NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES

BSTRAT. This paper is concerned with a variational formulation of a nonaxisymmetric water wave problem. The full set of equations of motion for the problem in cylindrical polar coordinates is derived. This is followed by a review of the current knowledge on analytical theories and numerical treatments of nonlinear diffraction of water waves by offshore cylindrical structures. A brief discussion is made on water waves incident on a circular harbor with a narrow gap. Special emphasis is given to the resonance phenomenon associated with this problem. A new theoretical analysis is also presented to estimate the wave forces on large conical structures. Second-order (nonlinear) effects are included in the calculation of the wave forces on the conical structures. A list of important references is also given.

In the theory of diffraction, it is important to distinguish between small and large structures (of typical dimension b) in comparison with the characteristic wavelength (2/k) and the wave amplitude a. Physically, when a/b is small and kb -I is large (the characteristic dimension b of the body is large compared with k k is the wavenumber) the body becomes efficient as a generator of dipole wave radiation, and the wave force on it becomes more resistive in nature.
This means that flow separation becomes insignificant while diffraction is dominant.
In other words, the body radiates a very large amount of scattered (or reflected) wave energy.
On the other hand, for small kb (the characteristic dimension b of the body is small compared with k-l) and a/b is large (a/b > 0(I)), the body radiates a very small amount of scattered wave energy.This corresponds to a case of a rigid lld on the ocean inhibiting wave scattering, that is, diffraction is insignificant.
Historically, Havelock [I] gave the l+/-nearlzed diffraction theory for small amplitude water waves in a deep ocean.Based upon this work, MacCamy and Fuchs [2] extended the theory for a fluid of finite depth.These authors successfully used the llnearlzed theory for calculation of wave loading on a vertical circular cylinder extending from a horizontal ocean floor to above the free surface of water.
However, the llnearized theory has limited applications since it is only applicable to water waves of small steepness.
In reality, ocean waves are inherently nonlinear and often irregular in nature.
Hence, water waves of large amplitude are of special interest in estimating wave forces on offshore structure or harbors.
,In recent years, there has been considerable interest in the study of the hydro- dynamic forces that ocean waves often exert on offshore structures, natural boundaries and harbors of various geometrical shapes.Historically, the wave loading estimation for offshore structures was based upon the classical work of Morlson et.al. [8] or on the linear diffraction theory of water waves due to Havelock [I] and MacCamy and Fuchs [2].Morlson's formula was generally used to calculate wave forces on solid structures in oceans.
According to Debnath and Rahman [9], Morlson's equation expresses the total drag D as a sum of the inertia force, C M V U associated with the irrota- coefficient, M is the added mass, V is the volumetric displacement of the body, a U is the fluctuating fluid velocity along the horizontal direction, A is the projected frontal area of the wake vortex, and C D is the drag coefficient.For cases of the flow past a cylinder or a sphere, these coefficients can be determined relatively simply from the potential flow theory.It is also assumed that inertia and viscous drag forces acting on the solid structure in an unsteady flow are independent in the sense that there is no interaction between them.
There are several characteristic features of the Morison equation.
One deals with the nature of the inertia force which is linear in velocity U.
The other includes the nonlinear factor U 2 in the viscous dra force term.It is generally believed that all nonlinear effects in experimental data are associated with drag forces.
However, for real ocean waves with solid structures, there is a significant nonlinear force associated with the irrotational component of the fluid flow because of the large amplitude of ocean waves.These waves are of special interest in the wave loading estimation.
It is important to include all significant nonlinear effects associated with the nonlinear free surface boundaryconditions in the irrotational flow component of the wave loading on the structures.
However, the effect of large amplitude waves on offshore structures of small mean diameter may be insignificant, but it is no longer true as the diameter increases in relation to the wavelength of the indident wave field.
Consequently the Morison equation is no longer applicable, and diffraction theory must be reformulated.
It is necessary to distinguish between the structures of small and large diameters, the diameter being compared to the characteristic wavelength and amplitude of the wave.
Several studies have shown that the Morison formula is fairly satisfactory.
However, several difficulties in using it in the design and construction of offshore structures have been reported in the literature.These are concerned with the drag force which has relativly large scale effects.The shortage of reliable full-scale drag data in ocean waves is another problem.
There is another significant question whether a linear theory of the Irrotational flow response is appropriate at all to water wave motions with a free surface.Despite these difficulties and short-comings, the use of the Morison equation had extensively been documented in the past literature throughplentiful data for determining the coefficients C M and C D.
Several recent studies indicate that the second-order theories for the diffraction of nonliner water waves by offshore structures provide an accurate estimate of the linearized analysis together with corrections approximating to the effect of finite wave amplitude.
This paper is concerned with a variational formulation of a non-axisymmetric water wave problem.The full set of equations of motion for the problem in cylindrical polar coordinates is derived.This is followed by a review of the current knowledge on analytical theories and numerical treatments of nonlinear diffraction of water waves by offshore cylindrical structures.
A brief discussion is made on water waves incident on a circular harbor with a narrow gap.Special emphasis is given to the resonance phenomenon associated with the problem.
A new theoretical analysis is also presented to estimate the wave forces on large conical structures.Second-order (nonlinear) effects are included in the calculation of the wave forces on the conical structures.
We consider an Inviscid irrotational non-axlsymmetrlc fluid flow of constant density p subjected to a gravitational field g acting in the negative z-axis which is directed vertically downward.The fluid with a free surface z n (r, , t) is confined in a region 0 < r < =, 0 < z < n, < < 7.There exists a velocity potential (r,,z) such that the fluid velocity is given by V ), and the potential is lying in between z 0 and z n (r,, t).
(r' 0' z Then the variational principle is 61 6 f f L dx dt 0 (2.1) where D is an arbitrary region in the (, t) space, and the Lagrangian L is n(r,O,t) ) L -[*t + (vo + gz] dz, (2.2) 0 and 0(r,0,z,t), (r,0,t) are allowed to vary subject to the restrictions 60 0, 0 on the boundary D of D.
According to the standard procedure of the calculus of variations, result (2.1) D + (0 t + O r 60 r +-00 600 + O z 60z dz} r drd0 dt 0 Integrating the z-integral by parts along with r and 0-integrals, it turns out that )2 In view of the fact that the first z-integral in each of the square brackets vanishes on the boundary D, we obtain We first choose 6n 0, [6O] z=0 [60] z=n 0; since 6O is arbitrary, we derive err + Or + O00 + Ozz 0, 0 < z < n (r,0,t), (2.3) qt + qr #r + r -q0#O #z 0, z (r,O,t) (2.5)   b z O, z 0 (2.6) Evidently, the Laplace equation (2.3), two free-surface conditions (2.4)-(2.5)and the bottom boundary conditon (2.6) constitute the non-axisymmetric water wave equations in cylindrical polar coordinates.This set of equations has also been used by several authors including Debnath [I0], Mohanti [II] and Mondal [12], for the initial value investigation of linearized axisymmetric water wave problems.The elegance of the variational formulation is that within its framework, the treatment of both linear and nonlinear problems become identical 3. DIFFRACrlOM OF NONLINEAR WATER WAVES IN AN OCEAN BY CYLINDERS.
Several authors including Charkrabarti [13], Lighthill [14], Debnath and Rahman [9], Rahman and his collaborators [15][16][17][18], Hunt and Baddour [19], Hunt and Williams [20], Sabuncu and Goren [21].Demirbilek and Gaston [22] have made an investigation of the theory of nonlinear diffraction of water waves in a liquid of finite and infinite depth by a circular cylinder.These authors obtained some interesting theoretical and numerical results.We first discuss the basic formulation of the problem and indicate how the problem can be solved by a perturbation method.
We formulate a nonlinear diffraction problem in an irrotational incompressible fluid of finite depth h.We consider a large rigid vertical cylinder of radius b which is acted on by a train of two-dimensional, periodic progressive waves of amplitude a propagating in the positive x direction as shown in Figure I.In the absence of the wave, the water depth is h and in the presence of the wave the free surface elevation is above the mean surface level.In cylindrical polar coordinates (r,0,z) with the z-axls vertically upwards from the origin at the mean free surface, the governing equation, free surface and boundary conditions at the rigid bottom and the body surface are given by 1 + 0, b<r <', -<0<, -h<z<n, (
We apply the Stokes expansion of the unknown functions # and in the form where is a small parameter of the order of the wave steepness For any given n the s of only the first n terms of the series (3.Tab) may be considered as the nth-order approximation to the solutions of the problem governed by (3.1)-(3.6).The nth-order approximation is the solution subject to the neglect of m terms when m > n.
We next carry out a Taylor-serles expansion of the nonlinear boundary conditions Equating the first powers of E leads to the following equations with the llnearlzed boundary conditions on z 0, valid for all r,8 and t: V251 (r,O,z,t) 0 (3.8) #It + gnl 0, for z 0, r > b (3.9) rtlt-@Iz 0, for z 0, r _> b where #I #II + #IR representing the total potential as the sum of the first order incident potential, #II and the flrst-order reflected potential.
Similarly, equating the second powers of E leads to the following system of equations that express the second-order terms #2 and 2 as functions of #I and i: #2r 0 where k 2 is the wavenumber corresponding to second-order wave theory, and #21 the second-order term of the incident potential The function q can be eliminated from (3.9)-(3.10 Similarly, n 2 can be eliminated from equations (3.15) and (3.16) to obtain -n The pressure p(r,8,z,t) can be determined from the Bernoulli equation Substituting # as a power series in into (3.22),we can express p as p -pgz The total horlzonatal force is (-b cos) dzd0 x r--b 0 -h (3.24)where q is given by the perturbation expansion (3.7b).
Substituting (3.23) into (3.24) and expressing the z-integral as the s of It is noted that condition (3.12) is used to derive (3.25).It is clear from (3.25) tha't the integral of gz, the hydrostatic term, up to z=0 contain no cos8 term and hence may be neglected.Also, the upper limit of the z-integral of the second-order terms may be taken at z=0 in place of z gql + 2q2 which would only introduce 3 hlgher-order terms e etc. Thus F may be written as x Fxl Fx2 where the flrst-order contribution is 27 0 Fxl bo [ (it)r_b_ dz] cos d0 MacCamy and Fuchs [2] solved the system (3.8)-(3.13)and obtained the first-order solution which can be expressed in the complex form mcosh k(z+h) It is noted that result (4.2) represents the complex form of a plane wave of -I amplitude k propagating in the x-direction and represented by the terms whose radial dependence is given by Jm(kr), together with a reflected component described by the terms whose radial dependence is given by Hm(1)(kr).The first order solution e includes the complex form of an incident plane wave of amplitude e.

SECOND-ORDER WAVE POTENTIAL AND FREE-SURFACE ELEVATION FUNCTION.
With known values for I and n given in (4.2) and (4.3), equation (3.2)   assumes the following form for z 0 and r > b: -2 imt g e Z B (kr) cos mD, and a prime in (5.3) denotes differentiation with respect to kr.
A tedious, but straight forward, algebraic manipulation gives the value of the complex quantity Bl(r) as m+l 2 coth kh m(m+l) A m Am+ (5.7) B l(r) 8 m=Or (-I) [Am,m+ + k2r2 Since the first-order problem described by (3.8)-(3.131 is linear, its real physical solution is given by (I + I where @I is the complex conjugate of I" However, since (3.211 is nonlinear in I' it is not possible to express the solution of (3.20 as (2 + 2 )" We next discuss physical meaningful second-order solution for 2" We first write a real solution (I + I from (4.1) in the form It is noted that for m, n 0 this expression is identical with (5.3) and hence the definition of A is consistent with the previous definition.Furthermore, , mn A A It can be verified that the double series in (5.14) contains terms -m,-n m,m that are independent of time t and hence correspond to standing waves.However, it can readily be checked that these terms add up to zero and hence there are no standing waves in the solution.
In the limit kh =, (6.[) corresponds to the result for deep water waves which are in agreement with Lighthill [14], and Hunt and Baddour [19].
We next summarize below the second-order contribution to the total horizontal force given by (].28).
We first put values of I and [ from (5.8) and (5.11)   into (].28) and then evaluate the z-integral to obtain the coefficient of cos0 in (].28), apart from the 2/t term, in the form In view of the subsequent 0-integration describe([ in (3.28), we need only the coefficient of cos0 in the double series (6.3) and hence obtain the following where Dl(k2) is related to Bl(kb) through (5.5).
Combining (6.3) and (6.9), integrating with respect to 0, the result can be expressed as the sum of steady and oscillatory componeuts: correspond to results (6.1), (6.2) and (6.3) obtained by Hunt and Williams [20] for the diffraction of nonlinear progressive waves in shallow water.The second-order contributions to the total horizontal force F on the vertical cylinder each consist of two components, a steady component x together with an oscillatory term having twice the frequency of the first-order term.
Hunt and Williams calculated the maximum value of F for various values of wave steepness, water depths and cylinder diameters.
The maxim[ value of F is found to be significantly higher than that predicted by the linear diffraction theory.
The second-order effects are found to be greatest in shallow water for slender cylinders, but in deep water they are greatest for cylinders of larger diameter.
These predictions are supported by some existing experimental results which, for finite wave steepness, shows an increase over the linear solution.
In order to establish the fact that (6.12) represents the oscillatory component of the second-order solution, it is necessary to evaluate the complex form of D1(k2) from the integral equation (5.5) for m=l and in conjunction with (5.7).Following the analyses of Hunt and Baddour [19] and Hunt and Williams [20], we obtain from (5.5)   and (5.7) that kk2 Al(k2r) Bl(r)dr Dl(k2) k 2 sinh k2h 4k tanh kh cosh k2h' (6.13) which san be written as D l(k2) sinh k2h k A l(k2r) B l(r)dr b 4k tanh kh (I)' k22 HI(1)ik2 b) k2(k2-a 2 H (k2b) (6.14) The nondimensional form of (6.14) is given by D l(bk2) sinh k2h It is noted that the function G2(bk2) is analytic near and at k 2 0. However, it is singular when k 2 is a root of the equation k 2 tanh k2h 4 k tanh kh (6.16) Clearly, k 2 4k for deep water case and k 2 2k for shallow water problem.The root k 2 of (6.16) lles between 2k and 4k and hence may be regarded as correspond- ing to an ocean of intermediate depth.
An argument similar to that of Grlffith [23] shows that the Integrand in the integral G(k 2) dk 2 is singular at k 2 4k for 0 a particular deep water wave, and at k 2 2k for a particular shallow water wave.
The non-dimensional forms of the first-order and the second-order forces can be expressed as where D is the diameter of the cylinder.
Thus the total horizontal force in nondimensional form is expressed as where H 2a is the total waveheight and L is the wavelength of the basic wave, and C M is defined to be the Morison coefficient due to linearlzed theory and is given by CM 4/[(kb)21H(ll)'(kb)l (6.22) In ocean engineering problems, wave forces on the structures depend essentially on three dimensionless parameters H/L, D/L and h/L.However, Hunt and Williams have pointed out that many experimental studies of wave forces have been published with such a variation of parameters that precise experimental verification is not possible.
Recent findings of Rahman and Heaps have been compared with experimental data collected by Mogridge and Jamieson [3].
An agreement between theory and experiment is quite satisfactory as shown in Figures 2 4.
Another comparison s made in Fig. 5 with the experimental data due to Raman and Venkatanarasaiah [24].The second-order results of Rahman and Heaps [17] seem to compare well with these experimental data.In Fig. 6, both the first-order and the second-order solutions are compared with force measurements of Chakrabarti [13] which are generally found to be closer to the second-order theory.
Experiment *** Second-Order Theory Comparion of linear and zecond-order wave forcee with experimental data of Ram=zn and Venk=tan=r=z=d=h [24].A flnal comment on the singular nature of G(k2) Is In order.For a cylindrical structure, the wavenumber k 2 of the second-order wave theory must not coincide with the root of the equation (6.16) unless the corresponding integral in (6.15) van- ishes.Otherwise, the structure will experience a resonant response at the wavenumber k 2. This kind of resonance is predicted by the second-order diffraction theory but not by the linear wave theory.
In real situations involving ocean waves, such non- linear resonant phenomenon is frequently observed.Hence the correct values of the wave forces on the offshore structures cannot be predicted by the linear wave theory.
According to Rahman and Heaps' analysis, the cylindrical structure will ex- perience a resonance when k 2 4k for the case of deep water waves, and when k 2 2k for the case of shallow water waves.
Obviously, there is a need for modification of the existing theories in order to obtain a meaningful solution at the resonant wave- number.
A partial answer to the resonant behavior related to the shallow water case has been given by Rahman [25].
Recently, Sabuncu and Goren [21]have studied the problem of nonlinear diffrac- tion of a progressive wave in finite deep water, incident on a fixed circular dock.
This study shows that the second-order contribution to the horizontal force is also highly significant.
Their numerical results for the vertical and horizontal wave forces on the dock are in excellent agreement with those of others.Demirbilek and Gaston [22] have also reported some improvements on the existing results concerning the nonlinear wave loading on a vertical circular cylinder.
In spite of various ana- lytical and numerical treatments of the problems, further study is desirable in order to resolve certain discrepancies of the predicted results.
Finally, we close this section by citing a somewhat related problem of waves incident on harbors.
A recent study of Burrows [26] on linear waves incident on a circular harbor with a narrow gap demonstrates that the wave amplitude inside the har- boT is significantly affected by the frequency of the incident waves.frequencies the harbor acts as a resonator and the wave amplitude becomes very large.If the harbor is closed and the damping neglected, the free-wave motion is the superposltlon of normal modes of standing waves with a discrete spectrum of char- acteristic frequencies.With a circular harbor wlth a narrow opening, a resonance occurs whenever the frequency of the Incident waves approaches to a characteristic frequency of the closed harbor.
A resonance of a different klnd is given by the so called Helmholtz mode when the oscillatory motion inside the harbor is much slower than each of the normal modes.Burrows determined the steady-state response of the harbor with a narrow gap of angular width 2 to an incident wave of a single frequency under the assumption of small width compared to the wavelength.
The re- sponse is a function of frequency and has a large value (a resonance) at the fre- quency of the Helmholtz mode and also near the characteristic frequencies of the closed harbor.The actual nature of the response near these frequencies depends on 2 .It is shown that the peak value at each resonance increases as decreases, that is the harbor paradox for a single incident wave frequency.However, the increase is slow.
The peak width also depends on , and decreases as decreases, but the decrease for the Helmholtz mode is less than for the higher modes.
Some authors including Lee [27] gave a numerical treatment of the resonance problem inside the harbor.
In approximate calculations it is assumed that the total flow through the gap will effectively determine the flow near the resonant frequency.This is correct near the Helmholtz resonance, but incorrect near the higher resonances where the through-flow is small.Most of the work on the subject of Helmholtz resonance was based on the linear theory.The question remains whether or not the circular harbor is a Helmholtz resonator for nonlinear water waves. 7. NONLI%RWAVE DIFFRACTION CAUSED BY LARGE CONICAL STECTRES.
We consider a rigid conical structure in waves as depicted in Figure 7.
With reference to this figure, the equation of the cone may be given by r (b-z) tan u where b is the distance between the vertex of the cone and undisturbed water sur- face, is the semi-vertical angle of the cone and r is the radial distance of the cylindrical coordinates (r,0,z).The fluid occupies the space (b-z) tan < r < =, < 0 < , h < z < (r,0,t), where h is the height of the undisturbed free surface from the ocean-bed and (r,0,t) is the vertical elevation of the free surface.
The governing partial differential equation for the velocity potential (r,0,z,t) is within the region (b-z) tan u < r < , < 0 < n, The free surface conditions are -h< z<. )) )) -f+ gn + [(-r + ('') + 0 ( for z and (b-z) tan a < r; an a an where n is the distance normal to the body sur- face.
There is another boundary condition which is needed for the unique solution of this boundary value problem.
This condition is known as the Sommefeld radiation condition which is discussed by Stoker [28].This is briefly deduced as follows: The velocity potential may be expressed as $(r,O,z,t) Re[(r,,z) e imt] (7.6)where Re stands for the real part and is the frequency.We assume where I and S are the incident wave and scattered wave potentials respectively.
Then the radiation condition is written as lira #V (_Sv, i k #S 0 (7.8) This condition may 1/_ be generally satisfied when ) S takes an asymptotic form proportional to (r) -v2exp(-ikr).Here k is a wave number.
The linear incident wave potential I may be obtained from the solution of the Laplace' s equation, The incident wave expression (7.10) can be written as cosh k(z+h) $I C l 13 J (kr) cos m(0-y) (7 12)   cosh kh m m m=0 We are now in a position to construct the scattered potential S which is given by S C -cShcoshk(Z+h)kh r. 13m Bm Hm(1) (kr) cos m(0-y) It can be easily verified that (7.13) satisfies the radiation condition (7.8).
The surface boundary condition (7.5) gives that 13 [J'm(kr) cosh k(z+h) cos a m + J (kr) sinh k(z+h) sin a] cos m(0-y) m (7.15)where y is the angle made by unit normal with the radial distance r.Similarly, we It is to be noted from (7.17) that the constant B turns out to be a function of z m instead of a constant.In order to overcome this difficulty, we estimate the constant B by taking the depth average value, which is obtained by integrating both sides of m (7.17 Once the scattered potential S is determined, we can formulate the wave forces on the structures.Therefore, the total complex potential may be written as cosh k(z+h.#y. [Jm(kr) + B H %(x,y,z) C cosh kh m m m 0 (1)(kr)] cos m(0-y) (7.19)The formulation of the wave forces is given in the next section.
Lighthill [14] demonstrated that second order wave forces on arbitrary shaped structures may be determined from the knowledge of linear velocity potential alone.
The exact calculation of second order forces on right circular cylinders has been obtained by Debnath and Rahman [9] using the Lighthill's technique.
The total potential has been obtained in the following form: where F is the linear force, F d is the second order dynamic force, Fw is the second order waterline force, and F is the quadratic force.These force components q are all functions of the linear diffraction potential .They may be obtained using the following formulas: The linear force is  (p2/g) 19 12 z=O z=0 dy.
Making reference to Rahman [15] the quadratic force may be written as where o 42/g 4k tanh kh and is the complex time independent potential gener- ated by the structure surging at a frequency of 2m.
The vertical particle velocity (z) on z 0 may be written in a series form for finite water depth 2K (mir Kl(mja) (m.h + sin m.h cos m.h) 3 2H2) (vr)   sinh 2vh + 2)' cos 0 (8.9)H (va)(vh + sinh h cosh vh) where (4m2/g) mj tan m.h for j 2 ., and o v tanh uh 4k tanh kh.J Expression (8.9) is valid only for right circular cylindrical structures.The wave drift forces on the structures may be obtained from the equations (8.5)Therefore, the horizontal and vertical forces can be obtained respectively as FX F cos a, FZ F sin a The resultant dynamic force can be calculated from (8.5) and is given by After extensive algebraic calculations, the dynamic force can be written as cos _) e-2 it 0 Fd (P 4   Re AA+l)dZ* (9.4)   (b-z) 2 tan2 Therefore, the horizontal and vertical dynamic forces can be obtained respectively as Fdx F d cos a, Fdz F d sin a The resultant waterline force can be obtained from (8.7) and is given by A A+ I) (9.8)   Th6refore, the horizontal and vertical drift forces may be respectively obtained as Therefore, the horizontal and vertical quadratic forces may be respectively obtained as FqX Fq cos , FqZ Fq sin (9.12ab) lO.CONCLUDING Second order nonlinear effects are included in the derivation of the wave forces on the large conical structures.
The second order theory is consistent because it satisfies all the necessary boundary conditions including the radiation condition.
Theoretical expressions for the wave forces have been obtained; the linear forces could be improved by adding to it the second order contributions namely, dynamic, waterline and quadratic forces.
It would be of considerable value if the theoretical results presented in this paper could be checked experimentally under laboratory condi- tions.
Plans are made in future research to check the accuracy of the predicted results with the experimental measurements.

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: component of the fluid flows under the assumption that the incident wave ) is the Morison(or inertial)

FIG. 1 .
FIG. 1. Schematic diagram of a cylindrical structure in waves

FF
force can be obtained from the formula (8.8) and is given by calculations, the quadratic force can be written as )x 2C2k 2 z [. k2r2+ 3 tanh 2 kh-l} AA+ + 2 AA+ I] e dO} =0