The effect of surface tension on trapped modes in water-wave problems

Harter, Robert; Abrahams, I. David; Simon, Michael

doi:10.1098/rspa.2007.0063

Cited by 16 publications

(15 citation statements)

References 15 publications

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“…Using Fourier transform technique, Panda et al [25] studied the scattering of water waves in a single layer fluid over a permeable bottom by considering the effect of surface tension. Harter et al [1] analyzed the effect of surface tension on trapped modes. They pointed out that the exclusion of surface tension during the formulation of the problem is not always justifiable, as its inclusion changes the qualitative nature of the streamline pattern.…”

Section: Introductionmentioning

confidence: 99%

Water-Waves Scattering by Permeable Bottom in Two-Layer Fluid in the Presence of Surface Tension

Panda

Martha

2017

Mathematical Modelling and Analysis

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In the present paper, reflection and transmission phenomena of water waves due to undulating permeable bottom in a two-layer fluid system are investigated using two-dimensional linearized theory. The effect of surface tension on the free surface is included in this work. In two-layer fluid system, there exist waves with two different wave numbers (modes). When a wave of a particular wave number encounters the undulating bottom, reflection and transmission phenomena occur in both the layers. The reflection and transmission coefficients in both layers due to incident waves of both modes are analyzed with the aid of perturbation analysis along with Fourier transform technique. It is found that these coefficients are obtained in terms of integrals which depend on the shape function of the undulating bottom. Two different kinds of undulating bottoms are considered to determine these coefficients. For a particular undulating bottom, namely sinusoidal bottom undulation the effect of various physical parameters such as number of ripples, surface tension and porous effect parameters are demonstrated graphically. The study further elaborates the energy balance relations associated with the reflection and transmission coefficients to ascertain the correctness of all the computed results.

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Section: Introductionmentioning

confidence: 99%

Water-Waves Scattering by Permeable Bottom in Two-Layer Fluid in the Presence of Surface Tension

Panda

Martha

2017

Mathematical Modelling and Analysis

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“…In the present study, the work of Harter et al (2007) for surface-piercing bodies has been extended to include two distinct choices of contact-line conditions, namely the fixed-edge and zero-slope conditions. For problems involving localized oscillations, these are appropriate conditions to apply since in both cases there is no dissipation of energy at the contact points.…”

Section: Resultsmentioning

confidence: 99%

“…We start by considering two sorts of potential. The first is the source derived in Harter et al (2007); this is given by…”

Section: Solution Of the Boundary-value Problemmentioning

confidence: 99%

“…In fact, to the authors' knowledge, the only investigations of this type have been by Groves (1998) who considered trapped modes in a horizontal channel, and Harter et al (2007) where the studies of McIver (1996McIver ( , 2000 were extended. However, Harter et al (2007) did not include a contact-line condition for the surface-piercing case; its main focus was on the submerged bodies found by McIver (2000), which does not require an additional condition when capillarity is considered. In the present study, the work of McIver (1996) will be extended to include surface tension while using two physically realistic contact-line conditions.…”

Section: Introductionmentioning

confidence: 99%

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The effect of surface tension on localized free-surface oscillations about surface-piercing bodies

2008

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It is well known that the inclusion of surface tension in the linear water-wave problem introduces an additional term in the free-surface boundary condition. Furthermore, if the fluid contains one or more partially immersed bodies, edge conditions describing the motion of the fluid at each contact line need to be applied. In this paper, an inverse procedure is used to construct examples of two-dimensional, surface-piercing trapping structures for non-zero values of surface tension. The problem is considered with two separate edge conditions that are appropriate for investigations involving trapped modes. The first edge condition fixes each contact line and the procedure used forces the bodies to be horizontal at the contact points; it is shown that results can be found for all values of surface tension. The second condition forces the free-surface slope to be zero at the contact points, and results are obtained for a restricted range of surface tension values.

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“…Very few attempts have been made to include the effect of surface tension in water wave problems in a single-layer fluid involving a fixed or floating structure (Evans [4,5], Harter et al [6,7] and Rhodes-Robinson [20,21]). An ocean contains more than one fluid layer (e.g., fresh water layer, salt water layer, mud layer) and the scattering of waves in a such fluid is quite important.…”

Section: Introductionmentioning

confidence: 99%

The effect of free-surface tension on scattering of water waves by small bottom undulation

Mohapatra

2017

ANZIAMJ

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We examine the influence of surface tension at the free-surface on the problem of scattering of normal incident waves by small bottom undulation in a two-layer fluid within the framework of linearised wave theory. In such a two-layer fluid, there exist two different modes of time-harmonic waves for the given frequency, one of which corresponds to a free-surface disturbance and the other to an interfacial wave motion. Using perturbation technique, the problem is reduced up to first-order to a coupled boundary value problem which is solved by a method based on Green's integral theorem with the introduction of appropriate Green's functions. The first-order approximations to the reflection and transmission coefficients evaluated in terms of computable integrals for both the modes. For a sinusoidal bottom topography, these coefficients are determined and the numerical results depicted graphically. When Contents E40the wave is incident due to the free-surface disturbance or the interfacial wave motion in a two-layer fluid, we always find energy transfer from the incident wave mode to the other wave mode. We conclude that realistic changes in surface tension on the free-surface have a significant effect on the problem of water wave scattering by bottom undulation in a two-layer fluid.

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The effect of surface tension on trapped modes in water-wave problems

Cited by 16 publications

References 15 publications

Water-Waves Scattering by Permeable Bottom in Two-Layer Fluid in the Presence of Surface Tension

Water-Waves Scattering by Permeable Bottom in Two-Layer Fluid in the Presence of Surface Tension

The effect of surface tension on localized free-surface oscillations about surface-piercing bodies

The effect of free-surface tension on scattering of water waves by small bottom undulation

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