ON GENERATION AND PROPAGATION OF TSUNAMIS IN A SHALLOW RUNNING OCEAN

A theory is presented of the generation and propagation of the two and the three dimensional tsunamis in a shallow running ocean due to the action of an arbitrary ocean floor or ocean surface disturbance. Integral solutions for both two and three dimensional problems are obtained by using the generalized Fourier and Laplace transforms. An asymptotic analysis is carried out for the investigation of the principal features of the free surface elevation. It is found that the propagation of the tsunamis depends on the relative magnitude of the given speed of the running ocean and the wave speed of the shallow ocean. When the speed of the running ocean is less than the speed of the shallow ocean wave, both the two and the three dimensional 374 L. DEBNATH & U. BASU free surface elevation represent the generation and propagation of surface -1/2 waves which decay asymptotically as t for the two dimensional case and as -I t for the three dimensional tsunamis. Several important features of the solution are discussed in some detail. As an application of the general theory, some physically realistic ocean floor disturbances are included in this paper.

investigation of tsunami generation due to some particular ocean floor distur- bances.The main emphasis of his analysis is to determine the applicability of the linear theory in the generation region.Although this model does con- sider the nonlinearity, the surface displacement integrals related to ocean floor disturbances have not been evaluated so that no conclusion can be drawn about the principal features of the wave motions.Braddock et al (4) have considered the problems of tsunami generation due to a sea floor disturbance which is described by series of orthogonal functions.Using the standard techniques of integral transforms and stationary phase methods, they have presented the asymptotic solution for the free surface flows produced by the applied ocean floor disturbance.It has been shown that tsunami consists of a dispersive wave train preceded by a nondispersive wave front traveling as a long ocean wave.The relative order of magnitude of the wave train and the wave front is found to depend on the degree of symmetry or asymmetry of the ocean floor disturbances.
In almost all models considered in the literature including the two men- tioned above, the dynamics of tsunamis was confined to shallow or deep oceans at rest.
In this paper, a study is made of the generation and propagation of tsunamis in a shallow running ocean due to the action of an arbitrary ocean floor or ocean surface disturbance.An asymptotic analysis for both two and three dimensional tsunamis is presented to investigate the principal features of the free surface elevation.It is shown that the propagation of the tsuna- mis depends on the relative magnitude of the speed of the running ocean and the critical wave speed in the shallow ocean.When the basic speed, U, of the running ocean is less than the speed of the shallow ocean wave, both the two and the three dimensional free surface elevation represent the generation and -1/2 for the two propagation of surface waves which decay asymptotically as t -I dimensional problem and as t for the three dimensional tsunamis.

MATHEMATICAL FORMULATION OF GENERAL PROBLEM.
We consider an inviscid incompressible fluid of infinite horizontal extent which is bounded above by the free surface at z 0 and bounded below by a solid bottom at z =-h.In its undisturbed state, the fluid flows with con- stant velocity.The wave motion is set up in the fluid by the combined action of a given bottom disturbance and the pressure distribution at the free surface.
It is convenient to formulate the initial value problem in a coordinate frame with respect to which the applied pressure is at rest.We thus take the Cartesian coordinate system Oxyz such that the x-y plane represents the undis- turbed free surface with the origin located on it, the z axis is directed vertically upward and the fluid moves in the Ox direction with uniform velocity U relative to this frame.
At time t > 0, the solid bottom of the ocean is subjected to move in a prescribed manner given by z -h + (_r, t) such that (_r, t) 0 as r when r (x, y).In addition, the pressure is prescribed on the free surface of the liquid so that the free surface flow is generated in the shallow running ocean by the action of the surface pressure or the bottom disturbance.
As the flow is generated by the disturbances in the uniform stream, the disturbance velocity potential (x,y,z;t) satisfies the Laplace equation V2 -= xx + yy + zz 0, (x, y) e (-, =) -h <__ z <__ 0, t > 0. (2.1)With (x,y;t) representing free surface displacement, the linearized kinematic boundary conditions on the free surface and the solid bottom are z nt + Unx on z 0, t > 0, ( z Ct + Ux on z 0, t > 0, (2.3) In the absence of surface tension of the fluid, the dynamic condition on the free surface in the linearized form is given by 1 t + Ux + gnp(x,y;t) on z 0, t > O, (2.4) where p is the constant density of the fluid and p(x,y;t) is the pressure prescribed on the face surface.
The initial conditions are (x,y,z;0) (x,y;0) (x,y;0) 0, ( Further, it is assumed that , and are generalized functions (or distributions) of x and y in the sense of Lighthill (5) so that their Fourier transforms exist with respect to x and y.

SOLUTION OF THE INITIAL VALUE PROBLEM.
The above wave problem can readily be solved by using the Laplace trans- form with respect to t and the generalized Fourier transform with respect to where k -= (k,) is the two-dimensional wave vector, the tilda and the bar denote the Laplace and the Fourier transforms respectively.
Application of (3.1) to the differential system (2.1) (2.5) gives the solutions for the transform functions #(k,z;s) and n(k,s).-(k,s), p(k,s) are the Laplace and the Fourier transforms of (r,t) and p(r,t), and b (glkltanh'Iklh) It is noted that in the absence of the surface pressure disturbance and the basic stream, the integral solution (3.2) with (3.3) reduces to that of Hammack (3) for the two dimensional case and to that of Braddock et al (4) for the three dimensional case.
The Laplace inversion of (3.3) can be carried out by means of a suitable complex contour integral combined with the theory of residues.The simple poles of the integral of (3.2) are at s ibl, -ib 2 where b I b Uk and b 2 b + Uk.These poles are on the imaginary axis of the complex s-plane and their residue contributions to (x,y;t) lead to the propagation of surface wave trains.The other singularities of (3.2), if any, related to some physi- cally realistic bottom or pressure disturbances are all poles located at the left of the imaginary axis in the s-plane.Their residue contributions to the solution decay exponentially in time and so are in general insignificant.
The residue contributions only from the simple poles at s ibl, -ib 2 give the oscillatory surface elevation in the form (x,y;t) where It is evident that the terms exp[i(k-r +_ tb )], n i, 2 represent the n surface waves with the complex amplitudes A(k, + ib respectively.n On the other hand, when -(k,s) and (k,s) have polar singularities on the imaginary axis in the s-plane, the Laplace inversion of (3.3) would be made up of these polar contributions including from ib I and-ib 2.
Of special interest are the following bottom and pressure disturbances" (r,t) 0(r_)f(t)H(t), p(_r,t) Pp(r)eitH(t), where 0' P are constants, H(t) is the Heaviside function of time t, (_r) (x,y), and p(_r) p(x,y) are functions of compact support in the x-y plane, f(t) is a suitable function of t and is the frequency of the applied pressure.
The surface displacement n(x,y;t) related to (3.7a,b) is obtained from )]e ----dk, ( In general, the exact evaluation of the integrals (3.5) and (3.8) is almost a formidable task and hence it is necessary to resort to asymptotic methods.
4. ASYMPTOTIC ANALYSIS FOR THE TWO DIMENSIONAL PROBLEM.
In the corresponding two dimensional wave problem, there is no y dependence.
Hence, the free surface displacement (x,t) corresponding to (3.5) is given by (x,t) where k is the one dimensional wave number, In order to make an asymptotic evaluation of (4.1) for large t such that Ixl << ut, we shall follow the method of Debnath and Rosenblat (1969).Writ- ing X for kh, the stationary points for the integrals in (4.1) as t + are approximately given by c' (X) YU where c g/ and () (X tanhx) 1/2.The necessary and sufficient condition for the existence of stationary points is U < c.And if U > c, there are no stationary points of the integrals in (4.1).
From a graphical method similar to that of Debnath and Rosenblat (6), it is easy to locate the roots of the equations c'() +U for U < c.Hence it follows that the first and the second integrals in (4.1) have stationary points at k -k2, (k 2 > 0) and k k I > 0 respectively for U <_ c.Invoking the standard formula for the stationary phase expansion and incorporating the existence condition for the stationary points through the Heaviside function, the asymptotic representation of (4.1) for large t is given by  It is also noted that result (4.4) includes the critical case U c in the -I sense that then k I k 2 0 and the contribution to (x,t) is of the order t Thus the wave system given by (4.4) decays more slowly than the wave front at the origin.
For the case U > c, the integrals do not have any stationary points and -I consequently they decay like t as t -.Physically, this means that the free surface elevation has not yet penetrated into the region In order to describe the ultimate wave system, we next evaluate the inte- where the integrals I (n i, 2, 3, 4), J (n i, 2, 3) are given by  It is convenient to write (4.6) in the form n(x,t) nl(X,t) + n2(x,t), (4.14)where n I consists of I 1 and Jl' and represents the free surface displacement due to the surface pressure distribution; 2 is made up of In(n 2, 3, 4) and J (n 2, 3), and describes also the surface displacement originated entirely n due to the bottom disturbance.
Integrals I 1 and Jl are exactly identical with those obtained by Debnath and Rosenblat (6).A detailed asymptotic analysis of these integrals was presented in that paper.It may therefore be fair to avoid duplication, and to quote some important results without any further elaboration.It follows from paper (6) that the asymptotic solution Dl(X,t) consists of the steady- state and the transient wave components.For large Ixl and t such that Ut >> Ixl, the latter is of the order t -1/2 for c > U, and decays asymptotically as t / =.Thus the ultimate steady-state wave system is established in the limit t + and can be written in terms of notations of (6) as and where Pi imt (-rE-) k:< The steady-state solution represents the propagation of either two or four surface waves according as U > c or U < c.
At the critical speed, U c, both the steady-state and the transient solutions asymptotically tend to infinity.Indeed, the former represents a wave whose amplitude increases like x while the latter is also a wave whose amplitude is of the order t 1/2 as t / .This clearly suggests that the llne- arized theory based on small amplitude disturbances fails to provide with a physically sensible solution at the critical speed.Naturally, it would be necessary to include nonlinear terms in the original formulation of the problem in order to achieve a physically reasonable solution.
We next turn our attention to the asymptotic evaluation of I and n J (n 2 3 4).These integrals have infinitely many pure imaginary poles at (n+1/2) , n 0, +i +2, and can readily be evaluated by using the residue theory over a suitable contour.The residue contributions from these poles are insignificant as t or Ixl / =.The integral 12 does not have any significant contribution to the free surface displacement.
From a graphical representation of (X)
It should be noted here that the wave front decays more slowly than the main transient wave system described by (4.2).
Integrals 14 and J3 have infinite sets of purely imaginary poles at +-iB n and +i n' where n and B n' satisfy the equations c(Btan) UB ha 0, c(B'tanB') + UB' ha 0, ( Evidently, as t / , these integrals do not have any sfgnificant contri- bution from these imaginary poles. On the other hand, J3 has the same stationary points as those of J2 and its contribution from the stationary points for large t can be written in the form 1 J3 0(t-1/2) or 0(t 3), according as c > U or c U. Thus the transient component of 2(x,t) decays asymptotically as t / and the ultimate steady-state is reached which takes the asymptotic form: -ik3x], when x > 0 + 2 (-k3)e  Hence all the wave integrals involved in the free surface displacement (x,t) given by (4.6) have been evaluated asymptotically for large Ix or t.
It follows from the above analysis that the transient component of n(x,t) decays asymptotically as t + .And the ultimate steady-state wave system is attained and consists of l(st) and 2(st) given by (4.15), (4.16) and (4.25a,b).
The integral representation for the free surface displacement (x,y;t) due 2. With these values of 9, and r x, (6.6a) reduces to the corresponding equations for the two dimensional problem.Hence the existence condition for the stationary points of (6.2) is the same as that discussed in Section 4.
Evidently, for U < c, the first equation in (6.6a) gives one non-negative stationary point at % %1 when 0 or 2; and the second equation in (6.6a)   has also a non-negative stationary point at % %1 when . Thus the first and the second integrals of (6.2) have stationary points at (%1' 0) and (%1' ) respectively.

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

( 4 .
5a,b)This represents the generation and propagation of surface wave trains which -1/2 decay asymptotically as t

5 .
SOME PARTICULAR DISTURBANCES FOR TSUNAMI GENERATION.It is of interest to mention some physically realistic form of disturbances for the generation of tsunami: ) -!-I exp(-)

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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