Water Wave Field Research Articles

Acoustic mode coupling induced by internal wave fields in shallow water

The problem of the propagation of acoustic waves in a stochastic ocean waveguide, for which the sound-speed variability within the water column is treated explicitly as a random variable, is addressed. The sound speed is composed of a deterministic, time-independent profile and two time-dependent stochastic components representing a (linear) background Garrett–Munk internal wave field and (nonlinear) internal wave soliton packets. A high-angle elastic parabolic equation method is used to compute single-frequency realizations of the pressure field using this representation of the sound-speed fluctuations. Transmission loss and scintillation index measures are estimated for both the full field and its modal decomposition at various ranges from the acoustic source, for different source depths and for both flat and sloping bottoms. These measures are incorporated in the analysis of acoustic modal coupling induced by the internal wave fields as a function of range; results support a recent prediction [D. Creamer, submitted to J. Acoust. Soc. Am.] that the scintillation index increases exponentially with range due to the competition between mode coupling and mode stripping found in shallow water waveguides. a)Permanent address: Forschungsanstalt der Bundeswehr für Wasserschall-und Geophysik, Klausdorfer Weg 2-24, 24148 Kiel, Germany.

The 1:2 resonance withO(2) symmetry and its applications in hydrodynamics

A geometrical study of the 1∶2 resonance in Hamiltonian systems withO(2) symmetry is presented. Several applications that can be found in the field of water waves are reviewed: two-dimensional capillary-gravity surface waves, gravity waves in a circular basin, two-dimensional gravity waves on a stably stratified two-layer fluid. All these waves can resonate, i.e., the fundamental mode can resonate with its second harmonic when the physical parameters approach some critical values. Use is made of the reduced Hamiltonian to formally derive a system of equations describing the dynamics of resonant waves. The resulting system does not depend on the physical application, except for the value of some coefficients. This system is then studied mathematically through the use of geometrical methods (projection on the orbit space and polar blowing-up of the vector field). It turns out that the additional symmetry of time reversibility plays an important role in the analysis of the system. The analysis shows that the presence of the0(2) symmetry brings nontrivial modifications to the classical 1∶2 resonance. For example, the persistence of the well-known one-parameter family of homoclinic orbits to “puremode” travelling waves is no longer obvious. Such homoclinic orbits also exist to “pure-mode” standing waves, but when the “tail” of the normal form truncation is no longer neglected, only two of them can be shown to persist, whereas an argument is given that strongly suggests the generic formation of a chaotic orbit nearby.

An Evaluation of the Interaction Between Surface Waves and a Simplified OTEC Intake Flow

The interaction between a steady linear water-wave field and a two-dimensional current, selected to approximate the surface intake of a large floating ocean thermal energy conversion (OTEC) plant, is examined. The mathematical formulation is based upon the use of averaged Lagrangians, and the determination of rays, of characteristics, numerical results in nondimensional form are obtained, for a headsea configuration, while relative intake strength (current velocity) is allowed to vary. Energy focusing is shown to occur in the vicinity of the intake walls, although this effect linearly depends upon relative intake strength. Simple dimensional arguments, however, indicate that such interaction effects are weak when a “typical” large OTEC plant is considered.

Energy branching of a subsonic flexural wave on a plate at an air–water interface: Transition radiation and the acoustic wave field in water

The diffraction of subsonic flexural plate waves due to a discontinuity in fluid-loading is experimentally investigated. A tone burst of flexural waves propagates down a plate, the lower section of which is submerged in water. Observations indicate that there occurs a branching of energy as the flexural wave passes through the air–water interface. A portion of the energy continues along the plate as a subsonic flexural wave with an associated evanescent wave. A second acoustic wave (which is termed transition radiation) originates at or near where the plate crosses the interface, and propagates in water to the far field. In the near field of the interface there exists an interference between the two acoustic waves in water that results in a series of pressure nulls. The pressure nulls are associated with a π phase change in the wave field and are indicators of wave front dislocations [P. L. Marston, ‘‘Geometrical and Catastrophe Optics Methods in Scattering,’’ Physical Acoustics (Academic, New York, 1992), Vol. 21, pp. 1–234]. A computation of the wave field in an unbounded fluid due to a line-moment excitation of a plate is comparable with the null pattern observed but differs in certain details. [Work supported by ONR.]

The Directional Response of Ocean Waves to Turning Winds

The directional response of ocean waves in turning wind situations has been studied with detailed wind and wave observations in open sea and with numerical simulations of the physical processes involved. The observations were acquired with pitch-and-roll buoys in the central and southern North Sea. They are selected and corrected to represent locally generated homogeneous wave fields in deep water. The response time scales thus obtained agree well with one published dataset. The disagreement with other published datasets is shown to be due to differences in analysis techniques, at least partially. The numerical simulations are carried out for homogeneous situations in which a constant wind suddenly shifts direction or rotates. These simulations show that the atmospheric input to the waves tends to rapidly turn the mean wave direction to the new wind direction. This, however, is opposed by whitecapping dissipation and nonlinear wave-wave interactions. The effect of whitecapping on the turning rate of the waves is of the same order of magnitude as that of the atmospheric input (but of opposite sign), but that of the nonlinear interactions is one order of magnitude smaller. Both observed and simulated time scales depend on the stage of development of the wave field, but the simulated time scales are considerably larger than the observed time scales.

Wave transmission and reflection characteristics of a rigid surface and submerged horizontal plate

Abstract The wave transmission and reflection characteristics of a rigidly fixed surface and submerged horizontal plate were investigated experimentally in detail for a wide range of incident wave steepnesses and for different depths of submerge of the plate in deep water conditions in regular water wave fields. The experiments were conducted at the Ocean Engineering Centre, Indian Institute of Technology, Madras, India, in a wave flume 10 m long, 0.3 m wide and in a constant water depth of 0.8 m. The horizontal plate is 0.22 m thick and 1.2 m in length, covering the enrire width of the flume. From the present investigation, it is found that for a rigid surface plate, the coefficient of transmission is a minimum and the coefficient of reflection is a maximum, but the maximum value of the coefficient of energy loss occurs for plates submerged closer to the still-water level (SWL) and not for the surface plate. It is also found that the value of the coefficient of reflection increases with the increase in the value of the Reddy-Neelamani ( RN ) number, the ratio of horizontal water particle excursion at the bottom of the plate in its absence to the length of the plate. The coefficient of transmission is found to decrease rapidly with increase in the value of RN number up to 0.1. The wave transmission is only 5% for RN from 0.1 to 0.2. It is also found that for RN number greater than 0.04, the minimum energy dissipation is consistently about 60% of the incident wave energy.

Vindication of mode-coupled descriptions of multiple-scale water wave fields

Herein we show that the modal description of deep-water waves on the sea surface Watson & West 1975) is independent of any reference surface around which expansions of the velocity potential and the surface velocity are done. We demonstrate by direct construction that the interaction between long and short waves does not lead to divergent terms in the equations of motion when this formalism is used.

Wind and Wave Induced Currents in a Rotating Sea with Depth-varying Eddy Viscosity

A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of a version of the Navier-Stokes equations in Lagrangian coordinates. The Coriolis effect and the effect of a depth-dependent eddy viscosity are included. (The eddy viscosity is taken to depend on the Lagrangian vertical coordinate ĉ.) Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. The vertical variation of the eddy viscosity causes an extra source term to appear in the equation for the evolution of the current profile. This additional source of momentum within the water column is exactly balanced by an extra term in the surface boundary condition, which in turn represents the contribution to wave dissipation caused by the eddy viscosity within the water column being different from its surface value. The equations were solved numerically, using a constant wind stress and monochromatic wave field simultaneously applied in the same direction at time t = 0. The eddy viscosity ν was assumed to be proportional to depth, using Madsen's relation (ν = −kKu*ĉ, where kK is von Kármán's constant, u* = (^&tau/ ρ)½ where ^τ is the wind stress and ρ is the water density), except for near the surface where it was modified to account empirically for the direct effects of breaking waves. Results for the Lagrangian mean current were in general agreement with observations; long-term average values ranged from 2.2% to 2.8% of the wind speed at the 10 m level, and were directed between 12° and 17° to the right of the wind and wave direction (in the Northern Hemisphere). The deviation of the current from the wind direction is closer to observed drift current observations than the corresponding results for a constant eddy viscosity. The Lagrangian mean current is surprisingly close to the current obtained from Madsen's theory, even though Madsen does not account explicitly for the effect of surface waves. The theory can easily take account of random sea states. There are good prospects for coupling it with the output of a numerical model for surface gravity waves, using the wave model's input and dissipation source terms.

A Theory for Steady and Variable Wind-and Wave-Induced Currents

Abstract A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of the Navier-Stokes equations in Lagrangian coordinates. The effects of rotation and of a constant eddy viscosity are included. Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. Some simple analytical solutions are presented. For small viscosities, a near-zero mean mass transport is obtained, in agreement with Ursell. Inertial oscillation are superimposed on the above mean solution, in agreement with Hasselmann and Pollard. In the case of a constant wind stress and a constant, horizontally homogeneous wave field, the steady-state results of Weber are reproduced (a surface drift current of about 3% of the wind speed, 23–30 deg to the right of the wind direction).

Water Waves and Circular Damping Regions

An analytical formulation is presented for the calculation of the water wave field in and around a circular region of wave damping, such as might occur over a kelp bed. Figures show the instantaneous wave field and wave amplitude along several transects across the circular damping region.

Statistical properties of water waves. Part 1. Steady-state distribution of wind-driven gravity-capillary waves

The Miles–Phillips model of the linear coupling between waves on the ocean surface and a fluctuating wind field is generalized to include the average effect of the nonlinear water-wave interactions in the dynamic equations for gravity–capillary waves. A statistical-linearization procedure is applied to the general problem and yields the optimum linear description of the nonlinear terms by linear terms. The linearized dynamic equations are stochastic with solutions that have stable moments, i.e. the average nonlinear interactions quench the linear instability generated by the coupling to the mean wind field. In particular, an asymptotic steady-state power-spectral density for the water-wave field is calculated exactly in the context of the model for various wind speeds.

Steady state spectral density of gravity‐capillary waves

The linear coupling model of Miles and Phillips for the air‐sea interaction is generalized to include the average effect of the nonlinear interactions in the dynamic equations for the gravity‐capillary waves. The linearized dynamic equations are stochastic with solutions that have stable moments. In particular, a steady state power spectral density for the water wave field is calculated exactly in the context of the model for various wind speeds.

Model of gravity wave growth due to fluctuations in the air‐sea coupling parameter

The linear Miles‐Phillips model of the coupling between water waves and a turbulent wind field is generalized to include fluctuations in the air‐sea coupling parameter in the dynamic equations. The model is solved exactly, assuming the fluctuations to be delta correlated in time. The energy spectrum of the water wave field is shown to grow exponentially at rates higher then those predicted by the linear Miles‐Phillips model. These increased initial growth rates are consistent with those observed in open ocean measurements.

Optical technique for measurement of random water wave surfaces

An optical system for measuring the slope of random wind-generated water waves is described. The instrument employs the refraction of a vertical light ray, which is a technique that has been previously used. The present system, however, employs a large-area array of individual photovoltaic cells to measure the refraction angle of the beam. Because the array provides a much larger active area with no preferred direction, it is much better suited to the measurement of random water wave fields, typical of those generated by wind.

Covariance Functions and Spectra of a Random Wave Field

Abstract Covariance functions and spectra of components of fluid particle velocity are obtained, taking into consideration the effects of free surface fluctuations, for a Gaussian, stationary and homogeneous random gravity wave field in deep water, using infinitesimal wave solutions. Approximate representations of the covariance functions and spectra are also derived. It is shown that the covariance functions and spectra presented in this paper differ from those when the effects of free surface fluctuations are ignored, especially at, around and above the equilibrium surface.

Analysis of inhomogeneous wave number spectra

As waves propagate from deep to shallow water, various effects of the shallowing depth of water spatially modify spectral characteristics of the sea surface. In such spatially inhomogeneous regimes, spectral 99putations based on a finite field size are subject to errors due to an inherent spatial smoothing in addition to resolution errors of a more conventional nature. The conflicting requirements of resolution and smoothing errors for an accurate analysis prescribe an optimal field size for which the combined magnitude of such errors is minimal. The optimal field size, the associated errors in resolution and smoothing, and the conditions for limiting accuracy are derived for computations in a shallow water wave field in which refraction and shoaling constitute the predominant inhomogeneity effects. Bottom friction, percolation, and reflection are neglected, and calculations are based on a first-order approximate shallow water wave theroy. entirely appropriate and effective in the study of the ocean sur- face. These techniques offer certain advantages over various other techniques for the computation of two-dimensional wave number spectra: the advantages of economy and of covering large areas in a short time..Stilwell ( 1969) and Stilwell and Pilon (1974) developed and demonstrated an operational photographic system capable of providing quantitative es- timates of surface wave spectra for nonstationary and spatially homogeneous wave fields and suggested further applications of these techniques in the study of wave propagation, interac- tion, and generation. Although several limitations resulting from wave slopes, solar angles, sky luminance, haze, etc., influence the accuracy of spectral estimates obtained with photographic techniques, the applicability of these methods is not severely restricted un- less a fundamental assumption embedded in such techniques is violated. This is the assumption of spatial homogeneity, i.e., the requirement that the statistical properties of the sea surface remain essentially the same over the area to be spectrally analyzed. The violation of this assumption means that if the size of the area to be photographed is too large, the spectral characteristics of a given location may be smoothed or smeared out by contamination from the neighboring areas in the estimation process. On the other hand, if the size of the area photographed is too small, the accuracy of the resulting estimates will be limited in terms of spectral resolution. Recently, in applying photooptical techniques in shallow water, Klemas et al. (1974) pointed out that the spatial ir- homogeneity of the shallow water wave field requires choosing an optimal field size. Co0sequently, questions arise as to the rational basis for such a selection, the criteria governing the accuracy of the associated shallow water spectral estimates, and the limiting conditions under which the applicability of photographic techniques becomes doubtful. Polis ( 1974), using Heisenberg's uncertainty principle, showed the possibility of choosing an optimal field size on the basis of a trade off that must be made between the intrinsic spatial variation of a wave number in shallow water and the increase in the resolution of the wave number from a larger field size. Although this con- cept is interesting as an optical analogy and obviously useful in examining a regular train of waves in shallow water, its ap.

Combined effects of current and waves on fluid force

Combined effects of current and waves on the force exerted on an element of a cylinder in a random gravity wave field in deep water are studied. Wave-current interactions are taken into account. Statistical quantities of the fluid force such as force spectrum and root mean square value of the force are obtained numerically and presented in graphical forms. Comparisons are made of the cases in which wave-current interactions are considered and ignored. It is shown that wave-current interactions contribute to changes in fluid force to an appreciable extent and therefore should be considered in the evaluation of fluid forces on objects.

REVISIONS TO HURRICANE DESIGN WAVE PRACTICES

The 1959 paper "Hurricane Design Wave Practices" (ref. 1) has been widely used in the past for obtaining design wave criteria. Additional wave data and revisions in wave forecasting procedures, including computing techniques, ideas and experience, make it possible to bring these techniques up to date. This paper should be considered also as an extension of the paper "A Non- Dimensional Hurricane Wave Model" (ref. 2) as well as revisions to the 1959 paper (ref. 1). Graphs, formulae and procedures are presented making it possible to calculate the entire deep water wave fields from model hurricane wind fields. The revisions have been applied to the U.S. East and Gulf coasts past historical hurricanes and also to the U.S. Weather Service standard project and probable maximum hurricanes for deep water conditions. The results of these calculations are presented in figures and tables and can serve as inputs for particular locations to calculate design storm surge and design wave criteria over the continental shelf to the coast line, making use of the material in the references listed at the end of this paper.

SHALLOW WATER WAVES A COMPARISON OF THEORIES AND EXPERIMENTS

A series of experiments were performed to determine the velocity field and other characteristics of large amplitude shallow water waves. The experimental results were compared with the predictions of a variety of wave theories including those commonly used in engineering practice. While no theory was found exceptionally accurate, the cnoidal wave theory of Keulegan and Patterson appears most adequate for the range of wavelengths and water depths studied.

Dynamics of Liquids in Moving Containers

M interest in the behavior of liquids in various containers is undoubtedly very old. The phenomena of liquid motion within a containing vessel presented a challenge as a natural curiosity to the hydrodynamicist of the past centu^, and continues to present difficulties for the engineer today. Civil engineers and seismologists have been very active, particularly since the early 1930's, in studying earthquake effects on large dams, oil tanks, reservoirs and elevated tanks. More recently, particularly with the advent of the jet age, the effects of fuel motion in partially full tanks have, of necessity, been included in some aircraft dynamic stability studies. With the appearance of the large liquid propelled rocket, where at least 90 per cent of the gross weight at liftoff is contributed by the liquid propellants, the forces and moments produced by the movement of large masses of liquids within their containers take on added importance. The exponential increase in the number of reports and papers on the subject of sloshing/' during the past decade is a reflection of this trend. It will be noted that, with the obvious exception of the work on centrifugal slosh/' only those papers have been surveyed here which consider the liquid to be subject to a gravitational force. Problems of bubble formation and growth, the behavior of liquids in free fall, vortex formation in draining tanks, and other related problems have not, in general, been included. Moreover, attention has been mainly restricted to the American literature, although some British, Russian and a few references in Japanese journals are included. The French, German and Italian literature, probably very rich in this field, remains relatively untouched. An extensive bibliography pertaining to the general field of water waves (136) * and two sloshing literature surveys, containing many abstracts, are available (141,142). Attention is also directed to (45) for special studies pertaining to specific rocket vehicles. It should be mentioned that a very large percentage of the reports cited here were carried out under Government contract and are available through the ASTIA offices.