THE DOCK PROBLEM REVISITED

The well-known semi-infinite dock problem of the theory of scattering of surface water waves is reexamined and known results are recovered by utilizing a Fourier type of analysis, giving rise to Carleman-type singular integral equations over semi-infinite ranges.


Introduction
The dock problem (cf.[2], [4,5]), which is that of understanding the scattering of surface water waves by a thin semi-infinite rigid plate floating on the free surface of water of infinite depth, gives rise to the following mixed boundary value problem, involving Laplace's equation in two dimensions, with (x, y) representing the rectangular Cartesian coordinates, assuming linearised theory of water waves: (1.1) Kφ + φ y = 0 on y = 0, x < 0, (1.2) φ y = 0 on y = 0, x > 0, (1.3) r ∂φ ∂r = 0 as r = x 2 + y 2 1/2 −→ 0, (1.4) φ and φ x are continuous at x = 0, y > 0, (1.5) φ −→ 0 as y −→ ∞, (1.6) Here Re{(g 2 /σ 3 )φ(x, y)e −iσt } denotes the velocity potential (actual) describing the fluid motion assumed irrotational, where σ is the circular frequency and g is the acceleration due to gravity, K = σ 2 /g, R is the unknown reflection coefficient due to a progressive wave train described by the complex velocity potential function e −K y+iKx incident on the semi-infinite rigid plate occupying the position y = 0, and x ≥ 0. A direct use of a Fourier type of analysis (cf.[7]) is shown to reduce the above boundary value problem to either of two possible singular integral equations of the Carleman type over a semi-infinite range.It is then shown that the closed form solutions of both of these Carleman equations are possible, giving rise to closed form solution of the problem under consideration.The associated reflection coefficient R is determined and is found to agree with the known result.The free surface profile and the pressure distribution on the dock are depicted graphically at initial time t = 0 against the distance.These figures coincide with those given in [2].
The present analysis is believed to be more straightforward and simple, as compared to the existing methods of [2] and [4,5] to handle this class of problems in the theory of surface water waves.
We emphasize, at this stage itself, that the representation (2.2) demands that we must have to help the integral in (2.2) converge.
The conditions (1.5) give the following relations: (2.5) Now there are two possible ways of handling the two relations (2.5) as far as the application of Fourier analysis is concerned.The first possibility is to use a Fourier cosine inversion formula to both the relations (2.5) to determine B(ξ) in terms of A(ξ), and the second possibility is to use the Havelock's expansion theorem (cf.[7]) to determine A(ξ) in terms of B(ξ).
Then by using the first of the above two approaches, we obtain that which on elimination of B(ξ), give rise to the following singular integral equation of the Carleman type: where (2.9) We observe that (2.8) contains the unknown constant R (the unknown reflection coefficient), and this can be determined by utilizing the convergence criterion (2.4).Also, by using the second approach, we obtain that provided that (2.12) provided that (2.13) The following generalized identities (cf.[1]) have been utilized in deriving the results (2.6), (2.7), (2.10), and (2.12): lim where u,ξ > 0 and δ(x) is the Dirac delta function.
Also the singular integral occurring in (2.8) and elsewhere in the present work is to be understood as its Cauchy principal value (cf.[3]).
Eliminating A(ξ) between the relations (2.10) and (2.12), we obtain a second singular integral equation of the Carleman type, as given by where c can be regarded as an unknown constant to be determined, along with the other unknown constant R (the unknown reflection coefficient), by using the two constraints (2.11) and (2.13).Now both the integral equations (2.8) and (2.15) are of the same type and each of them can be cast into a Riemann-Hilbert problem involving the complex-plane, with a cut along the positive real axis, which can finally be solved by using standard techniques available in [3] or [6].
The two Riemann-Hilbert problems for the two integral equations (2.8) and (2.15) are given by respectively, involving the two sectionally analytic functions Φ(ζ) and in the cut ζ plane, where The solutions of the above two Riemann-Hilbert problems are straightforward (cf.[3]) and we find that The solutions of the integral equations (2.8) and (2.15) can be finally determined by using the Plemelj's formulae as given by (2.21) Evaluating the various integrals appearing in the relations (2.18), by using standard techniques involving contour integration (cf.[1] and Appendix A to the present paper), we find that where D 1 is an unknown constant.We next use the relations (2.6) and (2.9) along with the result (2.22) and find that is obtained after evaluating several integrals by using appropriate contour integration procedures (see Appendix B).Then, by using the condition (2.4), we find that we must have (see Appendix C.1) obtained by using the relation (2.19a).Again, by using the result (2.23) in the two relations (2.11) and (2.13), and by evaluating the various integrals (see Appendix C.2), after using the relation (2.19b), we find that )

.27)
We thus find that the mixed boundary value problem under consideration gets solved completely either by the aid of the relations (2.24), (2.25), (2.22), and (2.9), or by the aid of the relations (2.23), (2.26), (2.27), and (2.12), which determine the unknown functions A(ξ) and B(ξ) and the unknown reflection coefficient R completely so that the complete knowledge of the potential φ(x, y) can be obtained by using the relations (2.1) and (2.2).
We find that the value of R is e −iπ/4 , as obtained by Holford [4,5], by using a completely different analysis.
It is rather interesting to verify (see Appendix D) that the two representations of B(ξ), as given by the relations (2.23) along with (2.27) and (2.24) along with (2.25), are identical.

Discussion
The exact form of φ(x, y) can be obtained from (2.1) for x < 0 and from (2.2) for x > 0 after substituting A(ξ) and B(ξ) in terms of Φ(ξ).This should coincide with the result for the potential function (except for a multiplying constant) given in [2] (Reχ R (z) given there).However, this is not verified here directly.Instead we obtain here the free surface depression η(x,t) and the pressure p(x,t) on the dock by using Bernoulli's equation, and depict them graphically against the nondimensional distance Kx at time t = 0 (actually Kη(x,0) and K p(x,0)/ρg) for x < 0 and x > 0, respectively.Using Bernoulli's equation, the free surface distribution η(x,t) (x < 0) is obtained as where Similarly, the pressure on the dock p(x,t) (x > 0) is obtained as where Figures 3.1 and 3.2 depict, respectively, the free surface profile η(x,0) against x(x < 0) and the pressure distribution p(x,0) on the dock also against x(x > 0); η, p, x being nondimensionalised as Kη, K p/ρg, Kx.These curves in Figures 3.1 and 3.2 can be identified with the curves given in [2] obtained from the potential function Reχ R (z) given there.This indirectly verifies that the solution for the potential function obtained here coincides with the solution Re χ R (z) given in [2] (except for a multiplying constant).

A. Determination of C(ξ)
In this appendix we will describe the contour integration procedure to obtain the value of C(ξ), as given by the relation (2.22).
The relation (2.20) gives The integrals appearing in (A.1) can be evaluated by considering integrals of the form with Γ a positively oriented closed contour consisting of a loop around the positive real axis and a circle of large radius with centre at the origin, in the complex τ-plane, and P(τ) and Q(τ) are polynomials in τ.If these polynomials are such that the contribution to the integral in (A.2) over the circle of large radius vanishes, then after using the relation Thus by using P(τ) = 1 and Q(τ) = −2iK (τ + iK), it is observed that Similarly, by choosing P(τ) = 1 and Q Using (A.7) and (A.8) in (A.1), C(ξ) is obtained as , ξ > 0. (A.9)

B. Proof of (2.24)
Here we will describe the details of the procedure to derive the relation (2.24).On using (2.21), we obtain The integrals in (B.1) can be evaluated by considering the integral where Γ is the same as in (A.2) and P(τ),Q(τ) are polynomials such that the contribution to the integral in (B.2) from the circle of large radius vanishes.We obtain where . Hence the use of Plemelj's formulae produces Using the results of (B.6) in (B.1), ultimately B(ξ) is obtained as Here we will prove the result (2.25).From (2.19a), we find that Hence we obtain that To determine the values of cD 1 /π = D (say) and R, the relation (2.23) is used in the relations (2.11) and (2.13).This gives rise to the relations The integrals in (C.4) can be evaluated by considering integrals of the form (A.2).Thus,

D. Equivalence of (2.23) and (2.24)
Here we prove the following results: so that the expressions as given by the relations (2.23) along with (2.27) and (2.24) along with (2.25) represent the same function B(ξ).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation