The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

1978 ◽  
Vol 85 (1) ◽  
pp. 7-31 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Shear Flow ◽  
Internal Waves ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Depth Interval ◽  
Plane Shear

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

Differential equations for long-period gravity waves on fluid of rapidly varying depth

1977 ◽  
Vol 83 (2) ◽  
pp. 289-310 ◽  
Author(s):  
James Hamilton
Keyword(s):  
Gravity Waves ◽  
Wave Equations ◽  
Velocity Potential ◽  
Taylor Series Expansion ◽  
Finite Amplitude ◽  
Necessary Condition ◽  
Exact Equation ◽  
Rapid Convergence ◽  
Long Period ◽  
Long Wave

The conventional long-wave equations for waves propagating over fluid of variable depth depend for their formal derivation on a Taylor series expansion of the velocity potential about the bottom. The expansion, however, is not possible if the depth is not an analytic function of the horizontal co-ordinates and it is a necessary condition for its rapid convergence that the depth is also slowly varying. We show that if in the case of two-dimensional motions the undisturbed fluid is first mapped conformally onto a uniform strip, before the Taylor expansion is made, the analytic condition is removed and the approximations implied in the lowest-order equations are much improved.In the limit of infinitesimal waves of very long period, consideration of the form of the error suggests that by modifying the coefficients of the reformulated equation we may find an equation exact for arbitrary depth profiles. We are thus able to calculate the reflexion coefficients for long-period waves incident on a step change in depth and a half-depth barrier. The forms of the coefficients of the exact equation are not simple; however, for these particular cases, comparison with the coefficients of the reformulated long-wave equation suggests that in most cases the latter may be adequate. This opens up the possibility of beginning to study finite amplitude and frequency effects on regions of rapidly varying depth.

scholarly journals Flexible plate and foundation modelling

2000 ◽  
Vol 4 (2) ◽  
pp. 103-110
Author(s):  
R. J. Hosking
Keyword(s):  
Critical Load ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Moving Load ◽  
Phase Speed ◽  
Flexible Plate ◽  
Engineering Analysis ◽  
Flexural Gravity Waves ◽  
Railway Engineering ◽  
Engineering Parameters

In the most common mathematical model for a moving load on a continuously- supported flexible plate, the plate is assumed thin and elastic. An exception is the inclusion of viscoelasticity in the theory for the response of a floating ice plate, where the deflexion at the critical load speed corresponding to the minimum phase speed of hybrid flexural-gravity waves consequently approaches a steady state. This is in contrast to the elastic theory, where the response is predicted to grow continuously at this critical load speed. In the theory for a floating ice plate, the dominant pressure due to the underlying water is inertial, introduced via a velocity potential and the Bernoulli equation (assuming non-cavitation at the plate-water interface). On the other hand, the classical Winkler representation used in early railway engineering analysis corresponds to retaining a term which is generally negligible in the ice plate context. Critical load speeds are consequently predicted to be much higher, at wavelengths correspondingly much lower, for commonly accepted railway engineering parameters. Other models might be considered.

On the stability of steep gravity waves

1984 ◽  
Vol 396 (1811) ◽  
pp. 269-280 ◽  
Keyword(s):  
Phase Shift ◽  
Normal Mode ◽  
Gravity Waves ◽  
Phase Speed ◽  
Numerical Calculations ◽  
Wave Steepness ◽  
High Wave ◽  
Pure Phase ◽  
The Stability

Previous calculations of the normal mode perturbations of steep gravity waves have suggested that the lowest superharmonic mode n = 2 becomes unstable at around ak = 0.436, where 2 a is the crest-to-trough height of the unperturbed wave and k is the wavenumber. This would correspond to the wave steepness at which the phase speed c is a maximum (considered as a function of ak ). However, numerical calculations at such high wave steepnesses can become inaccurate. The present paper studies analytically the conditions for the existence of a normal mode at zero limiting frequency. It is proved that for superharmonic perturbations such conditions will occur only for a pure phase-shift (corresponding to n = 1) or when the speed c is stationary with respect to the wave steepness, that is when d c = 0. Hence the limiting form of the instability found by Tanaka ( J. phys. Soc. Japan 52, 3047-3055 (1983)) near the value ak = 0.429 must be a pure phase-shift.

The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 489-505 ◽  
Keyword(s):  
Gravity Waves ◽  
Maximum Growth ◽  
Finite Amplitude ◽  
Wave Steepness ◽  
Maximum Value ◽  
Small Wave ◽  
New Type ◽  
Rates Of Growth ◽  
Good Agreement ◽  
Zero Frequency

Calculation of the normal-mode perturbation of steep irrotational gravity waves, begun in part I, is here extended to a study of the subharmonic perturbations, namely those having horizontal scales greater than the basic wavelength (2π/k). At small wave amplitudes a , it is found that all perturbations tend to become neutrally stable; but as ak increases the perturbations coalesce in pairs to produce unstable modes. These may be identified with the instabilities analysed by Benjamin & Feir (1967) when ak is small. However, as ak increases beyond about 0.346, these modes become stable again. The maximum growth scale of this type of mode in the unstable range is only about 14 % per wave period, which value occurs at ak ≈0.32. At values of ak near 0.41 a new type of instability appears which has initially zero frequency but a much higher growth rate. It is pointed out that this type might be expected to arise at wave amplitudes for which the first Fourier coefficient in the basic wave is at its maximum value, as a function of the wave height. The corresponding wave steepness was found by Schwartz (1974) to be ak = 0.412. A comparison of the calculated rates of growth are in rather good agreement with those observed by Benjamin (1967) in the range 0.07< ak <0.17

Finite-amplitude waves in inviscid shear flows

1982 ◽  
Vol 382 (1783) ◽  
pp. 389-410 ◽  
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Poisson Equations ◽  
Weakly Nonlinear Theory ◽  
Finite Difference Discretization ◽  
Finite Amplitude Waves ◽  
The Waves

This paper examines the existence and properties of steady finite-amplitude waves of cats-eye form superposed on a unidirectional inviscid, incompressible shear flow. The problem is formulated as the solution of nonlinear Poisson equations for the stream function with boundary conditions on the unknown edges of the cats-eyes. The dependence of vorticity on stream function is assumed outside the cats-eyes to be as in the undisturbed flow, and uniform unknown vorticity is assumed inside. It is argued on the basis of a finite difference discretization that the problem is determinate, and numerical solutions are obtained for Couette-Poiseuille channel flow. These are compared with the predictions of a weakly nonlinear theory based on the approach of Benney & Bergeron (1969) and Davis (1969). The phase speed of the waves is found to be linear in the wave amplitude.

Capillary-gravity waves over a sloping beach

1973 ◽  
Vol 74 (3) ◽  
pp. 539-547 ◽  
Author(s):  
D. A. Allwood
Keyword(s):  
Surface Tension ◽  
Standing Wave ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Surface Tension Force ◽  
Finite Amplitude ◽  
Wave Solutions ◽  
Linearized Theory ◽  
Sloping Beach ◽  
Effect Of Surface

AbstractIt is shown how the solution for the velocity potential may be determined when the effect of surface tension is included in the linearized theory of surface waves over a sloping beach. In particular, two independent standing wave solutions are found, both of which have finite amplitude at the shoreline. The results agree with those of previous writers when the surface tension force tends to zero.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

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