scholarly journals Model and experiments for resonant generation of second harmonic capillary–gravity waves

2019 ◽  
Vol 396 ◽  
pp. 12-17
Author(s):  
Adriano Alippi ◽  
Andrea Bettucci ◽  
Massimo Germano
Keyword(s):  
Gravity Waves ◽  
Second Harmonic ◽  
Resonant Generation

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.

Capillary–gravity Kelvin–Helmholtz waves close to resonance

1990 ◽  
Vol 217 ◽  
pp. 71-91 ◽  
Author(s):  
V. Bontozoglou ◽  
T. J. Hanratty
Keyword(s):  
Gravity Waves ◽  
Second Harmonic ◽  
Period Doubling ◽  
Current Speed ◽  
Period Doubling Bifurcation ◽  
Helmholtz Instability ◽  
Horizontal Pipe ◽  
Near Resonance ◽  
Gas Liquid Flow ◽  
Permanent Form

Capillary–gravity waves of permanent form at the interface between two unbounded fluids in relative motion are considered. The range of wavelengths for an internal resonance with the second harmonic and a period-doubling bifurcation are found to depend on the current speed. The Kelvin–Helmholtz instability of short waves becomes strongly subcritical near resonance. It is speculated that this instability is needed to trigger a period-doubling bifurcation. This notion is used to explain the development of waves at short fetch and the initiation of liquid slugs for gas–liquid flow in a horizontal pipe.

Resonant generation of the second harmonic of a surface wave on a plasma-metal boundary

1991 ◽  
Vol 34 (4) ◽  
pp. 385-391
Author(s):  
N. A. Azerenkov ◽  
A. N. Kondratenko ◽  
K. N. Ostrikov
Keyword(s):  
Surface Wave ◽  
Second Harmonic ◽  
Metal Boundary ◽  
Resonant Generation

Resonant generation of the second harmonic in calcium vapor

1981 ◽  
Vol 11 (5) ◽  
pp. 656-657 ◽  
Author(s):  
V G Arkhipkin ◽  
N P Makarov ◽  
A K Popov ◽  
V P Timofeev ◽  
V Sh Épshteĭn
Keyword(s):  
Second Harmonic ◽  
Calcium Vapor ◽  
Resonant Generation

Third-harmonic resonance in the interaction of capillary and gravity waves

1971 ◽  
Vol 48 (2) ◽  
pp. 385-395 ◽  
Author(s):  
Ali Hasan Nayfeh
Keyword(s):  
Gravity Waves ◽  
Multiple Scales ◽  
Second Harmonic ◽  
Travelling Wave ◽  
Wave Number ◽  
Third Harmonic ◽  
Resonant Wave ◽  
Near Resonance ◽  
Modulated Waves ◽  
Harmonic Resonance

The method of multiple scales is used to determine the temporal and spatial variation of the amplitudes and phases of capillary-gravity waves in a deep liquid at or near the third-harmonic resonant wave-number. This case corresponds to a wavelength of 2·99 cm in deep water. The temporal variation shows that the motion is always bounded, and the general motion is an aperiodic travelling wave. The analysis shows that pure amplitude-modulated waves are not possible in this case contrary to the second-harmonic resonant case. Moreover, pure phase-modulated waves are periodic even near resonance because the non-linearity adjusts the phases to yield perfect resonance. These periodic waves are found to be unstable, in the sense that any disturbance would change them into aperiodic waves.

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