# Water waves generated by disturbance at an inertial surface

TL;DR: In this paper, the Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles.

Abstract: Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles. Green's integral theorem produces the transformed potential function from which the form of the inertial surface is obtained as an infinite integral after taking Laplace inversion. The method of stationary phase is then employed to evaluate this integral approximately for large time and distance.

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TL;DR: In this article, the problem of generating water waves due to prescribed initial axisymmetric disturbances in a deep ocean with an ice cover modelled as a thin elastic plate was formulated as an initial value problem in the velocity potential describing the ensuing motion in the fluid.

Abstract: This paper is concerned with the generation of water waves due to prescribed initial axisymmetric disturbances in a deep ocean with an ice-cover modelled as a thin elastic plate. The initial disturbances are either in the form of an impulsive pressure distributed over a certain region of the ice-cover or an initial displacement of the ice-cover. Assuming linear theory, the problem is formulated as an initial-value problem in the velocity potential describing the ensuing motion in the fluid. In the mathematical analysis, the Laplace and Hankel transform techniques have been utilised to obtain the deformation of the ice-covered surface as an infinite integral in each case. The method of stationary phase is used to evaluate the integral for large values of time and distance. Figures are drawn to show the effect of the presence of ice-cover on the wave motion.

28 citations

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TL;DR: In this article, the authors present new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid, and prove the existence of nonlinear standing waves within the framework of an exact physical model.

Abstract: This survey presents new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid. A flotation liquid is a liquid on whose surface heavy particles are floating; the particles may consist of arbitrary materials or may be particles of frozen liquid. The first part of the article considers initial- and boundary-value problems in the theory, their solvability, and the behavior of the solutions over long periods. In the second part of the survey, theorems are proved on the existence of nonlinear standing waves within the framework of an exact physical model, and both internal and free waves are considered. Also, the fundamental equations for shallow flotation waves are derived and examined.

9 citations

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TL;DR: In this article, the surface wave generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface is analyzed for two-and three-dimensional cases.

Abstract: The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace–Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distance-to-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the inertial surface disappears.

9 citations

### Cites background from "Water waves generated by disturbanc..."

...The generation of the gravity waves due to the initial elevation and impulse at the surface of an inviscid fluid with an inertial surface was first considered by Mandal and Mukherjee [ 4 ,5] for two- and three-dimensional cases....

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TL;DR: In this paper, the effect of uniform current on the generation of flexural gravity waves resulting from initial disturbances at a point was analyzed in two dimensions, where the problem was formulated as an initial boundary value problem under the assumptions of the linearized theory of water waves by direct application of the Laplace transform and then the Fourier transform.

Abstract: The effect of uniform current on the generation of flexural gravity waves resulting from initial disturbances at a point was analyzed in two dimensions The problem was formulated as an initial boundary value problem under the assumptions of the linearized theory of water waves By direct application of the Laplace transform and then the Fourier transform, explicit expressions for the velocity potential and free surface elevation were obtained in integral forms; these were evaluated asymptotically for large distances and times by the application of the method of the stationary phase to obtain the far field behavior of the surface elevations in specific cases Simple numerical computations were performed to illustrate the effect of uniform current on the surface elevation, wavelength, phase velocity, and group velocity of the flexural gravity waves and on the far field behavior of the progressive waves in two different cases, namely, when there is an initial depression concentrated at the origin and an initial impulse concentrated at the origin

8 citations

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TL;DR: The Laplace and Fourier transform techniques have been utilized to obtain the depression of the ice-covered surface in the form of an infinite integral for the special case of initial disturbance concentrated at the origin taken on the ice cover.

Abstract: This paper is concerned with two-dimensional unsteady motion of water waves generated by an initial disturbance created at an ice sheet covering the water. The ice cover is modeled as a thin elastic plate. Using linear theory, the problem is formulated as an initial value problem for the velocity potential describing the motion in the liquid. In the mathematical analysis, the Laplace and Fourier transform techniques have been utilized to obtain the depression of the ice-covered surface in the form of an infinite integral. For the special case of initial disturbance concentrated at the origin, taken on the ice cover, this integral is evaluated asymptotically by the method of a stationary phase for a long time and large distance from the origin. The form of the ice-covered surface is graphically depicted for two types of initial disturbances.

5 citations

### Cites background from "Water waves generated by disturbanc..."

...Mandal [5] considered generation of water waves due to initial disturbances at such an inertial surface....

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TL;DR: On developpe une methode de transformee de Laplace as mentioned in this paper for resoudre les problemes de valeur initiale, le potentiel de vitesse decrivant la generation d'ondes de gravite capillaires infinitesimales dans un liquide au repos avec une surface inertielle composee de particules flottantes distribuees uniformement.

Abstract: On developpe une methode de transformee de Laplace pour resoudre les problemes de valeur initiale, le potentiel de vitesse decrivant la generation d'ondes de gravite capillaires infinitesimales dans un liquide au repos avec une surface inertielle composee de particules flottantes distribuees uniformement

26 citations

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TL;DR: In this paper, the velocity potential and the surface elevation are calculated for the three-dimensional motion of surface waves excited by any local disturbance of the surface of a sea with constant depth.

Abstract: The velocity potential and the surface elevation are calculated for the three-dimensional motion of surface waves excited by any local disturbance of the surface of a sea with constant depth. Approximate values of the resulting wave integrals are given for large values of time and distance. The results are illustrated using physically plausible distributions of the initial disturbance. Some features of the waves are discussed.

15 citations

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TL;DR: In this paper, the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth, is discussed.

Abstract: This note is concerned with the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth.

14 citations

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TL;DR: In this article, the technique of stationary phase is applied to solve an initial value problem describing water waves generated by surface disturbances under more general conditions, and asymptotic solutions are given for two special cases.

Abstract: The technique of stationary phase is applied to solve an initial value problem describing water waves generated by surface disturbances under more general conditions. The asymptotic solutions are new. Explicit forms are given for two special cases.

11 citations