On three-dimensional packets of surface waves

1974 ◽  
Vol 338 (1613) ◽  
pp. 101-110 ◽  
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Plane Waves ◽  
Three Dimensional ◽  
Phase Variation ◽  
Finite Depth ◽  
Amplitude Variation ◽  
The Stability ◽  
Small Disturbances

In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
You-Qi Tang
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Finite Support ◽  
Governing Equations ◽  
Parametric Stability ◽  
Stability Boundaries ◽  
Axial Acceleration ◽  
The Stability ◽  
Longitudinally Varying Tension

In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

ANALYSIS OF NONLINEAR MAGNETOELASTIC DYNAMICS AND STABILITY OF CONDUCTIVE CYLINDRICAL SHELLS

2010 ◽  
Vol 10 (01) ◽  
pp. 153-164
Author(s):  
YUDA HU ◽  
JIANG ZHAO ◽  
PI JUN ◽  
GUANGHUI QING
Keyword(s):  
Cylindrical Shell ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Approximate Analytical Solution ◽  
Transverse Magnetic ◽  
Field Equations ◽  
Thin Cylindrical Shell ◽  
The Stability ◽  
Magnetic Induction Intensity ◽  
Steady State Solutions

The nonlinear magnetoelastic vibration equations and electromagnetic field equations of a conductive thin cylindrical shell in magnetic fields are derived. The nonlinear principal resonances and dynamic stabilities of the cylindrical shell simply supported in a transverse magnetic field are investigated. Approximate analytical solution and bifurcation equations of the system with principal resonances are obtained by using the method of multiple scales. The stabilities and singularities of the steady-state solutions are analyzed and the stability criterion is given. The transition sets and bifurcation figures of unfolding parameters are also obtained. The variations of the resonance amplitudes with respect to the detuning parameter, the magnetic induction intensity, and the amplitude of excitations are presented. The corresponding phase trajectories in moving phase planes are given. The stabilities of solutions, characteristics of singular points, and bifurcation are analyzed. The impacts of electromagnetic and mechanical parameters on dynamic behaviors are discussed in detail.

He’s Multiple-Scale Solution for the Three-dimensional Nonlinear KH Instability of Rotating Magnetic Fluids

2019 ◽  
Vol 9 (1) ◽  
pp. 52-69 ◽  
Author(s):  
Yusry Osman El-Dib ◽  
Amal A Mady
Keyword(s):  
Multiple Scales ◽  
Velocity Ratio ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Stability Criteria ◽  
Helmholtz Instability ◽  
Vertical Field ◽  
Horizontal Flux ◽  
The Stability ◽  
Transition Curves

This paper elucidates a trend in solving nonlinear oscillators of the rotating Kelvin-Helmholtz instability. The system is constituted by the vertical flux or the horizontal flux. He’s multiple-scales perturbation methodology has been applied and therefore the system is represented by a generalized homotopy equation. This approach ends up in a periodic answer to a nonlinear oscillator with high nonlinearity. The cubic-quintic nonlinear Duffing equation is obligatory as a condition to uniformly answer. This equation is employed to derive the stability criteria. The transition curves are plotted to investigate the stability image. It's shown that the angular velocity suppresses the instability. The tangential flux plays a helpful role, whereas the vertical field encompasses a destabilizing influence. Within the existence of the rotation, the velocity ratio reduces stability configuration.

Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

Capillary–gravity simultons in a liquid layer

2002 ◽  
Vol 466 ◽  
pp. 1-15
Author(s):  
GUOXIANG HUANG ◽  
MANUEL G. VELARDE
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Second Harmonic ◽  
Harmonic Wave ◽  
Finite Depth ◽  
Harmonic Waves ◽  
Amplitude Equations ◽  
Solid Plate ◽  
Bounded Below ◽  
Harmonic Resonance

Simultaneous capillary–gravity solitary waves (simultons or quadratic solitons) are shown to be possible in a rectangular liquid channel of arbitrary finite depth bounded below by a solid plate and above with a free deformable surface with constant surface tension. A second-harmonic resonance between two waveguide modes (fundamental and second-harmonic waves) is studied with the inclusion of dispersion in the system. The nonlinearly coupled amplitude equations for the two slowly varying envelopes of the fundamental and the second-harmonic wave components are derived using the method of multiple scales. Two types of capillary–gravity simulton solutions are explicitly obtained and an experiment for observing such hydrodynamic simultons is suggested.

scholarly journals Transversally periodic solitary gravity–capillary waves

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130537 ◽  
Author(s):  
Paul A. Milewski ◽  
Zhan Wang
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Travelling Waves ◽  
Plane Waves ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Propagation Direction ◽  
The Stability ◽  
Dirichlet To Neumann Operator ◽  
The Waves

When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity–capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity–capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

Higher-Order Multiple Scales Analysis of Weakly Nonlinear Lattices With Implications for Directional Stability

2018 ◽  
Author(s):  
Matthew D. Fronk ◽  
Michael J. Leamy
Keyword(s):  
Acoustic Waves ◽  
Local Stability ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Plane Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Springs ◽  
Nonlinear Lattices ◽  
Multiple Scales Analysis ◽  
The Stability

Recent focus has been given to nonlinear periodic structures for their ability to filter, guide, and block elastic and acoustic waves as a function of their amplitude. In particular, two-dimensional (2-D) nonlinear structures possess amplitude-dependent directional bandgaps. However, little attention has been given to the stability of plane waves along different directions in these structures. This study analyzes a 2-D monoatomic shear lattice composed of discrete masses, linear springs, quadratic and cubic nonlinear springs, and linear viscous dampers. A local stability analysis informed by perturbation results retained through the second order suggests that different directions become unstable at different amplitudes in an otherwise symmetrical lattice. Simulations of the lattice’s equation of motion subjected to both line and point forcing are consistent with the local stability results: waves with large amplitudes have spectral growth that differs appreciably at different angles. The results of this analysis could have implications for encryption strategies and damage detection.

Nonlinear ripples of Kelvin–Helmholtz type which arise from an interfacial mode interaction

1997 ◽  
Vol 341 ◽  
pp. 295-315 ◽  
Author(s):  
M. C. W. JONES
Keyword(s):  
Gravity Waves ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Mode Interaction ◽  
Third Order ◽  
Two Fluids ◽  
Power Series Expansions ◽  
The Stability ◽  
Formal Power ◽  
The Waves

An analysis is made of the small-amplitude capillary–gravity waves which occur on the interface of two fluids and which arise out of the interaction between the Mth and Nth harmonics of the fundamental mode. The method employed is that of multiple scales in both space and time and a pair of coupled nonlinear partial differential equations for the slowly varying wave amplitudes is derived. These equations describe, correct up to third order, the progression of a wavetrain and are generalizations of the nonlinear Schrödinger-type equations used by many authors to model wave propagation. The equations are solved and formal power series expansions of the corresponding wave profiles obtained. Many different wave configurations can arise, some symmetric others asymmetric. It is found that an important influence on the type of waves which can occur is whether the ratio of the interacting wave modes is greater or less than two. Finally, an examination of the stability of the waves to plane wave perturbations is carried out.

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