In §§ 2 and 3 of the present paper the general conditions under which second-order pressure variations of this latter type will occur are considered. However, it was recently shown by Miche that in the second approximation to the standing wave there is a second-order pressure variation which is not attenuated with depth and which must therefore ultimately predominate over the first-order pressure variations. In the past it has been considered unlikely that ocean waves are capable of generating microseismic oscillations of the sea bed over areas of deep water, since the decrease of the pressure variations with depth is exponential, according to the first-order theory generally used. Results obtained by the second method for an ellipsoid also agree very well with existing results. For three-dimensional problems, test computations for a sphere in heave and sway motion by both methods agree well with others. Test results obtained by these methods for the case of a heaving circular cylinder in water of finite or infinite depth agree well with those obtained by others, except for the behavior of the added-mass coefficient at low frequency in water of finite depth. Both methods place no restrictions on the geometry of the body nor the bottom topography and are valid for either two- or three-dimensional problems. The second method, termed fundamental-singularity distribution method, works with the fluid boundary only. A modified variational method utilizing eigenfunctions expansions proves to be much more efficient. The first method, termed finite-element variational method, applies the conventional variational form for this type of boundary value problems. Two numerical methods for solving boundary-value problems related to potential flows with a free surface are introduced in this paper. This fact substantially accelerates theĬonvergence of the modal series and ensures the uniform convergence of the velocity field Sloping-bottom mode is included in the representation. Mode number, to O(n−4), when the additional Moreover, it is numerically shown that the rate of decay of the modal-amplitudeįunctions is improved from O(n−2), where n is the Satisfaction of the bottom boundary condition and, thus, it is energy conservative. To the previous extended mild-slope equations when the additional mode is neglected.Įxtensive numerical results demonstrate that the present model leads to the exact The coupled-mode system obtained in this way contains an additionalĮquation, as well as additional interaction terms in all other equations, and reduces This series consists of the vertical eigenfunctionsĪssociated with the propagating and all evanescent modes and, when the slope of theīottom is different from zero, an additional mode, carrying information about theīottom slope. In the present work, a consistent coupled-mode theory is derived fromĪ variational formulation of the complete linear problem, by representing the verticalĭistribution of the wave potential as a uniformly convergent series of local vertical & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to Over variable bathymetry regions, recently proposed by Massel (1993) and Porter During the plan-shape design of a harbor based on numerical simulation, attention can be focused on ordinary modes, and extreme modes can be ignored to a certain extent.Įxtended mild-slope equations for the propagation of small-amplitude water waves Hence, extreme modes can have a significant effect on the harbor only under remarkably ideal conditions. Therefore, for extreme modes, the widths of their resonant peaks in the amplification diagram are narrow. Only a few special wave frequencies can result in such a flow field. The wave energy inside the harbor increases very slowly because of the unusually weak flow field through the entrance excited by the extreme modes, thus resulting in an exceedingly long time required for the development of the extreme modes. Wave nonlinearity can mitigate the response of extreme modes significantly by transporting the energy to superharmonic components, thereby reducing the time required to reach a steady state. The smaller a harbor entrance is, the more severe the response of the extreme modes will be. The obtained results show that the entrance width has a significant effect on extreme modes, which causes a variation in the extreme modes for different harbor shapes. The difference between the extreme modes for various harbor shapes, the influence of nonlinearity, and the formation mechanism is comprehensively investigated. An extended mild-slope equation and a fully nonlinear Boussinesq equation are used to study those modes (termed as extreme modes in this paper). In the studies of harbor oscillations, some modes with extremely narrow amplification diagram are significantly common.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |