scholarly journals Elastic waves in periodically heterogeneous two-dimensional media: locally periodic and anti-periodic modes

2018 ◽  
Vol 474 (2215) ◽  
pp. 20170908
Author(s):  
Igor V. Andrianov ◽  
Vladyslav V. Danishevskyy ◽  
Graham Rogerson
Keyword(s):  
Elastic Waves ◽  
Plane Waves ◽  
Square Lattice ◽  
Periodic Case ◽  
Long Wave ◽  
Fibrous Medium ◽  
Heterogeneous Structures ◽  
Different Types ◽  
Classical Waves ◽  
Wave Limit

Propagation of anti-plane waves through a discrete square lattice and through a continuous fibrous medium is studied. In the long-wave limit, for periodically heterogeneous structures the solution can be periodic or anti-periodic across the unit cell. It is shown that combining periodicity and anti-periodicity conditions in different directions of the translational symmetry allows one to detect different types of modes that do not arise in the purely periodic case. Such modes may be interpreted as counterparts of non-classical waves appearing in phenomenological theories. Dispersion diagrams of the discrete square lattice are evaluated in a closed analytical from. Dispersion properties of the fibrous medium are determined using Floquet–Bloch theory and Fourier series approximations. Influence of a viscous damping is taken into account.

Multiwave interaction solutions for the (3+1)-dimensional extended Jimbo–Miwa equation

2020 ◽  
Vol 34 (35) ◽  
pp. 2050405
Author(s):  
Wenying Cui ◽  
Wei Li ◽  
Yinping Liu
Keyword(s):  
Algebraic Method ◽  
Periodic Waves ◽  
New Techniques ◽  
Interaction Solutions ◽  
Long Wave ◽  
Multiwave Interaction ◽  
Different Types ◽  
Wave Limit ◽  
Solving Strategy ◽  
New Algorithms

In this paper, for the (3+1)-dimensional extended Jimbo–Miwa equation, by the direct algebraic method, together with the inheritance solving strategy, we construct its interaction solutions among solitons, rational waves, and periodic waves. Meanwhile, we construct its interaction solutions among solitons, breathers, and lumps of any higher orders by an [Formula: see text]-soliton decomposition algorithm, together with the parameters conjugated assignment and long-wave limit techniques. The highlight of the paper is that by applying new algorithms and new techniques, we obtained different types of new multiwave interaction solutions for the (3+1)-dimensional extended Jimbo–Miwa equation.

On periodic water-waves and their convergence to solitary waves in the long-wave limit

1981 ◽  
Vol 303 (1481) ◽  
pp. 633-669 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Periodic Problem ◽  
Existence Theory ◽  
Periodic Waves ◽  
Equation Formulation ◽  
Long Wave ◽  
Integral Equation Formulation ◽  
Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

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 Propagation of Elastic Waves in Prestressed Media

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Inder Singh ◽  
Dinesh Kumar Madan ◽  
Manish Gupta
Keyword(s):  
Elastic Waves ◽  
Plane Waves ◽  
Dynamical Equations ◽  
P Waves ◽  
General Solutions ◽  
External Forces ◽  
Propagation Of Elastic Waves

3D solutions of the dynamical equations in the presence of external forces are derived for a homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves. Expressions for slowness surfaces and apparent velocities for these waves are derived analytically as well as numerically and represented graphically.

Lumps, breathers, rogue waves and interaction solutions to a (3+1)-dimensional Kudryashov–Sinelshchikov equation

2020 ◽  
Vol 34 (12) ◽  
pp. 2050117 ◽  
Author(s):  
Xianglong Tang ◽  
Yong Chen
Keyword(s):  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Long Wave ◽  
Wave Limit ◽  
Hirota Bilinear

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.

scholarly journals Collision patterns on mollusc shells

1997 ◽  
Vol 1 (1) ◽  
pp. 57-76 ◽  
Author(s):  
P. J. Plath ◽  
J. K. Plath ◽  
J. Schwietering
Keyword(s):  
Reaction Diffusion ◽  
Growth Front ◽  
Cell System ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Classical Waves ◽  
Cellular Behaviour ◽  
Mollusc Shells

On mollusc shells one can find famous patterns. Some of them show a great resemblance to the soliton patterns in one-dimensional systems. Other look like Sierpinsky triangles or exhibit very irregular patterns. Meinhardt has shown that those patterns can be well described by reaction–diffusion systems [1]. However, such a description neglects the discrete character of the cell system at the growth front of the mollusc shell.We have therefore developed a one-dimensional cellular vector automaton model which takes into account the cellular behaviour of the system [2]. The state of the mathematical cell is defined by a vector with two components. We looked for the most simple transformation rules in order to develop quite different types of waves: classical waves, chemical waves and different types of solitons. Our attention was focussed on the properties of the system created through the collision of two waves.

A model study of elastic waves in a layered sphere

1967 ◽  
Vol 57 (5) ◽  
pp. 959-981
Author(s):  
Victor Gregson
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
Elastic Waves ◽  
Wave Dispersion ◽  
Flat Space ◽  
Dispersion Curves ◽  
Body Waves ◽  
Surface Wave Dispersion ◽  
Steel Sphere ◽  
Long Wave

abstract Elastic waves produced by an impact were recorded at the surface of a solid 12.0 inch diameter steel sphere coated with a 0.3 inch copper layer. Conventional modeling techniques employing both compressional and shear piezoelectric transducers were used to record elastic waves for one millisecond at various points around the great circle of the sphere. Body, PL, and surface waves were observed. Density, layer thickness, compressional and shear-wave velocities were measured so that accurate surface-wave dispersion curves could be computed. Surface-wave dispersion was measured as well as computed. Measured PL mode dispersion compared favorably with theoretical computations. In addition, dispersion curves for Rayleigh, Stoneley, and Love modes were computed. Measured surface-wave dispersion showed Rayleigh and Love modes were observed but not Stoneley modes. Measured dispersion compared favorably with theoretical computations. The curvature correction applied to dispersion calculations in a flat space has been estimated to correct dispersion values at long-wave lengths to about one per cent of correct dispersion in a spherical model. Measured dispersion compared with such flat space dispersion corrected for curvature proved accurate within one per cent at long wave lengths. Two sets of surface waves were observed. One set was associated with body waves radiating outward from impact. The other set was associated with body waves reflecting at the pole opposite impact. For each set of surface waves, measured dispersion compared favorably with computed dispersion.

2D in-plane elastic waves in curved-tapered square lattice frame structure

2021 ◽  
pp. 1-12
Author(s):  
Rajan Prasad ◽  
Ajinkya Baxy ◽  
Arnab Banerjee
Keyword(s):  
Wave Propagation ◽  
Cross Section ◽  
Cross Sections ◽  
Elastic Waves ◽  
Dynamic Stiffness ◽  
Frame Structure ◽  
Square Lattice ◽  
Wave Localization ◽  
Finite Structure ◽  
Complete Bandgap

Abstract This work proposes a unique configuration of two-dimensional metamaterial lattice grid comprising of curved and tapered beams. The propagation of elastic waves in the structure is analyzed using the dynamic stiffness matrix (DSM) approach and the Floquet-Bloch theorem. The DSM for the unit cell is formulated under the extensional theory of curved beam considering the effects of shear and rotary inertia. The study considers two types of variable rectangular cross-sections, viz. single taper and double taper along the length of the beam. Further, the effect of curvature and taper on the wave propagation is analysed through the band diagram along the irreducible Brillouin zone. It is shown that a complete band gap, i.e. attenuation band in all the directions of wave propagation, in a homogeneous structure can be tailored with a suitable combination of curvature and taper. Generation of the complete bandgap is hinged upon the coupling of axial and transverse component of the lattice grid. This coupling emerges due to the presence of the curvature and further enhanced due to tapering. The double taper cross-section is shown to have wider attenuation characteristics than single taper cross-sections. Specifically, 83.36% and 63% normalized complete bandwidth is achieved for the double and single taper cross-section for a homogeneous metamaterial, respectively. Additional characteristics of the proposed metamaterial in time and frequency domain of the finite structure, vibration attenuation, wave localization in the equivalent finite structure are also studied.

Lump and interactional solutions of the (2+1)-dimensional generalized breaking soliton equation

2019 ◽  
Vol 34 (03) ◽  
pp. 2050037
Author(s):  
Yu-Pei Fan ◽  
Ai-Hua Chen
Keyword(s):  
Solitary Wave ◽  
Asymptotic Analysis ◽  
Bilinear Form ◽  
Lump Solution ◽  
Soliton Equation ◽  
Long Wave ◽  
Wave Limit

In this paper, by using the long wave limit method, we study lump solution and interactional solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional generalized breaking soliton equation without using bilinear form. The moving properties of the lump solution, and the interactional properties of a lump and a solitary wave, are analyzed theoretically and graphically with asymptotic analysis.

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