Shallow-water Cases Research Articles

Nonlinear Hydrodynamic Loading on Offshore Structures

This paper is mainly concerned with the investigation of second-order wave loading on offshore structures. The diffraction theory of hydrodynamics is used to predict the wave loading. Lighthill's second-order theory has been extended to include the investigation to the shallow water case. Wave forces on cylindrical structures are calculated using Lighthill's theory, and the numerical results are compared with the available experimental data. The present theory encompasses the entire spectrum of ocean waves, from deep to shallow water cases, and makes a significant step forward for determining the wave loads on submerged structures.

Weakly nonlinear internal waves in a two-fluid system

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin–Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

A second-order solution of waves passing porous structures

Only linear theoretical analyses of wave interaction with porous structures exist, mainly due to both the complexities of flows inside the porous medium, and the mathematical inhomogeneous boundary-value problem. Since the hydrodynamic flow mechanism is non-linear a non-linear analysis can better describe the characteristic nature of the problem. In this paper, a generalized potential theory is used to describe both the internal and external water flows. An implicit non-linear model is used to describe flow mechanism inside the porous medium. The perturbation method is used to solve the problem analytically up to the second order. The second-order solution is decomposed into time-dependent and time-independent parts. And, correspondingly, the inhomogeneous boundary-value problems are solved analytically. In the analysis, the second-order characteristics of the problem, including the dispersion equation, wave numbers and friction coefficient, as well as wave reflection and transmission, are investigated in detail. It is shown that the mode swapping of the second-order wave numbers only occurs among the evanescent modes. The second-order friction effects become important in shallow-water cases. The comparison of the results of present theory with experimental results shows that the second-order solution is good correction to the linear theory.

Edge waves revisited

For edge waves on an infinite beach of constant slope comparison is made between the full linear solution and the shallow water approximation. Modifications to the shallow-water edge-wave dispersion relation accounting for an offshore shelf are obtained and compared with an earlier approximation. A physically comprehensible description of edge waves is given, using a geometrical optics approach, leading to approximate expressions for their dispersion relation. Both shallow water and general depth are considered. For the shallow water case comparison is made with the exact relation for an exponentially seaward decreasing slope. Finally the maximum possible edge wave amplitude is estimated by heuristic arguments.

Localization of gravity waves on a channel with a random bottom

We present a theoretical study of the localization phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localization theory and applying it to the shallow-water case, we give the first study of the localization problem in the framework of the full potential theory; in particular we develop a renormalized-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localization length, which we compare with the viscous dissipation length. This allows the prediction of which cases localization should be observable in and in which cases it could be hidden by dissipative mechanisms.

On the application of HF ocean radar to the observation of temporal and spatial changes in wind direction

The application of a 30-MHz narrow-beam ground-wave ocean radar to the observation of wind directions is described. It is found that the \cos^{s} (\theta/2) model for wind-wave directions does not apply in a specific case of shallow water where swell waves are behaving nonlinearly. To experimentally extract unambiguous wind directions from this model requires sampling three different beam angles simultaneously. In practice some time and space stationarity is assumed. Detailed analysis in time and space reveals structure in the transition of the cold front from sea to land which, although unexpected, agrees with coastline observations where they are available. The nature of the structure is only briefly discussed. The response of the \lambda = 5 -m wind waves to the frontal change was two orders of magnitude faster than time constants for similar events previously modeled using pitch-and-roll buoy data. This discrepancy needs to be reconciled before lower frequency radars can be used without ground truth for wind-direction changes.

ON APPLICATIONS OF PARAMETRIC ACOUSTIC SOURCES TO SHALLOW WATER

As is shown by westervelt, the difference-frequency sound pressure of the parametric array in the far field is proportional to the second power of f/f0, the downshift ratio of the frequency. However, a different result may occure in the near field, especially, when the distence to the field point is much less than the length of the parametric array, but larger than R0, the Rayleigh length of the transducer. In this paper, our previous theory is applied to the case of shallow water and an expression for the parametric array design has been given. It is shown that the parametric pressure is proportional to Ro, instead of proportional to the length of the parametric array or to the second power of f/f0. Futhermore, an analysis for Moffett and Mellen's theory is also made in this paper and on identical expression has been obtained. Finally, an experiment is carried out in a pool and the results obtained support the theory.

A fast field model for three-dimensional wave propagation in stratified environments based on the global matrix method

A three-dimensional fast field program (FFP) model based on the global matrix method will be presented. Compared to traditional propagator matrix methods, the global matrix method allows for a more efficient, and, furthermore, numerically stable computation. The three-dimensional expansion does not restrict sources to be on the center axis, but allows for an arbitrary source geometry—as opposed to earlier two-dimensional versions. As a consequence, the solution includes simultaneously both vertically and horizontally polarized shear waves. A mathematical description will be given and the numerical aspects will be discussed. Some of the features of the model will be illustrated in two test cases: free-space and shallow water with strongly reflecting bottom. The free-space case shows that the three-dimensional solution gives results identical to those obtained by the two-dimensional model except for ranges close to the axis r≡0 of a cylindrical coordinate system. For the shallow water case, the well-known energy transportation in discrete modes above critical wavenumber is first demonstrated. Then the model is used to analyze the field radiated by a long horizontal array, and it is shown that different modes will propagate in slightly different directions.

Shallow Water Flow with Separation and Breaking Wave

The objective of the work described is not only to examine the flow in particular simple shallow water cases, but also to obtain a better fundamental understanding of the complicated flow field with separation and wave breaking.

MEASUREMENT OF SURFACE WAVES FROM SUBSURFACE GAGE

Wave measurement using subsurface pressure gage has many advantages, especially in coastal waters. However, there are some disadvantages associated with indirect measurement of surface waves. This paper deals with the problems in the recovery of surface waves from the subsurface pressure measurement. Two problems are examined: more exact recovery of the surface waves from subsurface pressure signal and a simple real time recovery of surface waves from the pressure record. The commonly used linear transformation from the subsurface pressure to surface elevation is evaluated and possible sources of error are examined. The effect of nonlinearity and effect of current are shown to be small in the intermediate water depth. For the case of shallow water where effects of nonlinearity and current are not negligible, a method of proper recovery of surface waves from the combination of pressure and current measurements is given. A simple method to recover the time series surface waves from subsurface record is proposed which enables speedy near real-time recovery of surface waves from pressure measurement using a microprocessor.

A Finite Element Approach to Steady Water Wave Problems in Deep Sea

Finite element formulations have been proposed by several authors for water waves caused by harmonic motions of a body. In any case, the water field is divided into two parts, the outer and inner domains, by a fictitious surface which surrounds the body under consideration. The solution in the outer domain, which will be called the outer solution, can be determined analytically within a number of unknown parameters, and it can be expressed by the eigenfunction expansion or the linear combination of Green's function and its derivatives. The solution in the inner domain, which will be called the inner solution, can be obtained by a numerical method such as the finite element method, and the unknown parameters in the outer solution are determined by the continuity condition for both solutions on the fictitious surface.In the case of shallow water, the formulation with the eigenfunction expansion in the vertical direction can be applied efficiently. In the case of deep water, however, Green's function has to be used as an analytical solution for the outer domain, which is inconvenient for treatments because of its complexity.In the present paper, a method of superposition of analytical and numerical solutions, which is proposed by Seto and the senior author1).2) for water wave problems, is modified for easy treatments of infinite or deep water field. The progressing part of the outer solution is determined for the actual depth of water, and the standing part is approximately determined by the eigenfunction expansion for a water depth restricted properly. The validity of this approximation for the outer solution can be acertained by the fact that the standing part decreases rapidly with the increase of depth. This analytical solution thus obtained does not necessarily satisfy the condition of continuity everywhere ; this difficulty can be solved by modifying the continuity condition properly on the fictitious surface.By this method, the inner domain is reduced to a great deal, and numerical computations can be performed efficiently. Example calculations by this method for two-dimensional or three-dimensional axi-symmetric wave propagation problems are performed by the aid of an input generator prepared, and the results obtained show good agreement with the previous authors'.

Mechanics of suspended sediment in random waves

The mechanics of suspended sediment under wave action are investigated. The results apply to regions well beyond the surf zone where the flow field, though irregular in appearance, is still amenable to mathematical expressions in gross terms. The motion of the suspended sediment particle and that of the fluid particle in a wave field are found to have an amplitude difference and a phase lag. The suspended sediment concentration is a hyperbolic trigonometric function of water depth. The relation reduces to a power law for shallow water cases. The value of power is found to be a function of particle settling velocity, fluid particle velocity, and wave number. Wave mass transport velocity is derived on the basis of turbulent boundary layer as the lower boundary condition. Finally, it is found that the rate of suspended sediment transport in a wind-generated random wave field can be approximated by a power function in terms of wind velocity.

Energy transfer between external and internal gravity waves

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.