scholarly journals Second harmonic resonance in magnetohydrodynamic jet

1994 ◽  
Vol 35 (3) ◽  
pp. 323-334 ◽  
Author(s):  
H. K. Khosla ◽  
R. K. Chhabra
Keyword(s):  
Steady State ◽  
Second Harmonic ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Resonant Interaction ◽  
Spatial Behaviour ◽  
Derivative Expansion ◽  
Modulated Waves ◽  
Harmonic Resonance ◽  
General Motion

AbstractCoupled nonlinear partial differential equations, which describe a nonlinear resonant interaction between the fundamental and its first harmonic on a magnetohydro-dynamic jet, are derived by the derivative expansion method. We investigate the spatial behaviour of the amplitude and phases. It is shown that the fluid surface is unstable in the neighbourhood of the first resonant wavenumber. In the steady state, it is observed that the general motion consists of both amplitude and phase modulated waves.

On Wilton's ripples: a special case of resonant interactions

1970 ◽  
Vol 42 (1) ◽  
pp. 193-200 ◽  
Author(s):  
L. F. McGoldrick
Keyword(s):  
Initial Conditions ◽  
Second Harmonic ◽  
Resonant Interaction ◽  
Finite Amplitude ◽  
Multiple Time ◽  
Resonant Interactions ◽  
Critical Choice ◽  
Harmonic Resonance ◽  
Special Case ◽  
Permanent Form

The phenomenon of second harmonic resonance for capillary-gravity waves is reconsidered here by the asymptotic method of multiple time and space scales. The periodic finite amplitude waves of permanent form found by Wilton in 1915 which correspond to this configuration are shown to be no more than a special case of the more general resonant interaction theory, and owe their existence to a critical choice of initial conditions. It is further suggested that the influence of viscous dissipation will render this solution virtually undetectable in a real liquid.

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.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

2021 ◽  
Vol 5 (4) ◽  
pp. 238
Author(s):  
Li Yan ◽  
Gulnur Yel ◽  
Ajay Kumar ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rational Function ◽  
Analytical Approach ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Analytical Scheme

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

Explicit solutions to nonlinear Chen–Lee–Liu equation

2021 ◽  
pp. 2150438
Author(s):  
Lanre Akinyemi ◽  
Najib Ullah ◽  
Yasir Akbar ◽  
Mir Sajjad Hashemi ◽  
Arzu Akbulut ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Phenomena ◽  
Explicit Solutions ◽  
Mathematical Algorithm ◽  
New Exact Solutions ◽  
Order Polynomial ◽  
Fourth Order Polynomial

In this work, a generalized [Formula: see text]-expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where [Formula: see text] follows the Jacobi elliptic equation [Formula: see text], and we let [Formula: see text] be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple.

Temporal Modulation of Steady-State Visual Evoked Potentials

2019 ◽  
Vol 29 (03) ◽  
pp. 1850050 ◽  
Author(s):  
Maciej Labecki ◽  
Maria Malgorzata Nowicka ◽  
Piotr Suffczynski
Keyword(s):  
Steady State ◽  
Evoked Potentials ◽  
Visual Evoked Potentials ◽  
Second Harmonic ◽  
Temporal Changes ◽  
Harmonic Frequency ◽  
Physiological Basis ◽  
Frequency Components ◽  
Over Time ◽  
Visual Evoked

Electroencephalographic responses to periodic stimulation are termed steady-state visual evoked potentials (SSVEP). Their characteristics in terms of amplitude, frequency and phase are commonly assumed to be stationary. In this work, we tested this assumption in 30 healthy participants submitted to 50 trials of 60[Formula: see text]s flicker stimulation at 15[Formula: see text]Hz frequency. We showed that the amplitude of the first and second harmonic frequency components of SSVEP signals were in general not stable over time. The power (squared amplitude) of the fundamental component was stationary only in 30% the subjects, while the power at the second harmonic frequency was stationary in 66.7% of the group. The phases of both SSVEP frequency components were more stable over time, but could exhibit small drifts. The observed temporal changes were heterogeneous across the subjects, implying that averaging results over participants should be performed carefully. These results may contribute to improved design and analysis of experiments employing prolonged visual stimulation. Our findings offer a novel characterization of the temporal changes of SSVEP that may help to identify their physiological basis.

scholarly journals Nonlinear interactions among standing surface and internal gravity waves

1974 ◽  
Vol 63 (4) ◽  
pp. 801-825 ◽  
Author(s):  
Terrence M. Joyce
Keyword(s):  
Energy Transfer ◽  
Steady State ◽  
Internal Wave ◽  
Surface Waves ◽  
Viscous Dissipation ◽  
Growth Rates ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Resonant Interaction ◽  
Initial Growth

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which \[ \sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0; \] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.

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