Long Water Waves Research Articles

Long Waves in Ocean and Coastal Waters

Long Waves in Ocean and Coastal Waters

Nonlinear tidal waves in channels: A perturbation method adapted to the importance of quadratic bottom friction

Bottom friction plays an important role in propagation and damping of long waves in shallow water. A perturbation method, well adapted to the importance of this bottom stress, is presented and applied for channels with constant mean depth and mean cross-section area. The main idea results from a development of quadratic bottom friction in a Fourier series up to the third order of approximation. A quasi-linearization of the damping effect of bottom stress is deduced from this expansion, which allows one to introduce the second-order damping effects of friction into the first-order system defining the fundamental component of the spectrum, and the main part of the third-order damping terms into the computation of the second-order harmonic components. From this expansion, the generating role of the harmonic played by friction is also identified and taken into account in the second-order approximation. Analytical and analytico-numerical solutions are presented and compared to numerical solutions of the full nonlinear equations obtained by the Lax-Wendroff finite-difference integration technique. These comparisons show that the analytical solutions limited to the second-order approximation fit very well with the L.W. solutions. DOI: 10.1111/j.2153-3490.1980.tb00942.x

Some properties of a hybrid element method for water waves

AbstractTwo properties of hybrid element method for diffraction and radiation of water waves are examined. For long waves in shallow water the method is shown to give a unique solution for all frequencies. Thus, unlike several other known methods, there are no irregular frequencies for which the approximating matrix equation is singluar. For a sea of arbitrary depth, it is shown that all known global identities such as reciprocity and energy theorems are preserved by the discrete solution. Thus, satisfaction of these identities by the numerical solution is only a necessary but by no means sufficient condition for accuracy.

A Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves

The surface water waves are modal waves in which the physical (t, x, y, z) is the product of a propagation space (t, x, y) and a cross space, the z-axis in the vertical direction. We have derived a new set of equations for the long waves in shallow water in the propagation space. When the ratio of the amplitude of the disturbance to the depth of the water is small, these equations reduce to the equations derived by Whitham (1967) by the variational principle. Then we have derived a single equation in (t, x, y)-space which is a generalization of the fourth order Boussinesq equation for one-dimensional waves. In the neighbourhood of a wave froat, this equation reduces to the multidimensional generalization of the KdV equation derived by Shen & Keller (1973). We have also included a systematic discussion of the orders of the various non-dimensional parameters. This is followed by a presentation of a general theory of approximating a system of quasi-linear equations following one of the modes. When we apply this general method to the surface water wave equations in the propagation space, we get the Shen-Keller equation.

On the virtual-mass and damping coefficients for long waves in water of finite depth

A half-immersed circular cylinder is making vertical oscillations on water of finite constant depth. The virtual-mass and damping coefficients are studied in the limit as the wavelength tends to infinity. It is found that the virtual-mass coefficient tends to a finite limit and that the amplitude ratio ultimately varies as the frequency. This behaviour differs from the behaviour for infinite depth, where the virtual-mass coefficient tends to infinity and the amplitude ratio ultimately varies as (frequency).

Tides in shallow water: Spectroscopy

Non-linear distortions of long waves (tides) in shallow water are analysod. The effects of finite wave amplitude, bottom slope, friction, and multi-chromatic interactions are investigated separately and in combination, using both perturbation analysis and numerical solutions of the full non-linear equations. The distortions caused by each source of non-linear behavior are presented in the time and frequency domains, and some implications of the results for local and ocean-wide tide predictions are discussed. DOI: 10.1111/j.2153-3490.1971.tb00581.x

The shallow water equations

In this paper a set of exact nonlinear equations is derived for gravity flows. By assuming the flow to be shallow and assuming the vertical acceleration to be small these equations reduce to the classical equations for long waves in shallow water. If only shallowness is assumed a set of equations results, which admits in the steady case periodic solutions for Froude numbers smaller than 1 and laminar jumps for Froude numbers larger than 1. In the last section potential flows are discussed.

Effect of Viscosity on Long Waves in Shallow Water

A linearized theory is used to investigate the effect of viscosity on long gravity waves in shallow water. The propagation of an initial disturbance to the surface is considered and an asymptotic solution to this problem is obtained.

Long waves on a beach

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

Note on the equations of Long waves over an uneven bottom

Equations for long waves in shallow water are derived systematically for an uneven bottom. With the basic assumption that the depth is small in comparison with a horizontal length scale, three regimes of approximation are presented according to the magnitude of the wave amplitude, (a) When the amplitude is of the same order of magnitude as the depth, the Airy equations are rederived as the first approximation, (b) When the amplitude is comparable to the cube of the depth, both lengths being nondimensionalized with respect to a common horizontal scale, two high-order nonlinear equations are obtained which include the classical cnoidal wave as a special solution for a horizontal bottom. These equations may be transformed to a set of first-order quasi-linear hyperbolic equations with the characteristic curves in the x-t plane directly related to the bottom profile. To facilitate numerical computations they are then written as differential equations along the characteristics. In the third regime (c) of extremely small wave amplitudes the appropriate linearized equations are given.

Propagation of long waves due to atmospheric disturbances on a rotating sea

Long waves in shallow water in a non-rotating system are not dispersive but in a rotating system they are. This paper investigates the generation and propagation of these dispersive waves in an infinite sea. The mode of generation is by air-pressure gradients or wind stresses applied to the surface. Bottom friction is neglected. The surface elevation due to a stationary force of constant amplitude suddenly applied and maintained at t = 0 over one-half of an infinite sea is shown to approach, through a series of oscillations approximating more or less to an inertia period, a steady-state amplitude decreasing with distance from the generating area, The longitudinal and transverse velocities are also given. The time elapsed from the initial disturbance at a point to the first maximum of elevation decreases with the distance of the point from the edge of the generating area. A generating area whose edge moves forward with the maximum group velocity of the waves is shown to lead to an elevation of ever-increasing height. The effect of a barrier placed at right angles to the direction of propagation is also briefly considered.

The long-wave paradox in the theory of gravity waves

ABSTRACTThe theory of long waves in shallow water under gravity employs two different approaches, which have given rise to a well-known paradox remarked by Stokes, but hitherto not fully resolved. On the one hand it was shown by Airy (1) that, if the pressure at any point in the fluid is equal to the hydrostatic head due to the column of water above it, then no wave form can be propagated without change hi shallow water of uniform depth; on the other hand, it was shown by Rayleigh's theory (8) of the solitary wave that this conclusion may be incorrect. In the present paper an attempt is made to elucidate the paradox. Waves of small amplitude η0 and large horizontal wave-length λ (compared with the depth h of the water) are studied, and it is shown that Airy's conclusion is valid if η0λ2/h3 ia large, whereas the solitary wave has η0λ2/h3 of order unity. Equations of motion are derived corresponding to large, moderate and small values of η0λ2/h3; these can be summarized in a single equation for the profile η(x, t):due to Boussinesq (3).It is also shown that the linearized theory of surface waves is valid only if η0/λ and η0λ2/h3 are both small. Some remarks are made on the generation of the solitary wave, and on the breaking of waves on a shelving beach.