Wave action on currents with vorticity

1999 ◽  
Vol 386 ◽  
pp. 329-344 ◽  
Author(s):  
BENJAMIN S. WHITE
Keyword(s):  
Phase Shift ◽  
Deep Water ◽  
Current Velocity ◽  
Wave Action ◽  
Ray Theory ◽  
Wkb Method ◽  
Interaction Of Waves ◽  
New Equation ◽  
Spatially Varying

The interaction of waves on deep water with spatially varying currents may be described by a ray theory, with the wave amplitudes determined by the principle of conservation of wave action (CWA). However, all previous deep water derivations of CWA are restricted to the case of an irrotational current. In this paper, both the ray theory and CWA are derived by a WKB method without the assumption of irrotationality. Also derived is a new equation for a spatially varying phase shift which is not predicted by the usual ray theory, and which, in general, displaces the positions of the wave crests by a distance on the order of a wavelength. This phase shift, which is caused by variations of the current velocity with depth, vanishes in the irrotational case, and so is in accord with the irrotational theory.

scholarly journals A Depth-Bistatic Bottom Reverberation Model and Comparison with Data from an Active Triplet Towed Array Experiment

2020 ◽  
Vol 10 (9) ◽  
pp. 3080
Author(s):  
Youngcheol Jung ◽  
Woojae Seong ◽  
Keunhwa Lee ◽  
Seongil Kim
Keyword(s):  
Time Series ◽  
Deep Water ◽  
Water Environment ◽  
Ray Theory ◽  
Reverberation Time ◽  
Environment Model ◽  
Line Array ◽  
Model Predictions ◽  
Towed Array ◽  
Good Agreement

In this paper, a depth-bistatic bottom reverberation model that employs the ray theory is presented. The model can be applied to an active towed array in the ocean. The reverberation time series are modeled under the depth-bistatic assumption and their Doppler shift is calculated based on the actual source–receiver geometry. This model can handle N × 2D range-dependent bathymetry, the geometry of a triplet array, and the Doppler motion of the source, targets, and receiver. The model predictions are compared with the mid-frequency reverberation data measured by an active triplet towed array during August 2015 in the East Sea, Korea. These data are collected with a variable depth source at mid-frequency and the triplet line array in a deep-water environment. Model predictions of the beam time series and its spectrogram are in good agreement with the measurement. In particular, we discuss the effects of the source and receiver depths on the reverberation in deep water observed in both the measured and modeled results.

The Equivalent Depth of Wave-Induced Scour Around Offshore Pipelines

2006 ◽  
Author(s):  
Mir Emad Mousavi ◽  
Abbas Yeganeh Bakhtiary ◽  
Nastaran Enshaei
Keyword(s):  
Deep Water ◽  
Numerical Models ◽  
Scour Depth ◽  
Offshore Pipelines ◽  
Equivalent Depth ◽  
Physical Experiments ◽  
Water Conditions ◽  
Installation Depth ◽  
New Equation ◽  
Wave Induced

Physical experiments have been conducted to study the scouring around a pipe over an erodible bed under waves motion. The results show that the depth of the wave-induced scour is reduced when the pipe is installed in a primary depth. It is also indicated that when the primary installation depth of the pipe, |e|, exceeds a specified depth, no scouring occurs underneath the pipe. This specific depth is called the Equivalent Depth of Wave-Induced scour, Se. The equivalent depth of wave-induced scour is estimated as a function of pipe diameter and the Keulegan Carpenter number. Also for prediction of the scour depth with respect to the primary installation depth of the pipe, a new equation is proposed. In addition, new equations are proposed for prediction of the scour width before the pipe is installed. The results and proposed equations are next simplified in the form of diagrams to be used in practice. The experiments cover small KC numbers (KC < 6) that represent deep water conditions. The results can be used either for the design purposes or for calibration of the numerical models.

The interaction of deep-water gravity waves and an annular current: linear theory

1993 ◽  
Vol 248 ◽  
pp. 153-172 ◽  
Author(s):  
Marius Gerber
Keyword(s):  
Linear Theory ◽  
Gravity Waves ◽  
Deep Water ◽  
Large Scale ◽  
Ray Theory ◽  
Velocity Parameter ◽  
Dimensional Parameters ◽  
The Waves ◽  
Angle Parameter

The interaction of linear, steady, axisymmetric deep-water gravity waves with preexisting large-scale annular currents has been investigated. Waves originating inside the annulus as well as waves approaching the annulus from the outside were studied. Exact linear ray solutions were obtained and involve two non-dimensional parameters, a radius-angle parameter and a velocity parameter. For opposing currents the linear solutions also allow the derivation of radii at which the waves are blocked, reflected at a linear caustic or stopped by the current. Various examples of rays interacting with an annular current are presented to illustrate aspects of the solutions obtained. In particular, the behaviour of the ray solutions at blocking, reflection and stopping is investigated. Linear ray theory is shown to fail at caustics and caustic solutions are briefly discussed.

Refraction of gravity waves by weak current gradients

2001 ◽  
Vol 442 ◽  
pp. 157-159 ◽  
Author(s):  
KRISTIAN B. DYSTHE
Keyword(s):  
Surface Waves ◽  
Group Velocity ◽  
Gravity Waves ◽  
Deep Water ◽  
Current Velocity ◽  
Water Surface ◽  
Simple Formula ◽  
Vertical Component ◽  
Weak Current ◽  
The Waves

When deep water surface waves cross an area with variable current, refraction takes place. If the group velocity of the waves is much larger than the current velocity we show that the curvature of a ray, χ, is given by the simple formula χ = ζ/vg. Here ζ is the vertical component of the current vorticity and vg is the group velocity.

scholarly journals A Deep-Water Bottom Reverberation Model Based on Ray Theory

2018 ◽  
Vol 06 (11) ◽  
pp. 2445-2452
Author(s):  
Longhao Wang ◽  
Jixing Qin ◽  
Zhenglin Li ◽  
Jianjun Liu
Keyword(s):  
Deep Water ◽  
Ray Theory ◽  
Model Based ◽  
Water Bottom

THE CONTAINMENT OF OIL SPILLED UNDER ROUGH ICE

1981 ◽  
Vol 1981 (1) ◽  
pp. 203-208 ◽  
Author(s):  
Jack C. Cox ◽  
Lawrence A. Schultz
Keyword(s):  
Laboratory Study ◽  
Deep Water ◽  
Current Velocity ◽  
Ice Cover ◽  
General Terms ◽  
Roughness Elements ◽  
Ice Conditions ◽  
Large Roughness ◽  
A Current

ABSTRACT This paper outlines the results of a laboratory study of the behavior of oil spilled beneath ice having various roughness characteristics in the presence of a current. It was found convenient to categorize ice conditions and ice features by addressing the problem in terms of oil interaction with smooth ice, ice having small roughness features, and ice having large roughness features. The small or large designation is determined by whether the height of the roughness elements is less than, or greater than, the equilibrium slick thickness of the oil beneath smooth ice. Further, large roughness elements can act individually, or they can act in combination, forming a cavity. In general terms, oil spilled beneath ice cover will remain stationary until the current velocity relative to the ice reaches 15 to 25 cm/sec. This holds true even for oil trapped upstream of large roughness elements. At the higher velocities, cavities have a somewhat greater oil containment capability than individual large roughness elements. The paper includes a procedure for quantitatively determining the transport or containment of oil spilled beneath ice in deep water.

scholarly journals Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

1988 ◽  
Vol 29 (4) ◽  
pp. 430-439 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Wave Packet ◽  
Wave Field ◽  
Gravity Waves ◽  
Deep Water ◽  
Short Wave ◽  
Wave Action ◽  
Leading Order ◽  
Order Interaction ◽  
Short Waves ◽  
Long Wave

AbstractThe leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

The Equivalent Depth of Wave-Induced Scour Around Offshore Pipelines

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Mir Emad Mousavi ◽  
Abbas Yeganeh Bakhtiary ◽  
Nastaran Enshaei
Keyword(s):  
Deep Water ◽  
Wave Motion ◽  
Scour Depth ◽  
Offshore Pipelines ◽  
Equivalent Depth ◽  
Physical Experiments ◽  
Water Conditions ◽  
Installation Depth ◽  
New Equation ◽  
Wave Induced

Physical experiments have been conducted to study the scouring around a pipe over an erodible bed under wave motion. The results show that the depth of the wave-induced scour is reduced when the pipe is installed in a primary depth. It is also indicated that when the primary installation depth of the pipe, |e|, exceeds a specified depth, no scouring occurs underneath the pipe. This specific depth is called the equivalent depth of wave-induced scour, Se. The equivalent depth of wave-induced scour is estimated as a function of pipe diameter and the Keulegan–Carpenter number. With respect to the primary installation depth of the pipe, a new equation is suggested to predict the scour depth before installation of the pipe. And the scour width is predicted in two other new equations. The proposed equations are then simplified in the form of diagrams. The experiments cover small KC numbers (KC<6) that represent deep water conditions.

A modified WKB method for the improved phase shift at a turning point

1994 ◽  
Vol 12 (3) ◽  
pp. 443-452 ◽  
Author(s):  
Feng Xiang ◽  
Gar Lam Yip
Keyword(s):  
Phase Shift ◽  
Turning Point ◽  
Wkb Method
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