Small Amplitude Linear Wave Research Articles

Hydrodynamic performance of shallow-water waveguides subject to nonlinear waves

The idea of water wave control for ease of shipping or energy exploitation has been the subject of extensive research. Yet, most studies are limited to two-dimensional simulations focusing primarily on transformations of small-amplitude linear waves. This paper presents a theoretical and numerical investigation of the hydrodynamic performance of a type of shallow-water waveguides. The concept of an effective refractive index for water waves is proposed through analogy with electromagnetic waves, based on which the wave-controlling mechanism is explained. The precise wave field is revealed using computational fluid dynamics simulations. The numerical model is validated by comparing the numerical predictions with experimental data. Subsequently, the nonlinear wave fields around the waveguide are systematically studied by increasing the incident wave height. The numerical results confirm the significant increase in the wave height above the waveguide platform, as a result of wave refraction and reflection. Such an amplification effect slightly reduces as the nonlinearity of the incident wave increases.

Odd surface waves in two-dimensional incompressible fluids

We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing shear viscosity. In this limit, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. Within the small surface angle approximation, the equation of motion leads to a new class of non-linear chiral dynamics governed by what we dub the chiral Burgers equation. The chiral Burgers equation is identical to the complex Burgers equation with imaginary viscosity and an additional analyticity requirement that enforces chirality. We present several exact solutions of the chiral Burgers equation. For generic multiple pole initial conditions, the system evolves to the formation of singularities in a finite time similar to the case of an ideal fluid without odd viscosity. We also obtain a periodic solution to the chiral Burgers corresponding to the non-linear generalization of small amplitude linear waves.

Wave force on the submerged inclined thin plate in intermediate depth of water

The paper presented is a solution of wave force, using small amplitude, linear wave theory on two-dimensional inclined submerged rectangular thin plates. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. In this paper, analytical solution is derived and the result is validated with numerical results. Simpson’s 1/3 rule is used here for the validation. The variations of horizontal and vertical wave forces on the plate with respect to the wave amplitudes are obtained in the intermediate depth of water and at a different plate angle. It is observed that the wave forces on the plate are high at the free surface for d/L = 0.24 compare to the other relative depth. It is also observed that the wave forces of the two types of plate are gradually converging with the decreasing value of the relative depth. Laplace’s equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory.

Effect of large deformation pre-loads on the wave properties of hexagonal lattices

We study linear wave propagation in nonlinear hexagonal lattices capable of undergoing large deformations, under different levels of pre-load. The lattices are composed of a set of masses connected by linear axial and angular springs, with the nonlinearity arising solely from geometric effects. By applying different levels of pre-load, the small amplitude linear wave propagation response can be varied from isotropic to highly directional. Analytical expressions for the stiffness of a unit cell in the deformed configuration are derived and they are used to analyze the dispersion surfaces and group velocity variation with pre-load. Numerical simulations on finite lattices demonstrate the validity of our unit cell predictions and illustrate the wave steering potential of our lattice.

Variation of Wave Force on Submerged Object at Shallow Water: Fourier Series Technique

This paper is presented a solution of shallow water wave force, using Fourier series approximation method on two dimensional vertically submerged rectangular thin plates under two different configurations: (1) a surface-piercing rectangular thin plate, and (2) a bottom-standing rectangular thin plate. A propagating wave of different amplitude and at different time period is made to impose normal wave force on submerged object. Bernoulli's equation is used for the determination of wave force which is based on the linear wave theory. The plate is submerged in water near the shore under different depth of water. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. Laplace's equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory. The variation of horizontal force with respect to the wave amplitude is obtained at different depth of water and at different wave period. The wave force is measured at four different depths on the submerged object. It is observed that the forces are converging with the increase of wave period and the gradients of forces with respect to the wave amplitude are extremely high for lower wave period. Wave forces are maximums on the surface of water and it decreases gradually towards the bottom of water. The purpose of this study is based on to made an innovative design of a low cost wave energy converter (WEC) where the kinetic energy obtainable from the ocean surface wave is converted in the form of potential energy.

Amplification of electromagnetic waves excited by a chain of propagating magnetic vortices in YBa2Cu3O7−δ Josephson-junction arrays at 77 K and above

When a soliton propagates in a discrete lattice it excites small-amplitude linear waves in its wake. In a dc current-biased Josephson-junction (JJ) array these manifest as electromagnetic (EM) waves excited by a (magnetic field induced) chain of propagating magnetic vortices. When the vortex velocity and the phase velocity of one of the excited EM waves match, phase-locking occurs. This produces resonant steps in the current–voltage characteristics where amplification of EM radiation occurs. We report the first observation of phase-locking-induced amplification of EM radiation at 77 K and above in JJ arrays made of high temperature superconductors.

Flexural gravity wave over a floating ice sheet near a vertical wall

The behavior of flexural gravity waves propagating over a semi-infinite floating ice sheet is studied under the assumptions of small amplitude linear wave theory. The vertical wall is assumed to be either fixed or harmonically oscillating with constant horizontal displacement, in which case the problem is analogous with a harmonically oscillating plane vertical wavemaker. The potential flow approach is adhered to and the higher-order mode–coupling relations are applied to determine the unknown coefficients present in the Fourier expansion formula of the potential functions. The ice sheet is modeled as a thin semi-infinite elastic beam. Three different edge conditions are applied at the finite edge of the floating ice sheet. The effects of different edge conditions, the thickness of the ice sheet and the water depth on the surface strain, the shear force along the ice sheet, the horizontal force on the vertical wall, and the flexural gravity wave profile are analyzed in detail.

Fast electromagnetic response of a thin film of resonant atoms with permanent dipole

We consider the propagation of extremely short pulses through a dielectric thin film containing resonant atoms (two-level atoms) with permanent dipole. Assuming that the film width is less than the field wavelength, we can solve the wave equation and reduce the problem to a system of generalized Bloch equations describing the resonant atoms. We compute the stationary solutions for a constant irradiation of the film. Superimposing a small amplitude linear wave we calculate the reflection and transmission coefficients. From these, one can then deduce the different parameters of the model. We believe that this technique could be used in experiments to obtain the medium atomic and relaxation parameters.

Undular bores and the initial-boundary value problem for the modified Korteweg-de Vries equation

Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value (IBV) problem for the mKdV equation is also examined. The solutions of the initial value problem are used to construct approximate analytical solutions of the IBV problem. An alternative analytical solution for the IBV problem, based on the assumption of an uniform train of solitons, is also developed. The parameter regimes, in which the different types of solution occur, for both the initial value and IBV problem are identified and excellent comparisons are obtained between the numerical and approximate solutions, for both problems.

When Shil'nikov Meets Hopf in Excitable Systems

This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh–Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a U-shaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these “CU systems.” A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov–Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov–Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

Wave force on vertically submerged circular thin plate in shallow water

The paper presented is a solution of shallow water wave force, using small amplitude linear wave theory on two-dimensional vertically submerged circular thin plates under three different configurations: (1) a surface-piercing circular thin plate, (2) a submerged circular thin plate, and (3) a bottom-standing circular thin plate. Finally Morison's equation is used for the determination of wave force which is based on the linear wave theory. The plate is submerged in water near the shore on uniformly sloping bottom. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. Laplace's equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory. The variation of horizontal force by single particle, total horizontal force and moment with respect to the wave amplitude are obtained at different depth of water and at different wave period. It is observed that the force and moment are converging with the increase of wave period and the gradients of force and moment with respect to the wave amplitude are extremely high for lower wave period.

Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation

We analyze the damped, driven, discrete sine-Gordon equation with periodic boundary conditions and constant forcing. Analytical and numerical results are presented about the existence, stability, and bifurcations of traveling waves in this system. These results are compared with experimental measurements of the current-voltage ( I–V) characteristics of a ring of N = 8 underdamped Josephson junctions. We find two types of traveling waves: low-velocity kinks and high-velocity whirling modes. The kinks excite small-amplitude linear waves intheir wake. At certain drive strengths, the linear waves phase-lock to the kink, generating resonant steps in the I–V curve. Steps also occur in the high-velocity region, due to parametric instabilities of the whirling mode. We analyze the onset of these instabilities, then numerically study the secondary bifurcations and complex spatiotemporal phenomena that occur past the onset. In all cases, the measured voltage locations of the resonant steps are in good agreement with the predictions.

Control Time for Gravity-Capillary Waves on Water

This paper considers distributed open-loop control of small-amplitude linear waves on a fluid in which both surface tension and gravity are significant. It formulates the control system as a first-order evolution equation, reducing the null controllability problem to a moment problem involving frequency exponentials. For simple geometries in which eigenvalues can be calculated explicitly, controllability in arbitrary finite time is established, and aa relationship between mode frequencies and controller norm is noted.

On drag and wave-resistance

It is shown that for steady-state small-amplitude linear wave motion the drag exerted on the airsteam by a mountain, in an inviscid isothermal atmosphere, as a function of the basic current is quite similar to the wave-resistance experienced by a steady disturbance, moving along the surface of an infinitely deep homogeneous stream, as a function of the translation speed. However the maintenance of both types of resistance by the flux of energy is different in each case.