Self-similar solutions describing the propagation of solitary waves above underwater ridges and troughs
Tóm tắt
The nonlinear ray method [1] is used to investigate the propagation of solitary waves over an uneven bottom. In the process of nonlinear evolution of the wave front, singular points develop in it; these are treated in the given model as discontinuities [2, 3]. In contrast to earlier studies, it is not assumed here that the intensity of the discontinuity is weak. Boundary conditions at the discontinuities are introduced on the basis of the results of Miles and Bakholdin [4–6], and this makes it possible to take into account the energy loss at a discontinuity and the effects of wave reflection and construct a number of new self-similar solutions for the propagation of a wave above a ridge and trough. The main attention is devoted to considering how the type of solution depends on the parameters of the wave and the relief. For certain values of the parameters, the self-similar solution of the encounter of a homogeneous wave with a ridge is not unique. The reason for this is the singularity of the relief at the end of the ridge. A numerical investigation has therefore also been made of the encounter of a wave with a ridge having a smooth relief at its end. For an under-water trough and a ridge—trough system, self-similar solutions with complete or partial reflection or transmission of the wave energy into the trough are found. A reflected wave can also arise from an encounter with a ridge.
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Tài liệu tham khảo
R. Grimshaw, “The solitary wave in water of variable depth. Pt. 2,” J. Fluid Mech.,46, 611 (1971).
V. A. Reutov, “Behavior of perturbations of solitary and periodic waves on the surface of a ponderous liquid,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 156 (1976).
A. G. Kulikovskii and V. A. Reutov, “Propagation of nonlinear waves above semi-infinite underwater troughs and ridges,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 53 (1980).
J. W. Miles, “Obliquely interacting solitary waves,” J. Fluid Mech.,79, 157 (1977).
J. W. Miles, “Resonantly interacting solitary waves,” J. Fluid Mech.,79, 171 (1977).
I. B. Bakholdin, “Discontinuities of the variables characterizing the propagation of solitary waves in a layer of liquid,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 87 (1984).