Imaginary Wave Numbers Research Articles

Bandgap formation in topological metamaterials with spatially modulated resonators

Within the field of elastic metamaterials, topological metamaterials have recently received much attention due to their ability to host topologically robust edge states. Introducing local resonators to these metamaterials also opens the door for many applications such as energy harvesting and reconfigurable metamaterials. However, the interactions between phenomena from local resonance and modulation patterning are currently unknown. This work fills that gap by studying multiple cases of spatially modulated metamaterials with local resonators to reveal the mechanisms behind bandgap formation. Their dispersion relations are determined analytically for infinite chains and validated numerically using eigenvalue analysis. The inverse method is used to determine the imaginary wavenumber components from which each bandgap is characterized by its formation mechanism. The topological nature of the bandgaps is also explored through calculating the Chern number and integrated density of states. The band structures are obtained for various sources of modulation as well as multiple resonator parameters to illustrate how both local resonance and modulation patterning interact together to influence the band structure. Other unique features of these metamaterials are further demonstrated through the mode shapes obtained using the eigenvectors. The results reveal a complex band structure that is highly tunable, and the observations given here can be used to guide designers in choosing resonator parameters and patterning to fit a variety of applications.

Raman, ATRIR and NMR spectra, conformational stability, vibrational assignments, normal coordinates analysis and DFT calculations of hydrochlorothiazide drug

The Raman (4000-50 cm−1), attenuated total reflectance (ATR) infrared (4000-400 cm−1) spectra of Hydrochlorothiazide (C7H8ClN3O4S2; HCTZ) have been recorded in the solid state in addition to LCMS in the dimer and trimer ranges. The 1H, 13C and 15N NMR spectra were also recorded in addition to DEPT (45, 90 and 135) of HCTZ sample dissolved DMSO‑d6. Initially, three conformers resulting from the rotation of -SO2NH2 moiety around C9-S11 were proposed, namely gauche, trans and cis conformers. And then Density Functional Theory (DFT) calculations for an isolated molecule were carried out employing both B3LYP and P3PW91 methods at 6-31G(d), 6-311+G(d), 6–311++G(d,p) basis set. The cis isomer is a transition state owing to the prediction of an imaginary wavenumber, while the gauche being more stable than trans by ∼438–538 cm−1 (∼1.25–1.54 kcal) employing the afore-mentioned computations. Moreover, we carried out a potential surface scan for the gauche to determine SO2NH2 barriers to internal rotations and to ensure the optimal gauche is the most stable conformer. Aided by normal coordinate analysis (NCA) in addition to the atomic displacements in Cartesian coordinates (ADCC), confident vibrational assignment of the observed infrared and Raman bands are proposed for all fundamentals of the gauche rotamer. Nevertheless, force constants (FCs) in internal coordinates were also estimated using the above-mentioned methods and basis sets as well. The 1H, 13C and 15N chemical shifts were also correlated to their corresponding atoms. The results are reported herein and compared with HCTZ and its cocrystals whenever appropriate.

COMPUTATION OF DISPERSION CURVES OF RAYLEIGH-LAMB WAVES BY NEWTON-RAPHSON METHOD

The paper treats the numerical construction of dispersion curves which belong to the propagation of Rayleigh-Lamb waves in isotropic elastic media. The Newton-Raphson method.is applied for computation of real, imaginary and complex wavenumbers in dependence on the frequency of the wave. Due to multivaluedness, unexpected jumps between modes of dispersion curves occur in branch tracing, which prevents successful direct application of the method. This shortcoming is eliminated in the article by introducing the arc-length continuation method. The efficiency of this supplement to the Newton-Raphson method is shown on the example of an aluminum plate with constructions of dispersion curves for real, imaginary and complex wave numbers.

Characterization of the full complex-valued stiffness tensor of orthotropic viscoelastic plates using 3D guided wavefield data

A two-stage inversion scheme is proposed to determine the full set of 9 complex-valued stiffness properties of orthotropic viscoelastic plates using their 3D surface velocity response to a broadband vibrational excitation. A combination of the hybrid TLS-ESPRIT and IWC method is employed to the extract complex-valued wavenumber–frequency pairs corresponding to relevant Lamb wave and shear horizontal plate waves from the data from surface velocity response. Particle swarm optimization is chosen to inversely determine the plate’s orthotropic viscoelastic properties using the semi-analytical finite element method as a forward model to compute the real and imaginary parts of the wavenumbers. The inversion procedure first identifies the elastic parameters by matching the real wavenumber values, followed by the characterization of the viscoelastic properties which are merely linked to the imaginary wavenumber values. To validate the accuracy of the proposed method, a series of numerical simulations of 3D wavefield data using a finite element model (COMSOL) is conducted, as well as broadband experimental measurements obtained by means of a 3D Infrared Scanning Laser Doppler Vibrometer. It is shown that the inverted orthotropic viscoelastic properties based on the virtual wavefield data are in close agreement with the target values, showing a mean relative error of less than 2% and 5% error on the elastic and viscous properties, respectively. Additionally, the median absolute deviations are negligible, showing the robust convergence of the inversion procedure to a global minimum, and giving confidence to apply the method on experimental wavefield data in order to reveal the full (unknown) C-tensor for material characterization. The proposed characterization method can be used for various orthotropic viscoelastic materials, including metal sheets, carbon/epoxy laminates, and wooden plates.

Variation on propagation of Rayleigh waves in thermo-poroelastic solids

We analyse the dispersion and attenuation of Rayleigh-wave propagation along on isothermal free surface of a thermo-poroelastic medium. The thermo-poroelastic theory, including the Lord-Shulman (LS) and Green-Lindsay (GL) theories, predicts three compressional waves and one shear wave, which induce two Rayleigh-type surface waves, R1 and R2 modes along the interface by evanescent potential functions specified by vertical imaginary wavenumbers. The GL model can give a stronger P1 and P2 wave thermal attenuation and consequently a stronger velocity dispersion than the LS model. We investigate the influence of frequency and surface boundary condition. The additional thermal diffusion exhibits the inverse dispersion in R1 waves, and makes the pseudo surface-wave transition under the closed (CP) and partial permeable (PP) surface toward high frequencies. Another T wave affects the energy inference along the surface and induce the other R2 waves under the open surface (OP). The thermal dispersion occurs at characteristic frequencies around the thermal relaxation peaks, which induces the high-velocity T wave toward low frequencies when the Maxwell–Vernotte–Cattaneo relaxation time increases. Until the T wave propagates faster than the P2 wave, the Rayleigh equations lose the R2 poles making the prohibition of R2-wave propagation.

Carrier confinement and interband optical transitions in lead chalcogenide quantum wells, nanosheets, and nanoplatelets.

Analytic equation for energy dispersion of electronic states in lead chalcogenide nanosheets is derived within an effective mass model. Selection rules for interband optical transitions are analyzed and expressions for interband optical matrix elements are obtained. It is shown that the main effect of the lateral confinement in nanoplatelets can be accounted for in terms of the quantized in-plane wave vector.

Perturbation analysis of nonlinear evanescent waves in a one-dimensional monatomic chain.

This paper investigates evanescent waves in one-dimensional nonlinear monatomic chains using a first-order Lindstedt-Poincaré approach. Perturbation approaches applied to traveling waves in similar chains have predicted weakly nonlinear phenomena such as dispersion shifts and amplitude-dependent stability. However, nonlinear evanescent waves have received sparse attention, even though they are expected to serve a critical role in nonlinear interface problems. To aid in their analysis, the nonlinear evanescent waves are categorized herein as either complete or transitional evanescent waves. Complete evanescent waves, including linear evanescent waves, attenuate to zero amplitude in the far field. Transitional evanescent waves, only occurring in softening systems, attenuate to a nontrivial amplitude in the far field, regardless of the initial amplitude, resulting in a saturation effect. For both cases, the presented perturbation approach reveals that the imaginary wave number in the evanescent field is a function of space, rather than a constant value as in its linear counterpart. It also reveals that hardening and softening nonlinearity slow and accelerate the near-field decay, respectively. The predictions obtained from the perturbation approach are verified using numerical simulations with both initial-condition and boundary-continuous excitation, documenting strong agreement.

Keto-enol tautomerism, spectral (infrared, Raman and NMR) studies and Normal coordinate analysis of 4-Methyl-2-hydroxyquinoline using quantum mechanical calculations

The Raman (3500−100 cm−1) and mid-infrared (4000−400 cm−1) spectra of 4-Methyl-2-hydroxyquinoline (MHQ) have been recorded. Owing to the keto-enol tautomerism and rotation of methyl group around C4–C13 single bond, six structures (S1–S6) were hypothetically proposed. Full geometry optimization and frequency calculations were performed for keto (quinolone) and enol (hydroxyquinoline) structures using the DFT methods including B3LYP, ωB97XD and mPW1PW91 with basis sets up to 6–311+G(2d,p). In addition, 1H/13C NMR chemical shifts (δ, ppm) were estimated utilizing GIAO approach and IEF-PCM solvation model. The computational outcomes favor keto tautomer, 4-Methylquinolin-2-one (MQO) while the methyl group is staggered to C4–C5 bond (S5). However, other structures were excluded owing to high energy differences besides having an imaginary wavenumber for the transition state structures S2, S4 and S6. These results agree with the observed/calculated vibrational spectral data along with 1H/13C NMR chemical shifts. Aided by the calculated wavenumbers and normal coordinate analysis, complete assignments for the observed infrared/Raman bands to their corresponding vibration modes were quantitatively reported herein for keto- MQO. Employing B3LYP/6–311+(2d,p) structural parameters, the thermo kinetic parameter F-number (5.3936707 cm−1) is estimated along with V3 barrier (1067±48/1219±62 cm−1, 3.05±0.14/3.48±0.18 kcal/mol) using the calculated/observed methyl torsion at 217/233 cm−1.

An Accurate Integral Equation Formulation to Scattering of Periodic Grating Structures

Broadband wave transmission grating structures have potentials in many engineering applications. Theoretical predictions of the transmission spectra and near field characteristics of such structures are crucial for the understanding and design of wave functional gratings. However, accurate theoretical modeling of the wave interactions with such periodic grating structures turns out to be difficult, especially at near grazing incidence angles. In this paper, an efficient and accurate method to solve the scattering of periodic grating structures is proposed based upon integral equation formulations utilizing periodic lattice Green's functions. The method of moments is used to discretize the integral equation with the pulse basis function and the point matching scheme. The imaginary wavenumber extraction technique and the integral transformation approach are combined to efficiently and accurately evaluate the period Green's function. Meanwhile, an over-determined testing scheme on the integral equation is devised to overcome the intrinsic internal resonance of surface integral equations. Such strategy has significantly improved the numerical accuracy of the proposed approach. Calculation results are compared with Comsol simulations for various scenarios, and the proposed approach is found to be predicting physically sound results when Comsol fails at certain frequencies and incident angles. The proposed method is applicable to grating periodic structures of various shapes and it provides a useful reference and tool for researchers and designers.

Inertial amplification band-gap generation by coupling a levered mass with a locally resonant mass

Inertial amplification has been utilized in phononic media as a mechanism for the generation of large band gaps at low subwavelength frequencies. A unique feature in an inertial-amplification band gap is that it may exhibit two coupled peaks in the imaginary wavenumber portion of its band diagram. This unique double-attenuation band gap has been shown to emerge from a periodic arrangement of a levered mass whose motion is directly connected to that of an independent degree of freedom in the system through the motion of the lever base. Here we demonstrate a double-attenuation band gap emerging specifically from a modal coupling of the levered mass with a conventional local-resonance mass separately attached to the base. This presents a fundamentally distinct mechanical mechanism for the shaping of inertially-amplified band gaps and provides a pathway for realising a combination of strength and breadth in the wave attenuation characteristics. We theoretically present this concept, analytically identify critical conditions for the coupling of the attenuation peaks, and provide a series of parametric sweeps to further highlight the phenomenon and guide design. For example, we find a design with a relatively elevated level of minimum attenuation over practically the entire width of a band gap with a relative size of 130%, and another design with a smaller band gap but a 15-fold increase in the minimum attenuation strength compared to a pure IA chain.

Fast multipole method for 3-D Poisson-Boltzmann equation in layered electrolyte-dielectric media

In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson–Boltzmann (PB) equation in layered electrolyte-dielectric media. We will extend our previous work on FMMs for Helmholtz and Laplace equations in layered media [1,2] to the case of electrolyte and dielectric layered media for applications arising from biophysics and colloidal fluids, such as ion channel transport and Helmholtz double layers. Two key mathematical formulas are developed for this purpose: Firstly, a Funk–Hecke formula for purely imaginary wave numbers is derived, which facilitates the derivation of multipole expansions of the potential far fields of charges in layered electrolyte-dielectric media. Secondly, a recurrence formula is constructed for run-time computations of the Sommerfeld-type integrals used in the FMM algorithm. Numerical results show that the proposed FMM for interactions of charges embedded in layered media under screened PB potentials has the same accuracy and the O(N) computational complexity as the classic FMM for charge interactions in the free space.

The Experimental Registration of the Evanescent Acoustic Wave in YX LiNbO3 Plate.

Evanescent acoustic waves are characterized by purely imaginary or complex wavenumbers. Earlier, in 2019 by using a three dimensional (3D) finite element method (FEM) the possibility of the excitation and registration of such waves in the piezoelectric plates was theoretically shown. In this paper the set of the acoustically isolated interdigital transducers (IDTs) with the different spatial periods for excitation and registration of the evanescent acoustic wave in Y-cut X-propagation direction of lithium niobate (LiNbO3) plate was specifically calculated and produced. As a result, the possibility to excite and register the evanescent acoustic wave in the piezoelectric plates was experimentally proved for the first time. The evanescent nature of the registered wave has been established. The theoretical results turned out to be in a good agreement with the experimental ones. The influence of an infinitely thin layer with arbitrary conductivity placed on a plate surface was also investigated. It has been shown that the frequency region of an evanescent acoustic wave existence is very sensitive to the changes of the electrical boundary conditions. The results obtained may be used for the development of the method of the analysis of thin films electric properties based on the study of evanescent waves.

A NOVEL FRACTAL GEOMETRY DOPING FOR GRAPHENE NANORIBBON AND THE OPTIMIZATION OF CRYSTAL: A DENSITY FUNCTIONAL THEORY (DFT) STUDY

Chemical doping is a promising route to engineering and controlling the electronic properties of the zigzag graphene nanoribbon (ZGNR). By using the first-principles of the density functional theory (DFT) calculations at the B3LYP/ 6-31G, which implemented in the Gaussian 09 software, various properties, such as the geometrical structure, DOS, HOMO, LUMO infrared spectra, and energy gap of the ZGNR, were investigated with various sites and concentrations of the phosphorus (P). It was observed that the ZGNR could be converted from linear to fractal dimension by using phosphorus (P) impurities. Also, the fractal binary tree of the ZGNR and P-ZGNR structures is a highlight. The results demonstrated that the energy gap has different values, which located at this range from 0.51eV to 1.158 eV for pristine ZGNR and P-ZGNR structures. This range of energy gap is variable according to the use of GNRs in any apparatus. Then, the P-ZGNR has semiconductor behavior. Moreover, there are no imaginary wavenumbers on the evaluated vibrational spectrum confirms that the model corresponds to minimum energy. Then, these results make P-ZGNR can be utilized in various applications due to this structure became more stable and lower reactivity.

Fast and broadband computation of the Green's function in a cavity resonator of irregular shape using an imaginary wave number extraction technique.

A semi-analytical approach for rapid calculation of the Green's function inside a cavity of irregular shape over a broad range of frequency is presented. The method is based on the extraction of the Green's function at an imaginary wave number from itself to obtain a rapidly convergent hybrid spatial-spectral expansion of the Green's function. The method is applied to a V-grooved cavity, and the results are compared with the integral equation method.

Broadband Green's Function (BBGFL) Method With Imaginary Wavenumber Extractions for Simulations of Radiated Emissions From Irregular Shaped Printed Circuit Board

In this article, we apply the broadband Green's function for fast full-wave simulations of the radiated emissions from the printed circuit boards (PCBs) of irregular shape. Changes were made on the previous version that improve the computational efficiency and accuracies of the solutions. The improvements include imaginary wavenumber extractions for the acceleration of convergence of modal expansions without the need of high-order quadrature, and the fast-exact normalization of modes. Results of the radiated fields are illustrated over a broad frequency range up to 10 GHz and for irregular shaped PCB up to 4 in in sizes. The results are in good agreement with the method of moment for both the modal solutions and the radiated emissions. The results are shown to be more accurate than the previous version of broadband Green's functions with low wavenumber extractions. The significant computational efficiency makes this technique useful for electronic design automation to electromagnetic interference/electromagnetic compatibility applications.

Schwarzschild quantum fluctuations from Regge-Wheeler scattering

We apply a multichannel variable phase method to scattering from Regge-Wheeler potentials. Using a reduced version of the Wentzel-Kramers-Brillouin subtraction developed by Candelas and Howard, this approach allows for efficient numerical calculations of scattering data for imaginary wave number, making it possible to compute quantum expectation values in a Schwarzschild curved spacetime background through Wick rotation to the imaginary frequency axis. These scattering theory techniques are also potentially applicable to a variety of other problems involving wave propagation in curved spacetime.

Numerical solution of Rayleigh-Lamb frequency equation for real, imaginary and complex wavenumbers

Guided waves, especially Lamb waves or shear-horizontal waves, are widely used types of waves for ultrasonic inspection of large structures. Well known property of guided waves is their dispersive character, which means that the propagation velocity of the particular wave mode is not only a function of physical properties of the material, in which the wave propagates or the wave´s frequency, but also depends on the geometry of the structure in itself. Dispersion curves provide us the information related to the dependency between the wavenumber and the frequency of the particular mode and can be obtained by a numerical solution of Rayleigh-Lamb frequency equation. A solution of Rayleigh-Lamb frequency equation forms for a given frequency and plate thickness a set of a finite number of real and pure imaginary wavenumbers and an infinite number of complex wavenumbers. Proposed paper presents a complete procedure of how to obtain all three kinds of wavenumbers for a given geometry and frequency interval. The main emphasis is placed on the effectiveness of the procedures, which are used for finding the roots of dispersion equation for all three kinds of wavenumbers.

Beam driven temporal growth and spatial amplification of electrostatic waves in a partially spin polarized degenerate magnetized plasma

The temporal growth and spatial amplification of electrostatic waves due to the interaction of an electron beam with a partially spin polarized degenerate magnetized plasma are studied. Using Fourier analysis, a generalized linear dispersion relation is derived for parallel propagating electrostatic waves in the presence of electron spin polarization. The electron spin polarization gives birth to a new spin-dependent wave (spin electron acoustic wave) in the real wave spectrum. Furthermore, increasing spin polarization reduces the temporal growth rate of the unstable mode. Regarding spatial amplification (magnitude of the imaginary wave number), it is found that spin polarization significantly affects the magnitude of both the real and the imaginary wave numbers. Additionally, how the various other parameters such as electron beam density, beam thermal speed, and streaming speed of the beam electron influence the temporal growth and spatial rates is also discussed. These effects would have a strong bearing on instability phenomena in the degenerate metallic and semiconductor plasmas and in astrophysical plasmas.

Propagating and non-propagating waves in infinite plates and rectangular cross section plates: orthogonal polynomial approach

The wave field near defect, edge or load represents a superposition of propagating and non-propagating modes. It is necessary to investigate the non-propagating wave characteristics if the wave scatter and edge resonance should be taken into consideration. In this paper, a modified orthogonal polynomial approach was developed to computationally solve, for non-dissipative materials, the propagating waves given by pure real roots and non-propagating waves given by pure imaginary and complex roots. The dispersion curves giving real part wavenumber and imaginary part wavenumber as functions of the angular frequency are obtained. The results for an isotropic plate and a cubic plate are calculated to make the comparison with exact results. The dispersion curves for a plate with a rectangular cross section are also illustrated. The displacement amplitude variations are illustrated in the vicinity of the frequency where the non-propagating wave becomes a propagating one.

The theory of a trapped degenerate mode resonator.

Resonance based biosensors are used in the detection of biological molecules for medical diagnostics. Sensing in a liquid environment is very desirable for this application, but presents a significant challenge for resonators based upon conventional technologies. In this paper, the major originality lies in the development and exposition of a fundamental theory enabling design of an original elastic resonant sensor whose modes are engineered to simultaneously possess three separate but complementary dynamical properties: namely, (1) in-plane displacement of the free interface whereby the SH waves are uncoupled from the SV and P waves; (2) intrinsic modal trapping; and finally, (3) cyclic symmetry and modal degeneracy. The modal trapping is due to the physical configuration of the resonator resulting in an imaginary wavenumber for one region of the resonator. The wave will be evanescent in this region and propagating elsewhere. The fundamental principles are elucidated, and analytical techniques are presented that facilitate the efficient design of this unique class of device.