Abstract
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.
Highlights
In recent times, many researchers have shown their interest in the field of scattering of water waves in a multi-layered fluid system instead of a single layer fluid system
The relevance of such consideration is mainly due to the following associated significance: (a) the sinusoidal bottom has the ability to reflect the incident wave energy which is important in coastal protection [10], and (b) the sinusoidal bottom undulation closely resembles some naturally occurring obstacle formed at the bottom due to sedimentation and ripple growth of sands
The above observations reveal that the effect of surface tension can be neglected in practice as the results presented in Figures 7 and 6 are smaller than one percent
Summary
Many researchers have shown their interest in the field of scattering of water waves in a multi-layered fluid system instead of a single layer fluid system. There is a possibility that some of the incident wave energy may transfer from one mode to another In this context, Linton and McIver [15] studied the problem involving scattering of water waves in the presence of a long horizontal cylinder present in either of the two layers in a two-layer fluid. Linton and Cadby [14] have extended the problem of Linton and McIver [15] to the scattering of oblique waves. They have studied the reflection and transmission phenomena with the help of multipole expansion method. It should be noted that, these kind of problems arise while modelling an underwater pipe bridge across one of the Norwegian fjords which consists of a layer of fresh water on top of a deep layer of salt water
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