Construction of wave-free potentials and multipoles in a two-layer fluid having free-surface boundary condition with higher-order derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.     

三层流体中被淹没球体的波浪散射研究（英文）

Using linear water wave theory, three-dimensional problems concerning the interaction of waves with spherical structures in a fluid which contains a three-layer fluid consisting of a layer of finite depth bounded above by freshwater of finite depth with free surface and below by an infinite layer of water of greater density are considered. In such a situation timeharmonic waves with a given frequency can propagate with three wavenumbers. The sphere is submerged in either of the three layers. Eac...     

线性水波理论中远场无波势的构建(英文)

Various water wave problems involving an infinitely long horizontal cylinder floating on the surface water were investigated in the literature of linearized theory of water waves employing a general multipole expansion for the wave potential. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation (for non-oblique waves in two dimensions) or two-dimensional Helmholz equation (for oblique waves) satisfying the free surface condition and decaying rapidly away from the point of singularity. The method of constructing these wave-free potentials is presented here in a systematic manner for a number of situations such as deep water with a free surface, neglecting or taking into account the effect of surface tension, or with an ice-cover modelled as a thin elastic plate floating on water.