Within the framework of two-dimensional linear water wave theory, we consider the problem of normal surface wave propagation over small undulations in a channel flow consisting of a two-layer fluid with the upper layer bounded by a fixed wall, considered as an approximation to the free surface, and the lower one by a bottom surface which has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first order corrections to the velocity potentials in the two-layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient is one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. But the theory breaks down at resonance and it is applicable only to infinitesimal reflection when the reflection coefficient cannot assume large value, and away from resonance. Hence, the results demonstrated here is valid for up to the near resonant cases only. Again, when a patch of sinusoidal ripples having two different wave numbers is considered, the resonant interaction between the bed and the interface attains in the neighborhood of two (singular) points along x-axis (when the ripple wave numbers of the bottom undulation become twice as large as interface wave number). The theoretical observations are presented in graphical form.

Engineering materials are vulnerable targets for damage by chemical agents. This results in various types of irregular cavities which may subsequently change their shape under the combined action of loads and chemical attack. Such shape evolutions are subject to certain constraints. This paper explores the evolution in stresses as a result of an evolution in the shape of an isolated irregular hole in an infinite elastic plate subjected to remote uniform stress. The constraint employed here is a fixed area for the irregular hole with variable perimeter as a result of the evolution. Increase in perimeter implies decrease in strain energy on account of increased surface energy. Such phenomena could also occur in polymeric sheets on account of viscoelasticity even in the absence of chemical agents. This paper presents the evolution in boundary stresses as the cavity evolves to take different shapes. Complex variable methods are developed to tackle three cases of remote loading: (a) hydrostatic tension, (b) uniaxial tension, and (c) pure shear state. Of the above three cases, the first case of hydrostatic loading leads to a remarkably simple result for the boundary stress as shown in this paper. The last case is obtained by superposing a uniaxial tension and uniaxial compression along orthogonal directions.

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.

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