Chree Equation Research Articles

High accuracy solutions for the Pochhammer–Chree equation in elastic media

This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

Optical soliton solutions of generalized Pochammer Chree equation

This research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study’s findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures.

Double reductions and traveling wave structures of the generalized Pochhammer–Chree equation

Symmetry methods are always very useful for discussing the classes of differential equation solutions. This article focuses on traveling wave structures of the generalized Pochhammer–Chree (PHC) equation. First, we will discuss Lie point symmetries of the PHC equation to classify the solutions. Then, we formulate traveling wave structures considering the reduced differential equations (DEs) by using sech method and the new extended direct algebraic (EDA) method. In the end, we will sketch some of the traveling wave structures.

The exact solutions of the conformable time fractional version of the generalized Pochhammer–Chree equation

The time-fractional version of the generalized Pochhammer–Chree equation is analyzed. In this paper, the equation is converted into an ordinary differential equation by applying certain real transformation, then the discrimination of polynomials system is used to find exact solutions depending on the fractional order derivative. The obtained solutions are graphically illustrated for different values of the fractional order derivative keeping the other parameters fixed.

Manual for Calibrating Sound Speed and Poisson’s Ratio of (Split) Hopkinson Bar via Dispersion Correction Using Excel® and Matlab® Templates

This manual presents a procedure to calibrate the one-dimensional sound speed (co) and Poisson’s ratio (ν) of a (split) Hopkinson bar using the open-source templates written in Excel® and Matlab® for dispersion correction. The Excel® template carries out the Fourier synthesis and one-time dispersion correction of a traveling elastic pulse under a given set of co and ν. The MATLAB® template performs the Fourier synthesis and iterative dispersion correction of a traveling elastic pulse for a range of co and ν sets. In the case of the iterative dispersion correction, a set of co and ν is assumed at each iteration step, and the sound speed vs. frequency (cdc vs. fdc) relationship necessary for dispersion correction is obtained under the assumed set by solving the Pochhammer–Chree equation. Subsequently, dispersion correction is carried out by using the cdc vs. fdc relationship. The co and ν values of the bar are determined in the iteration process when the dispersion-corrected pulse profiles are reasonably consistent with the measured ones at two travel distances (2103 and 4000 mm) in the bar. In the case of the experimental profile considered herein, the ν and co values were calibrated to six and four decimal places, respectively. The calibration algorithm is described with the tips for using the open-source templates, which are available online in a publicly accessible repository.

Sound Speed and Poisson’s Ratio Calibration of (Split) Hopkinson Bar via Iterative Dispersion Correction of Elastic Wave

AbstractA process of calibrating a one-dimensional sound speed (co) and Poisson’s ratio (ν) of a (split) Hopkinson bar is presented. This process consists of Fourier synthesis and iterative dispersion correction (time-domain phase shift) of the elastic pulse generated by the striker impact on a circular bar. At each iteration, a set of co and ν is assumed, and the sound speed versus frequency (cdc versus fdc) relationship under the assumed set is obtained using the Pochhammer–Chree equation solver developed herein for ground state excitation. Subsequently, each constituting wave of the overall elastic pulse is phase shifted (dispersion corrected) using the cdc–fdc relationship. The co and ν values of the bar are determined in the iteration process when the dispersion-corrected overall pulse profiles are reasonably consistent with the measured profiles at two travel distances in the bar. The observed consistency of the predicted (dispersion-corrected) wave profiles with the measured profiles is a mutually self-consistent verification of (i) the calibrated values of co and ν, and (ii) the combined theories of Fourier and Pochhammer–Chree. The contributions of the calibrated values of co and ν to contemporary bar technology are discussed, together with the physical significance of the tail part of a traveling wave according to the combined theories. A preprocessing template (in Excel®) and calibration platform (in matlab®) for the presented calibration process are openly available online in a public repository.

Pochhammer–Chree equation solver for dispersion correction of elastic waves in a (split) Hopkinson bar

A robust algorithm for solving the Bancroft version of the Pochhammer–Chree (PC) equation is developed based on the iterative root-finding process. The formulated solver not only obtains the conventional n-series solutions but also derives a new series of solutions, named m-series solutions. The n-series solutions are located on the PC function surface that relatively gradually varies in the vicinity of the roots, whereas the m-series solutions are located between two PC function surfaces with (nearly) positive and negative infinity values. The proposed solver obtains a series of sound speeds at exactly the frequencies necessary for dispersion correction, and the derived solutions are accurate to the ninth decimal place. The solver is capable of solving the PC equation up to n = 20 and m = 20 in the ranges of Poisson’s ratio ( ν) of 0.02 [Formula: see text] ν [Formula: see text] 0.48, normalised frequency ( F) of F [Formula: see text] 30, and normalised sound speed ( C) of C [Formula: see text] 300. The developed algorithm was implemented in MATLAB®, which is available in the Supplemental Material (accessible online).

On the integrability of the generalized Pochhammer–Chree (PC) equations

In this paper, the generalized Pochhammer–Chree (PC) equations which the propagation of longitudinal deformation waves in an elastic rod is considered. By using the Weierstrass elliptic function method, we find exact traveling wave solutions. As a result, some previously known solutions are recovered, and at the same time some new ones are also given. Moreover, three-dimension and two-dimension graphics of some solutions have been plotted.

Dispersion correction in split-Hopkinson pressure bar: theoretical and experimental analysis

The paper presents experimental and mathematical analysis of the dispersion effect of pulses, which propagate along an elastic bar whose cross section has a finite radius. In some cases, dispersion can influence significantly the interpretation of the experimental data, obtained in experimental schemes, which are based on using measuring bars for investigation material behavior at high strain rates (the split-Hopkinson pressure bar method and its modifications). Solutions for the Pochhammer–Chree equation are obtained for various measuring bars used in experimental setups in the Research Institute for Mechanics of Lobachevsky State University of Nizhny Novgorod. The procedure of pulse shift accounting for dispersion has been implemented and tested on the basis of our experimental data and the data available in the existing scientific literature. This procedure was employed to analyze the influence of pulse shape on the degree of its change in shape during its propagation along a measuring bar. It is shown how the procedure of dispersion shift improves the quality of interpretation of primary experimental data for some materials.

On degeneracy of dispersive waves at the bulk wave velocities

Degeneracy of the linear dispersion wave equation at the phase velocities coinciding with the bulk wave velocities is observed and analysed. Spectral analysis of Pochhammer – Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.

Polarization of the Longitudinal Pochhammer–Chree Waves

Abstract The exact solutions of the linear Pochhammer – Chree equation for propagating harmonic waves in a cylindrical rod, are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear wave speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer – Chree wave can propagate at this phase speed.

Polarization of Pochhammer–Chree Waves: Axisymmetric Longitudinal Modes

Exact solutions to the Pochhammer–Chree equation describing harmonic wave propagation in an elastic cylindrical bar are analyzed. For the first time, spectral analysis of the matrix of dispersion equation is performed for axisymmetric longitudinal modes. Analytic expressions for wave polarization are derived. For the fundamental axisymmetric longitudinal mode, polarization coefficients of the corresponding waves at the bar surface are determined and frequency dependent variations of the coefficients are analyzed. It is found that, when the phase velocity of the fundamental axisymmetric longitudinal mode coincides with the velocity of transverse waves, all the displacement components determined at the lateral bar surface simultaneously vanish, which is significant for acoustic waveguide design.

Pochhammer–Chree waves in rods: degeneracy at the bulk wave velocities

Degeneracy of the linear Pochhammer–Chree wave equation at phase velocities coinciding with the bulk wave velocities is observed and analyzed. Spectral analysis of Pochhammer–Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.

Longitudinal Pochhammer — Chree Waves in Mild Auxetics and Non-Auxetics

ABSTRACTThe exact solutions of Pochhammer — Chree equation for propagating harmonic waves in isotropic elastic cylindrical rods, are analyzed. Spectral analysis of the matrix dispersion equation for the longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of the wave polarization due to variation of Poisson’s ratio for mild auxetics (Poisson’s ratio is greater than -0.5) is analyzed and compared with the non-auxetics. It is observed that polarization of the waves for both considered cases (auxetics and non-auxetics) exhibits abnormal behavior in the vicinity of the bulk shear wave speed.

Propagation of longitudinal Pochhammer – Chree waves in cylindrical piles

The exact solutions of the linear Pochhammer – Chree equation for propagating harmonic waves in cylindrical piles, are analysed and reviewed emphasizing the role of both propagating and non-propagating wave modes. The notion of limiting wave velocities for the Pochhammer – Chree waves is discussed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes of the Pochhammer – Chree waves is performed, revealing existence of the non-propagating mode at the phase velocity coinciding with the c2 shear wave velocity. Analytical expressions for displacement fields are derived for both interior and exterior domains.

Pochhammer–Chree waves: polarization of the axially symmetric modes

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed ( $$c_2$$ ) all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree wave can propagate at $$c_2$$ phase speed.

Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of C2 phase speed

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed and analytical expressions for displacement fields are obtained. The variation of wave polarization on the free surface due to the variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree waves can propagate at this phase speed.

A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

AbstractIn this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

Direct Observation of Gigahertz Coherent Guided Acoustic Phonons in Free-Standing Single Copper Nanowires.

We report on gigahertz acoustic phonon waveguiding in free-standing single copper nanowires studied by femtosecond transient reflectivity measurements. The results are discussed on the basis of the semianalytical resolution of the Pochhammer and Chree equation. The spreading of the generated Gaussian wave packet of two different modes is derived analytically and compared with the observed oscillations of the sample reflectivity. These experiments provide a unique way to independently obtain geometrical and material characterization. This direct observation of coherent guided acoustic phonons in a single nano-object is also the first step toward nanolateral size acoustic transducer and comprehensive studies of the thermal properties of nanowires.

Investigation of dynamic Saint-Venant's principle in a cylindrical waveguide – Analytical results

An analytical approach has been carried out to analyze the wave propagation in the cylindrical waveguide which was hit by four impactors with different head shapes. The Pochhammer–Chree solution, including evanescent wave modes, was applied to describe the stress field in the vicinity of the proximal end of the waveguide. Two strategies were used to derive the Pochhammer–Chree equation convergent at the vicinity of the loaded end, i.e., 1) Fast Fourier Transformation of the loading boundary condition, and 2) derivation of complex wave-numbers for the evanescent modes. The length of the affected zone, which confirms a dynamic Saint-Venant's principle (DSVP), was about 2.2 times of radius based on the analytical results, which coincide with the experimental and numerical results. Parametric study has also been conducted with respect to different frequency ranges of the pulse wave.