scholarly journals From metaphors to equations: How can we find the good ones?

2000 ◽  
Vol 23 (3) ◽  
pp. 409-410
Author(s):  
Gottfried Mayer-Kress
Keyword(s):  
Boundary Conditions ◽  
Forest Fire ◽  
Water Waves ◽  
Standing Waves ◽  
Musical Instruments ◽  
Oscillator Model ◽  
Mechanical Oscillator ◽  
Target Article ◽  
Mechanical Oscillators

Among the metaphors used in the target article are “musical instruments,” “water waves,” and other types of mechanical oscillators. The corresponding equations have inertial properties and lead to standing waves that depend on boundary conditions. Other, physiologically relevant quantities like refractory times are not contained in the mechanical oscillator model but occur naturally, for instance, in biological forest fire metaphors.

Concentrating standing waves for Davey–Stewartson systems

2021 ◽  
pp. 1-40
Author(s):  
Yi He ◽  
Xiao Luo
Keyword(s):  
Water Waves ◽  
Elliptic Equations ◽  
Standing Waves ◽  
Regularity Theory ◽  
Local Minima ◽  
Shallow Water Waves ◽  
Penalization Method ◽  
Finite Set ◽  
New Ideas ◽  
Concentrating Solutions

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

scholarly journals ASSESSMENT OF DYNAMIC FEATURES OF MECHANICAL OSCILLATORS FOR A ROOT-BALLING MACHINE WITH A VIBRATING BLADE

2009 ◽  
Vol 40 (4) ◽  
pp. 1
Author(s):  
Dario Friso ◽  
Marco Bietresato
Keyword(s):  
Theoretical Analysis ◽  
Motion Analysis ◽  
Dynamic Features ◽  
Mechanical Oscillator ◽  
Strong Phase ◽  
Harmonic Motion ◽  
Phase Displacement ◽  
Air Motion ◽  
Mechanical Oscillators ◽  
Oscillating Motion

A theoretical analysis of the oscillating motion produced by a mechanical oscillator for the vibrating blade of a root-balling machine was carried out. The result was harmonic motion with a strong phase displacement with respect to the forcing torque of the eccentric masses; however, the amplitude can be approximately calculated by in-air motion analysis. Experiments were also carried out in order to determine the cutting torque of the blade in typical treenursery soil. These results, together with the indications of other literature reports, were used to propose a possible design directive.

Theory of water waves derived from a Lagrangian. Part 1. Standing waves

2000 ◽  
Vol 423 ◽  
pp. 275-291 ◽  
Author(s):  
MICHAEL S. LONGUET-HIGGINS
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Fourier Coefficients ◽  
Equations Of Motion ◽  
Standing Waves ◽  
System Of Equations ◽  
Practical Applications ◽  
Low Degree ◽  
Sharp Corners ◽  
New System

A new system of equations for calculating time-dependent motions of deep-water gravity waves (Balk 1996) is here developed analytically and set in a form suitable for practical applications. The method is fully nonlinear, and has the advantage of essential simplicity. Both the potential and the kinetic energy involve polynomial expressions of low degree in the Fourier coefficients Yn(t). This leads to equations of motion of correspondingly low degree. Moreover the constants in the equations are very simple. In this paper the equations of motion are specialized to standing waves, where the coefficients Yn are all real. Truncation of the series at low values of [mid ]n[mid ], say n < N, leads to ‘partial waves’ with solutions apparently periodic in the time t. For physical applications N must however be large. The method will be applied to the breaking of standing waves by the forming of sharp corners at the crests, and the generation of vertical jets rising from the wave troughs.

scholarly journals SPH Simulation of Interior and Exterior Flow Field Characteristics of Porous Media

Water ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 918 ◽  
Author(s):  
Shijie Wu ◽  
Matteo Rubinato ◽  
Qinqin Gui
Keyword(s):  
Numerical Simulation ◽  
Porous Media ◽  
Flow Field ◽  
Boundary Conditions ◽  
Water Waves ◽  
Flow Behavior ◽  
Simulation Method ◽  
Interface Problem ◽  
Ocean Engineering ◽  
Flow Field Characteristics

At the present time, one of the most relevant challenges in marine and ocean engineering and practice is the development of a mathematical modeling that can accurately replicate the interaction of water waves with porous coastal structures. Over the last 60 years, multiple techniques and solutions have been identified, from linearized solutions based on wave theories and constant friction coefficients to very sophisticated Eulerian or Lagrangian solvers of the Navier-Stokes (NS) equations. In order to explore the flow field interior and exterior of the porous media under different working conditions, the Smooth Particle Hydrodynamics (SPH) numerical simulation method was used to simulate the flow distribution inside and outside a porous media applied to interact with the wave propagation. The flow behavior is described avoiding Euler’s description of the interface problem between the Euler mesh and the material selected. Considering the velocity boundary conditions and the cyclical circulation boundary conditions at the junction of the porous media and the water flow, the SPH numerical simulation is used to analyze the flow field characteristics, as well as the longitudinal and vertical velocity distribution of the back vortex flow field and the law of eddy current motion. This study provides innovative insights on the mathematical modelling of the interaction between porous structures and flow propagation. Furthermore, there is a good agreement (within 10%) between the numerical results and the experimental ones collected for scenarios with porosity of 0.349 and 0.475, demonstrating that SPH can simulate the flow patterns of the porous media, the flow through the inner and outer areas of the porous media, and the flow field of the back vortex region. Results obtained and the new mathematical approach used can help to effectively simulate with high-precision the changes along the water depth, for a better design of marine and ocean engineering solutions adopted to protect coastal areas.

Numerical Simulation of Nonlinear Water Wave Propagation Over Rippled Bed

2007 ◽  
Author(s):  
Gerasimos A. Kolokythas ◽  
Athanassios A. Dimas
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Water Waves ◽  
Stokes Equations ◽  
Surface Flow ◽  
Surface Boundary ◽  
Computational Domain ◽  
Time Step ◽  
Outflow Boundary ◽  
Rippled Bed

In the present study, numerical simulations of the free-surface flow, developing by the propagation of nonlinear water waves over a rippled bottom, are performed assuming that the corresponding flow is two-dimensional, incompressible and viscous. The simulations are based on the numerical solution of the Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and the suitable bottom, inflow and outflow boundary conditions. The equations are properly transformed so that the computational domain becomes time-independent. For the spatial discretization, a hybrid scheme with finite-differences and Chebyshev polynomials is applied, while a fractional time-step scheme is used for the temporal discretization. A wave absorption zone is placed at the outflow region in order to efficiently minimize reflection of waves by the outflow boundary. The numerical model is validated by comparison to the analytical solution for the laminar, oscillatory, current flow which develops a uniform boundary layer over a horizontal bottom. For the propagation of finite-amplitude waves over a rigid rippled bed, the case with wavelength to water depth ratio λ/d0 = 6 and wave height to wavelength ratio H0/λ = 0.05 is considered. The ripples have parabolic shape, while their dimensions — length and height — are chosen accordingly to fit laboratory and field data. Results indicate that the wall shear stress over the ripples and the form drag forces on the ripples increase with increasing ripple height, while the corresponding friction force is insensitive to this increase. Therefore, the percentage of friction in the total drag force decreases with increasing ripple height.

scholarly journals A Note on Sign-Changing Solutions to the NLS on the Double-Bridge Graph

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 161
Author(s):  
Diego Noja ◽  
Sergio Rolando ◽  
Simone Secchi
Keyword(s):  
Boundary Conditions ◽  
Standing Waves ◽  
Nls Equation ◽  
Sign Changing Solutions

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle.

scholarly journals Standing Waves in One-Dimensional Resonator Containing an Ideal Isothermal Gas Affected by the Constant Mass Force

2017 ◽  
Vol 42 (2) ◽  
pp. 263-271
Author(s):  
Anna Perelomova
Keyword(s):  
Boundary Conditions ◽  
Acoustic Waves ◽  
Standing Waves ◽  
Ideal Gas ◽  
Excess Pressure ◽  
Mass Force ◽  
Constant Mass ◽  
Total Field ◽  
One Dimensional ◽  
Isothermal Gas

Abstract The study is devoted to standing acoustic waves in one-dimensional planar resonator which containing an ideal gas. A gas is affected by the constant mass force. Two types of physically justified boundary conditions are considered: zero velocity or zero excess pressure at both boundaries. The variety of nodal and antinodal points is determined. The conclusion is that the nodes of pressure and antinodes of velocity do not longer coincide, as well as antinodes of pressure and nodes of velocity. The entropy mode may contribute to the total field in a resonator. It is no longer isobaric, in contrast to the case when the external force is absent. Examples of perturbations inherent to the entropy mode in the volume of a resonator are discussed.

On the observability of finite-depth short-crested water waves

1996 ◽  
Vol 322 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ioualalen ◽  
A. J. Roberts ◽  
C. Kharif
Keyword(s):  
Standing Wave ◽  
Water Waves ◽  
Numerical Study ◽  
Standing Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Wave Fields ◽  
Progressive Waves ◽  
Wave Limit ◽  
Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

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