Phase Velocity Parameters Research Articles

Bi-directional solitons of dual-mode Gardner equation derived from ideal fluid model

In this article, the main goal is to find the exact solution to the dual-mode Gardner equation derived from the ideal fluid model. To achieve this, two distinct methods will be employed: the tan/cot method and the tanh/coth method. Using these methods, the problem can be seen through different angles, providing a full understanding of how the equation works and the various solutions it provides. Furthermore, the periodic, kink and singular solutions will be illustrated through the use of 3D and 2D graphs. This visualization will clearly show how the phase velocity parameter affects the solutions, improving our understanding of wave dynamics within the dual-mode Gardner equation. The findings from this study help in understanding how both solitons and periodic solutions influence wave behavior in shallow water and how light travels through optical fibers.

Novel Optical bi-directional solutions to the new dual-mode derivative nonlinear Schrödinger equation

In recent times, a novel category of nonlinear physical models known as dual-mode nonlinear equations has emerged. These equations include various real-valued dual-mode equations linked to widely-known single-mode equations like KdV, mKdV, Schrödinger and Burger’s. Extensive research has been conducted to establish and investigate these equations. This study presents a novel dual-mode derivative nonlinear Schrödinger equation that incorporates new parameters for dissipative effects, nonlinearity, and interaction phase velocity. Various methods such as the tanh-coth scheme, extended exponential method, Kudryashov-scheme and the sine-cosine function methods are employed to investigate the solutions of the model. The obtained solutions are illustrated through graphical 2D and 3D and to demonstrate their dynamics and shapes. Furthermore, the interaction of the dual-waves is correlated with changes in the phase-velocity parameter. This model describes propagation of two simultaneously directional waves instead of as in standard Schrödinger equation. For the propagation of solitons in nonlinear optics, the solutions found in this study have important significance. All the resulting solutions can help to comprehend the underlying mechanisms for numerous nonlinear phenomena in diverse domains, including nonlinear optics, plasma physics, Bose–Einstein condensates and others.

Propagation of two-wave solitons depending on phase-velocity parameters of two higher-dimensional dual-mode models in nonlinear physics

Nonlinear evolution equations exhibit a variety of physical behaviours, which are clearly illustrated by their exact solutions. In this view, this article concerns the study of dual-mode (2 + 1)-dimensional Kadomtsev-Petviashvili and Zakharov-Kuznetsov equations. These models describe the propagation of two-wave solitons traveling simultaneously in the same direction and with mutual interaction dependent on an embedded phase-velocity parameter. The considered nonlinear evolution equations have been solved analytically for the first time using the Paul-Painlevé approach method. As a result, new abundant analytic solutions have been derived successfully for both the considered equations. The 3D dynamics of each of the solution has been plotted by opting suitable constant values. These graphs show the dark-soliton, bright-soliton, complex dual-mode bright-soliton, complex periodic-soliton and complex dual-mode dark-soliton solutions.

Symmetric and asymmetric binary-solitons to the generalized two-mode KdV equation: Novel findings for arbitrary nonlinearity and dispersion parameters

In this paper, the two-mode version of the generalized KdV (gTMKdV) equation is presented. The new model arises in weakly dispersive and nonlinear wave medium and describes the motion of either symmetric or asymmetric binary-waves regarding each wave’s height, which mainly depends on the nonlinearity and dispersion parameters. The overlapping of such binary-waves is affected by the phase-velocity parameter. Two recent effective schemes are proposed to extract novel explicit solutions for arbitrary values of the nonlinearity and dispersive factors. In addition, comprehensive graphical analysis is conducted to explain the nonlinear dynamics of the obtained bidirectional solitary solutions of gTMKdV. Finally, all the recovery solutions reported in this work are verified by direct substitution in the governing model.

Propagations of symmetric bidirectional nonlinear waves in two-mode foam drainage model

In this paper, a new two-mode version of the foam drainage equation is presented. The new model describes the motion of instantaneous symmetric double-waves and their interaction depends on two involved parameters known as nonlinearity and phase-velocity. Two recent methods are used to obtain solutions with interesting variety of physical structures such as kink-type and periodic shape. In addition, the influence of both nonlinearity and phase-velocity parameters are investigated by using graphical analysis. Finally, all the obtained solutions to the TMFD are confirmed by direct substituting and verifying the correctness of the governing equation.

Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics

In this work, we present a new two-mode version to the complex Hirota equation. This new model arises in many applications in the field of optics, communications and other engineering sciences. It describes the propagation of two-symmetric solitary waves moving simultaneously with dependent phase-velocity parameter. The extended tanh-coth expansion method and the polynomial-function method are used to extract novel new cusp-type solutions to the Hirota model. A comprehensive graphical analysis is conducted to show physical aspects of this new type of nonlinear equations. The findings of this work are from mathematical point of view and we propose some benefits regarding the concept of two-mode where it could be visualized and practically applied.

Construction of Solitary Two-Wave Solutions for a New Two-Mode Version of the Zakharov-Kuznetsov Equation

A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter.

The dynamics of new dual-mode Kawahara equation: interaction of dual-waves solutions and graphical analysis

In this paper, we introduce a new dual-mode Kawahara equation with new dissipative, nonlinearity and interaction phase velocity parameters. Also, we study the solutions of this model by using the tanh-scheme, Kudryashov-scheme and the sine–cosine function methods. Dynamics and shapes of the obtained solutions are illustrated by using graphical 2D and 3D plots. Finally, the interaction of the obtained dual-waves has been linked with the change of the phase-velocity parameter.

Multiplicative of dual-waves generated upon increasing the phase velocity parameter embedded in dual-mode Schrödinger with nonlinearity Kerr laws

In this paper, we introduced a new dual-mode nonlinear Schrodinger (DMNLS) equation with nonlinearity Kerr of types square-root law and dual-power law. The new model consists of three parameters defined as dissipative, nonlinearity and the phase velocity. Also, this model describes propagations of two simultaneously directional waves instead of single wave as in the standard Schrodinger model. We determined the necessary conditions on the dissipative nonlinearity parameters that produce soliton solutions of DMNLS. Finally, a graphical analysis regarding the effect of the phase velocity on the shapes of the obtained dual-waves is accomplished.

Metamaterial Dispersion Engineering Concepts and Applications

Metamaterial dispersion engineering is presented as a general concept for engineering the phase versus frequency response of microwave materials and devices. Two categories of metamaterials are considered, composite right/left-handed (CRLH) transmission line and multiscale ferromagnetic nanowire (FMNW) metamaterials. The dispersive Drude properties of CRLH metamaterials are derived and corresponding application examples are described in terms of CRLH dominant Taylor dispersive parameters: a tight broadband coupled-line coupler (phase velocity parameter), an ultra-wideband pulse position modulator transmitter (group velocity parameter), and a leaky-wave antenna based real-time spectrum analyzer (group velocity dispersion parameter). FMNW metamaterials are discussed as a double-Lorentz example of a multiscale metamaterial with unique properties, and their applications are illustrated with the example of a dual-band edge-mode isolator based on the recently discovered double ferromagnetic resonance.

Predictions of the modified Biot–Attenborough model for the dependence of phase velocity on porosity in cancellous bone

The modified Biot–Attenborough (MBA) model for acoustic wave propagation in porous media has been found useful to predict wave properties in cancellous bone. The present study is aimed at applying the MBA model to predict the dependence of phase velocity on porosity in cancellous bone. The MBA model predicts a phase velocity that decreases nonlinearly with porosity. The optimum values for input parameters of the MBA model, such as compressional speed c m of solid bone and phase velocity parameter s 2, were determined by comparing the predictions with previously published measurements in human calcaneus and bovine cancellous bone. The value of the phase velocity parameter s 2 = 1.23 was obtained by curve fitting to the experimental data for 53 human calcaneus samples only, assuming a compressional speed c m = 2500 m/s of solid bone. The root-mean-square error (RMSE) of the curve fit was 15.3 m/s. The optimized value of s 2 for all 75 cancellous bone samples including 22 bovine samples was 1.42 with a value of 55 m/s for the RMSE of the curve fit. The latter fit was obtained by using of a value of c m = 3200 m/s. Although the MBA model relies on the empirical parameters determined from experimental data, it is expected that the model can be usefully employed as a practical tool in the field of clinical ultrasonic bone assessment.

Dependence of phase velocity on porosity in cancellous bone: Application of the modified Biot-Attenborough model

This study aims to apply the modified Biot-Attenborough (MBA) model to predict the dependence of phase velocity on porosity in cancellous bone. The MBA model predicted that the phase velocity decreases nonlinearly with porosity. The optimum values for input parameters of the MBA model, such as compressional speed cm of solid bone and phase velocity parameter s2, were determined by comparing the prediction with the previously published measurements in human calcaneus and bovine cancellous bones. The value of the phase velocity parameter s2=1.23 was obtained by curve fitting to the experimental data only for 53 human calcaneus samples with a compressional speed cm=2500 m/s of solid bone. The root-mean-square error (rmse) of the curve fit was 15.3 m/s. The optimized value of s2 for all 75 cancellous bone samples (53 human and 22 bovine samples) was 1.42 with the rmse of 55 m/s. The latter fit was obtained by using cm=3200 m/s. Although the MBA model relies on empirical parameters determined from the experimental data, it is expected that the model can be usefully employed as a practical tool in the field of clinical ultrasonic bone assessment.

High-accuracy absorbing boundary conditions for high-order parabolic wave equations

Using a new systematic approach, a novel set of arbitrarily high-accuracy absorbing boundary conditions (HABC) is introduced for high-order parabolic equations. HABC is derived by finite-element discretization of the boundary, followed by applying a special integration scheme and imaginary stretching. The resulting boundary condition is then coupled with the interior and can be solved with the same numerical method as the interior, e.g., finite differences. The accuracy of the HABC is controlled by two parameters: (1) the number of absorbing layers (the order of the HABC); and (2) a reference phase velocity for each layer. The latter controls the wave number for which the HABC absorbs the wave field exactly. Hence, the absorption range, i.e., the distribution of the reflection coefficient, can be controlled by the coefficient matrices of the HABC. A specially designed explicit finite-difference scheme is used to solve the HABC. The added computational cost due to the HABC is negligible for practical purposes. The performance of the proposed boundary condition is shown through numerical examples for high-order parabolic equations, for different orders of the HABC and different sets of phase velocity parameters. Effective absorption of both propagating and evanescent waves is illustrated in the results. [Work supported by NSF.]

Propagation and Structural Interpretation of Non-Plane Waves

SUMMARY As a model for the 2-D horizontal propagation of seismic surface waves, we study the propagation of non-plane acoustic waves in homogeneous and inhomogeneous media. We find that their phase velocity depends not only on the medium but also on the local geometry of the wavefield, especially on the distribution of amplitudes around the point of observation. the phase velocity of a wave is therefore conceptually and in most cases numerically different from the phase velocity parameter in the wave equation, which is determined by the elastic properties of the medium. the same distinction must be made for seismic surface waves. Although it is a common observation that waves of the same period can propagate with different phase velocities over the same path, the fundamental character of this observation has apparently not been recognized, and the two phase velocities are frequently confused in the seismological literature. We derive a local relationship between the two phase velocities that permits a correct structural interpretation of acoustic waves in inhomogeneous media, and also of non-plane seismic surface waves in laterally homogeneous parts of the medium.

Synthetic aperture radar imaging of ocean waves from an airborne platform: Focus and tracking issues

There has been a controversial issue of many years standing in airborne Synthetic Aperture Radar (SAR) ocean imaging which this paper addresses and resolves. Investigators have been strongly divided on the reasons for apparent improvement in wave contrast in response to processor focus adjustment. The dispute has centered on two parameters of wave dynamics: orbital velocity and phase velocity. This paper shows that both orbital velocity and phase velocity are of fundamental importance in the SAR wave imaging problem. The first affects the phase of the received signal, leads to velocity bunching, and is scaled by the ratio of sensor altitude to sensor velocity. The second affects the magnitude of the received signal, leads to translation of wave features during image formation (observed as blurring in the image), and is scaled by the ratio of wave phase velocity to sensor velocity, thus becoming significant for airborne radars. This treatment of the phase velocity parameter is new. It is shown that focus adjustment, as a side effect, shifts image position. This explains why experiments have appeared to “prove” that focus adjustment may be optimised for wave movement. The paper shows that better performance is obtained by compensating individual looks for wave movement before look summation, while using nominal perfect focus just as for static scatterers. The work is applicable to a full ocean wave spectrum and does not depend on the details of the scattering mechanism itself.

Influence of geometric taper on the derivation of the true propagation coefficient using a three point method

We studied the effect of geometric taper on the derivation of the true propagation coefficient from three pressures determined 10 cm apart (‘three-point method’). For this purpose the true propagation coefficients of a uniform latex tube (length 50 cm, outer diameter 12.73 mm, Womersley phase velocity 6.23–6.42 ms −1 (1–10 Hz), Womersley damping coefficient 0.05–0.14 m −1 (1–10 Hz)) and of a tapered latex tube (length 50 cm, outer diameter varying from 15.88 to 9.45 mm, in the middle section with same properties as the uniform tube) were determined. The real part of the true propagation coefficient (the damping coefficient) was compared with apparent damping, and with the damping coefficient calculated using Womersley's theory. The imaginary part of the true propagation coefficient (the phase coefficient) was expressed in terms of phase velocity. True phase velocity was compared with measurements of apparent phase velocity, foot-to-foot velocity, and calculations of phase velocity parameters using Womersley's theory and the Moens-Korteweg equation. The results show that in the uniform tube the three-point propagation coefficient is in agreement with all other estimates. Taper causes an error in the three-point propagation coefficient. At some frequencies the damping is reversed to amplification (values up to −2 m −1) and the phase velocity may be both overestimated or underestimated (up to 50%). The overestimation of true damping as reported in the literature cannot be explained from vessel taper.

Imaging of VSP Data

Migration of VSP data can be effectively accomplished using a focused-lag sum imaging technique, similar to the Kirchoff 'diffraction stack' method. The imaging procedure is sufficiently flexible to handle the irregular source-receiver geometry associated with walkaway-source and deviated hole VSP. Signal dispersion is readily accommodated by admitting both group and phase velocity parameters. The migration can be performed on transmission data (downgoing wave) to yield a velocity map or on reflection data (upgoing wave) to produce a reflector map. The accuracy of both reflector mapping and velocity mapping depends on the nature and form of the signal. Pulse compression and multiple suppression are desirable pre-processing operations. The technique has been tested on synthetic VSP data and successfully applied to shallow VSP data collected in the Sydney Basin coalfields. For the real data, average velocities of the coal measure sequence and blurred images of the main seams were obtained. An equivalent section migration technique, based on leastsquares theory, is formulated in the Appendix.