen to pass through the singular point so that the point of singularity is situated either at (O,q) or (0,-) ( > 0).The strength of the llne source is assumed to vary sinusoidally with time as well as with z.Let Pl, P2 be the densities of the lower and upper fluids respectively so that Pl > P2- The motion is assumed to be irrotational and is of small amplitude, and can be described, by velocity potentials Re {.(x,y,z) exp(-it} (j-l,2), where is 3 the circular frequency.
's satisfy the three-dimensional Laplace's equation in respective fluid regions except at the point of singularity where it exists.The linearized SS conditions are K >I +-y s (K + 3--), y 0, grad iI 0 as y =.
Further, I, z satisfy the radiation condition that both represent outgoing waves in the far fiels as Ixl .
Assuming the z-variation of the strength of the line source as exp(iz), it is possible to extract the z-variation completely from the functions .(x,y,z).Thus we can write j (x,y,z) j (x,y) exp (iz) j 1,2 where now j's satisfy the two-dimensional modified Helmholtz's equation  when the lower fluid is of infinite depth; also IVY21 0 as y +-=; (2.5) and finally, 'I' 2 satisfy the radiation condition in the far field as Thus I, 92 satisfy a boundary value problem (BVP) described by (2.1) to (2.6).In section 3 we will decompose this BVP into two BVP's by defining two sets of component potentials where the first set accounts for the singularity in the medium but die out in the far field while the second set is non-singular but accounts for the radiation condition in the far field as Ix =.In sections 4 and 5 we will obtain solutions to these BVP's assuming the lower fluid to be of infinite and finite depth respect- ively, thereby deriving the source potentials in the two fluids completely.We define potentials 9j @j j + j j--1,2 where @j satisfy {n DI, D2 respectively.Thus @I, 9Z satisfy the BVP described by (3.2) to (3.6) (hereinafter Pl).Then Xl, A2 satisfy the BVP (hereinafter P2) described by g (I +-y (I + hl) s {Kx 2 +-y (W 2 + X2)} y 0; and finally, Xl, X2 satisfy the radiation condition in the far field as In the conditions (3.8) and (3.10a), 91 and 2 are assumed to be known (solution of pl).(i) Wave Source in the Lower Fluid.In this case we seek a solution to the BVP described by (2.1) to (2.6) where @i K0(r) as r {x 2 + (y-B)2} I/2/ 0. Thus in PI the precise form of (3.2) and the condition (3.3) are (V -91 0, y > 0 except at (O,Q), Clearly @I, 2 as given above satisy the equations (4.1) and the conditions (4.2) and (3.6).We choose cl and c2 such that the conditions (3.4) and (3.5) are satisfied.
The following integral representations will be needed in our calculation K0(vr)  T K(r)k---cosx exp (k(y-q) dk (4.7)where the contour is indented below the pole at k M to ensure the radiation condi- tion at infinity.To establish this, we replace 2 cos Cx in the integrals by exp (i [xl)+ exp (-i Ixl).The contour in the integral involving exp (i is deformed into a line from to X (where X is a large positive number) on the real axis with an indentation below the pole at k M, the quarter of a circle of radius X in the first quadrant, the imaginary axis from iX to 0 and a line from 0 to just above the cut from k to in the complex k-plane.
It is being assumed inc tat v < M. (In fact if we assume an incident wave field represented by I exp {-My + i (M cos x + M sin z)}, y < 0 then M sin .However see section 6).
In this case there will be a contribution from the pole at k M. As X (R), the contribution from the circular arc will be exponentially small.Similarly in the (-i[x[) the contour is deformed into a llne from 0 to integral involving exp below the cut, a line from to X on the real line with an indentation below the point k M, the quarter of a circle of radius X in the fourth quadrant, and the imaginary axis from -iX to 0. In this case as the point k M lles outside the closed contour, there will be no contribution to the integral from this.As X the contribution from the circular arc will be exponentially small.The contribution from the real line from 0 to above and below the cut from k to will cancel out.Comb-ining the two integrals we will finally obtain the alternative representations (which account for the radiation condition in the far field as ix for @I, 2 as l-s K0(r) --s K0(r*) where N (M (4.9) (4.o) (ii) Wave Source in the Upper Fluid.In this case 2 K0(r*) as r* 0 so that 2 K0(r*) as r* O.
By writing WI cl K0(r*), 2 c2 K0(r) + K0(r*) and AI, A2 the same integrals as in (4.4) (with different A and B) we will.slmilarly where the contour is indented below the pole at k M to ensure the radiation condi- tion at infinity.
Alternative representation for 91, 2 can be obtained following the same method mentioned above as 2s i l--s [K0(r*) the terms in the square bracket in (4.8)], l-s 2s 2 K0(r*) + l--s K0(r) + l-s [the terms in the square bracket in (4.9)].

LOWER FLUID OF FINITE DEPTH. (i)
Wave Source in the Lower Fluid.
In this case @l, 2 are the same as in Section 4(i), while XI, xz satisfy P2 with the condition (3.10a) in place of (3.10b).
XI, 2 given above obviously satisfy (3.7) and (3.11).The SS conditions (3.8), (3.9) and the bottom condition (3.10a) yield the following three equations for the derivation of A, B, C.

CONCLUSION.
We have derived in the present paper source potentials for the two-dimenslonal modified Helmholtz's equation in a two-fluid medium.The parameter in the Helmholtz's equation has been assumed to be less than the wave parameter M (for infinite depth of the lower fluid) or k (for finite depth of the lower fluid).
However, if u is greater than the wave parameter then the potentials will no longer represent outgoing waves in the far field, rather they will die out in the far field (see the corresponding one-fluid case with surface tension in the FS in [16]).
Making s 0 in the above results, source potentials in an one-fluid medium ( [16] with surface tension put equal to zero) can be recovered.Making v O, potentials due to only time-harmonic line source in a two-fluid medium [3] can be recovered.
One can also include the effect of surface tension of the SS in these results.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: g being the gravity and s P2/Pl < I.If the lower fluid is of depth 'h' below the mean SS,

3 ( 2
.1) except at a point of singularity, where DI,D 2 denote respectively the regions occu- 2 pied by lower and upper fluids and V is the two-dimensional Laplacian operator.Near a point of singularity the potential behave as K (UR) which is a typical singu- lar solution of Helmholtz's equation, K (z) being the modified Bessel function of second kind and R being the distance from the poln.t.The boundary conditions are Kgl + --s (K2 + ,-17-)

3 .
DECOMPOSITION INTO TWO BOUNDARY VALUE PROBLEMS.(x,y), A_. (x,y)(j--1.2) such that at a point of singularity, and near a singularity the appropriate conditions are Wj Ko(R) as R 0.

First
Round of Reviews May 1, 2009