Resonant Wavenumber Research Articles

A Review on Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers Using Research Based Learning

Belajar adalah proses perolehan pengetahuan atau keterampilan melalui studi atau pengalaman. Salah satu metode pembelajaran untuk meningkatkan pemahaman siswa adalah Research-based Learning (RBL), sebuah metodologi yang bertujuan menanamkan berfikir kritis, kreativitas, dan keterampilan memecahkan masalah. Pada penelitian ini, metode RBL digunakan untuk meningkatkan pemahaman dalam percobaan Chladni dengan mengamati fenomena resonansi ketika pelat diberi getaran dengan frekuensi tertentu serta menentukan besar frekuensi yang menghasilkan pola-pola Chladni. Pada metode ini, siswa akan bekerja dalam kelompok yang terdiri dari 3 orang. Ada lima langkah yang akan dilakukan pada RBL ini: (i) merumuskan masalah dan tujuan, membuat hipotesis, (ii) melakukan eksperimen, mengoleksi data hasil eksperimen (iii) mencari informasi mengenai eksperimen, memproses data (iii) menganalisis data, menjawab rumusan masalah (iv) membuat kesimpulan, (v) mempresentasikan hasilnya kepada dosen/penguji. Pada akhir eksperimen ini siswa mampu membuktikan hukum Chladni.

Resonance of Low-frequency Electromagnetic and Ion-sound Modes in the Solar Wind

Parker Solar Probe measurements have recently shown that coherent fast magnetosonic and Alfvén ion-cyclotron waves are abundant in the solar wind and can be accompanied by higher-frequency electrostatic fluctuations. In this Letter we reveal the nonlinear process capable of channelling the energy of low-frequency electromagnetic to higher-frequency electrostatic fluctuations observed on board Parker Solar Probe. We present Hall-MHD simulations demonstrating that low-frequency electromagnetic fluctuations can resonate with the ion-sound mode, which results in steepening of plasma density fluctuations, electrostatic spikes, and harmonics in the electric field spectrum. The resonance can occur around the wavenumber determined by the ratio between local sound and Alfvén speeds, but only in the case of oblique propagation to the background magnetic field. The resonance wavenumber, its width, and steepening timescale are estimated, and all indicate that the revealed two-wave resonance can frequently occur in the solar wind. This process can be a potential channel of energy transfer from cyclotron resonant ions producing the electromagnetic fluctuations to Landau resonant ions and electrons absorbing the energy of the higher-frequency electrostatic fluctuations.

Plasmonic excitations in double-walled carbon nanotubes

The interactions of charged particles moving paraxially in multi-walled carbon nanotubes may excite electromagnetic modes. This wake effect has recently been proposed as a potential novel method of short-wavelength high-gradient particle acceleration and for obtaining brilliant radiation sources. In this work, the excitation of wakefields in double-walled carbon nanotubes is studied by means of the linearized hydrodynamic theory. General expressions have been derived for the excited longitudinal and transverse wakefields and related to the resonant wavenumbers which can be obtained from the dispersion relation. In the absence of friction, the stopping power of the wakefield driver, modelled here as a charged macroparticle, can be written solely as a function of these resonant wavenumbers. The dependencies of the wakefields on the radii of the double-walled carbon nanotubes and the driving velocity have been studied. Double-walled carbon nanotubes with inter-wall distances much smaller than the internal radius may be a potential option to obtain higher wakefields for particle acceleration compared to single-walled carbon nanotubes.

CARS.jl: Efficient fitting of dual‐pump multi‐species CARS spectra using compressed libraries

AbstractIn this publication, we demonstrate the extension of our loss‐less library approach previously limited to fitting ro‐vibrational N2 CARS spectra for thermometry purposes to accurate and efficient fitting of dual‐pump CARS spectra with multiple species. The fundamental reasoning behind the compression approach is to make use of finite resolution effects arising from the laser linewidths, which are typically larger than the transitional linewidths. For accurate reconstruction at runtime of the fit, in addition to the squared modulus and real component, also the imaginary component as well as intra‐region species cross terms have to be compressed and tabulated for dual‐pump multi‐species spectra. The resulting libraries do not require to tabulate combinations of mole fractions, leading to a near linear growth of the library with respect to the number of species. The accuracy and computational efficiency of the method is benchmarked using a reference implementation provided alongside this publication. Library generation times, mainly depending on the underlying spectral model, are typically in the order of seconds to a few minutes. The simulation of a single dual‐pump spectrum containing three species is sped up by a factor of >5000 without significantly affecting the accuracy of reconstruction. In a simulated experiment with Stokes‐noise imposed data, the time required to it temperature, mole fraction of three resonant species and wavenumber shift on a single spectrum required 90 ms.

Stability of longitudinal sediment waves formed by turbidity currents: linear and weakly nonlinear perspectives

We perform the linear and weakly nonlinear stability analyses of longitudinal sediment waves triggered by the interaction of turbidity currents with an erodible bed. The mathematical framework is based on the two-dimensional flow equations, advection–diffusion equation for sediment concentration and Exner equation for bed evolution. Using the standard linearization, the linear analysis offers the resonant wavenumber that maximizes the growth rate of sediment waves. We study the influence of the key parameters, such as the gravitational parameter, longitudinal bed slope, Rouse number, shear Reynolds number, relative roughness number and erosion coefficient, on the growth rate, resonant wavenumber and phase velocity. The results reveal that the longitudinal sediment waves migrate both upstream and downstream. By plotting the gravitational parameter against the dimensionless wavenumber, we obtain a stability diagram that accurately captures the experimental data plots within the unstable zone. Additionally, we explore the perturbation fields of velocity components, pressure and sediment concentration. Employing the centre manifold projection, the weakly nonlinear analysis provides the equilibrium amplitude of sediment waves and its sensitivity to the key parameters. The predicted wavelength and amplitude of sediment waves are comparable with field observations in the Atlantic Ocean, the Sea of Japan and the Pacific Ocean.

NUMERICAL MODELLING OF ELECTROMAGNETIC WAVE SCATTERING ON GRATINGS WITH ONE-DIMENSIONAL QUASI-FRACTAL PERIOD STRUCTURE

Numerical modelling of the properties of E-polarised and H-polarised waves scattered on periodic screened quasi-fractal gratings is carried out. The location of the band system at each period is determined by the principle of constructing a generalized symmetric Cantor set at a certain step of the algorithm. A mathematical model of the problems based on systems of boundary singular integral equations of the first kind was used in the study. These systems of equations were obtained using the method of parametric representations of singular and hypersingular integral operators. The systems of singular integral equations were solved numerically using the computational schemes of the method of discrete singularities. The solutions of these equations are used to obtain the main characteristics of the electric and magnetic fields. This experiment proved the possibility of using the MDS computational scheme to analyse systems containing 8 – 16 bands at different distances from each other. Graphs of the dependence of harmonic amplitudes on the wavenumber, point plots of absolute values of all non-zero harmonics at resonant wavenumber values, and maps of electric and magnetic field components in the region above the grating were obtained. It is confirmed that the overall field structure in the case of normal incidence is significantly influenced by all harmonics with absolute numbers from 0 to 50. The harmonics had a large number of resonances that were observed at different values of the wavenumber. This led to a complex structure of the isolines of absolute values of the scattered electric and magnetic field amplitudes in the region above the structure, and a significant difference in amplitude values with small changes in coordinates. In the future, it is planned to carry out computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case considered in this paper. The proposed structure may be of interest for the design of multimode broadband antennas.

A topology optimization of open acoustic waveguides based on a scattering matrix method

This study presents a topology optimization scheme for realizing a bound state in the continuum along an open acoustic waveguide comprising a periodic array of elastic materials. First, we formulate the periodic problem as a system of linear algebraic equations using a scattering matrix associated with a single unit structure of the waveguide. The scattering matrix is numerically constructed using the boundary element method. Subsequently, we employ the Sakurai–Sugiura method to determine resonant frequencies and the Floquet wavenumbers by solving a nonlinear eigenvalue problem for the linear system. We design the shape and topology of the unit elastic material such that the periodic structure has a real resonant wavenumber at a given frequency by minimizing the imaginary part of the resonant wavenumber. The proposed topology optimization scheme is based on a level-set method with a novel topological derivative. We demonstrate a numerical example of the proposed topology optimization and show that it realizes a bound state in the continuum through some numerical experiments.

Nonlinear light scattering spectra in water: Dependence on the content of dissolved gas and effects of non‐stationarity

AbstractHere, we study stimulated nonlinear scattering in the spectral range near the second optical harmonic of a laser pump in water samples containing gas nanobubbles. It was found that the spectral and spatial characteristics of nonlinear scattering radiation undergo significant changes upon settling of initially degassed water samples in a flask, open to the atmospheric air. It turned out that the contact with the atmospheric air results in the growth of the volume number density of gas nanobubbles. It appeared that the physical mechanism of nonlinear scattering changes upon increasing the nanobubble content. In fact, four‐photon parametric scattering in degassed water, which occurs due to the cubic nonlinear susceptibility χ(3) of medium, transforms upon increasing the content of nanobubbles into stimulated hyper‐Raman scattering controlled by the constant (∂χ(2)/∂q), where χ(2) is the quadratic nonlinear susceptibility and q is coordinate of a nucleus. The physical nature of the observed phenomenon is the generation of a surface plasmon arising due to the optical breakdown inside a gas nanobubble, which leads to an effective buildup of vibrations of molecules/ions localized in the surface layer of nanobubble. The plasmon bandwidth Δω was theoretically estimated. It turned out that Δω ≈ ω0, where ω0 is the resonant plasmon wavenumber, that is, in our case, the quality factor of the plasmon oscillator is very low.

Instability of a meandering channel with variable width and curvature: Role of sediment suspension

In this Perspective, we explore the instability of a meandering channel with variable width and curvature. The analysis employs the depth-averaged formulations for the flow and sediment transport. Unlike the conventional instability analysis that considers mainly the bedload transport, we consider the generic sediment transport including both the bedload and suspended load. The analysis addresses the variations of the near-bank excess azimuthal velocity and the bed topography deviation with the meander wavenumber for different pertinent parameters, such as Shields number, relative roughness number, channel aspect ratio, width-variation amplitude, and shear Reynolds number. The analysis detects a resonance phenomenon for certain critical values of the pertinent parameters and explores the sensitivity of the resonant wavenumber to the pertinent parameters. In a hydraulically smooth flow regime, the sediment suspension is found to play a stabilizing role. On the contrary, in hydraulically transitional and rough flow regimes, the sediment suspension offers a destabilizing effect. The stability diagrams reveal that the stable zone enlarges as the Shields number and relative roughness number increase, while it contracts with an increase in width-variation amplitude. For a given shear Reynolds number in a hydraulically smooth flow regime, the stability diagram predicts various stable zones. By contrast, for a given shear Reynolds number in hydraulically transitional and rough flow regimes, a unique stable zone exists.

Three-Photon Ionization with One-Photon Resonance between Excited Levels

Within the framework of a three-level model, the process of three-photon ionization with one-photon resonance between two excited levels (with the lower one being initially unpopulated) is considered using the density matrix method. It is shown that such resonance can result in the appearance of a maximum in the three-photon ionization spectrum when detuning between the resonance wavenumber and the wavenumber of the transition responsible for the lower excited level being populated exceeds the laser radiation linewidth by more than three orders of magnitude.

Understanding longitudinal optical oscillator strengths and mode order

A classical way of describing dielectric functions employs sums of contributions from damped harmonic oscillators. We derive the corresponding dispersion relations for the dielectric function and the inverse dielectric function to investigate and establish the relationship between the transversal and longitudinal optical oscillator strengths, which can differ, according to experimental results, by up to three orders of magnitude. We demonstrate that transversal and longitudinal oscillator strengths should be identical for a single oscillator and that the experimental differences are in this case due to the introduction of a dielectric background in the dispersion formula. For this effect we derive an exact correction. Based on this correction we further derive a modified Kramers-Kronig sum rule for the isotropic case as well as for the components of the inverse dielectric function tensor. For systems with more than one oscillator, our model provides oscillator strengths and LO resonance wavenumbers for uncoupled LO modes.

Modulation-resonance mechanism for surface waves in a two-layer fluid system

We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly varying, long-wavelength baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant $k_{res}$, can be generated locally at the leading edge of the underlying, slowly varying, long-wavelength baroclinic flow. Here, the resonant $k_{res}$ satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation, ray-based, theory for the spatiotemporal asymmetric behaviour of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber $k_{res}$ in Fourier space.

Point-dipole approximation for small systems of strongly coupled radiating nanorods

Systems of closely-spaced resonators can be strongly coupled by interactions mediated by scattered electromagnetic fields. In large systems the resulting response has been shown to be more sensitive to these collective interactions than to the detailed structure of individual resonators. Attempts to describe such systems have resulted in point-dipole approximations to resonators that are computationally efficient for large resonator ensembles. Here we provide a detailed study for the validity of point dipole approximations in small systems of strongly coupled plasmonic nanorods, including the cases of both super-radiant and subradiant excitations, where the characteristics of the excitation depends on the spatial separation between the nanorods. We show that over an appreciable range of rod lengths centered on 210 nm, when the relative separation kl in terms of the resonance wave number of light k satisfies klgtrsim pi /2, the point electric dipole model becomes accurate. However, when the resonators are closer, the finite-size and geometry of the resonators modifies the excitation modes, in particular the cooperative mode line shifts of the point dipole approximation begin to rapidly diverge at small separations. We also construct simplified effective models by describing a pair of nanorods as a single effective metamolecule.

On resonant interactions of gravity‐capillary waves without energy exchange

AbstractWe consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.

Single-pass high-gain tapered free-electron laser with transverse diffraction in the postsaturation regime

It has been well known that the resonant interaction of an ultrarelativistic electron beam and the radiation field in the single-pass high-gain free electron laser (FEL) amplifier leads to the optical gain guiding. The transverse Laplacian term of the slowly varying wave equation in the linear regime can be approximated as a constant detuning parameter, i.e., |∇⊥2|∼kR/zR where kR is the resonant wave number and zR is the Rayleigh range of the laser. In the post-saturation regime, the radiation power begins to oscillate about an equilibrium for the untapered case while continues to grow by undulator tapering. Moreover, in this regime the gain guiding decreases and the simple constant detune is no longer valid. In this paper we study the single-pass high-gain FEL performance in the post-saturation regime with inclusion of diffraction effect and undulator tapering. Our analysis relies upon two constants of motion, one from the energy conservation and the other from the adiabatic invariant of the action variable. By constructing a two-dimensional axisymmetric wave equation and the coupled one-dimensional electron dynamical equations, the performance of a tapered FEL in the postsaturation regime can be analyzed, including the fundamental mode profile, the power efficiency and the scaled energy spread. We begin the analytical investigation with two different axisymmetric electron beam profiles, the uniform and bounded parabolic ones. It is found that the tapered FEL power efficiency can be smaller but close to the taper ratio provided the resonant phase remains constant and the beam-wave is properly matched. Such a tapered efficiency is nearly independent of transverse electron beam size before significant electron detrapping occurs. This is essentially different from the untapered case, where the power extraction efficiency is around the essential FEL gain bandwidth (or ρ, the Pierce or FEL parameter) and depends on the beam size. It is also found that the power enhancement due to undulator tapering is attributed more by the field increase outside the transverse electron beam than that inside the transverse electron beam. Several scaling properties on the taper ratio and the transverse electron beam size are also discussed in this paper.3 MoreReceived 29 January 2018DOI:https://doi.org/10.1103/PhysRevAccelBeams.21.060702Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsSynchrotron radiation & free-electron lasersAccelerators & Beams

Hydrodynamic instability of meandering channels

In this paper, we explore the hydrodynamic instability of meandering channels driven by the turbulent flow. The governing equations of channel dynamics with suitable boundary conditions are closed with the fluid and granular constitutive relationships. A regular expansion of the fundamental variables is employed to linearize the parent equations by superimposing the perturbations on the basic unperturbed flow. The channel dynamics reveal a resonance phenomenon which occurs when the key variables fall in the vicinity of the distinct critical values. The resonance phenomenon preserves its distinctive signature in different flow regimes which are guided by the characteristic values of the shear Reynolds number. The hydrodynamic analysis indicates that the fluid friction and the volumetric sediment flux play a decisive role to characterize the channel instability in different flow regimes. The growths of azimuthal velocity perturbation in phase with curvature, bed topography perturbation, bend amplification rate, and meander propagation speed in different flow regimes are investigated by varying the meander wavenumber, Shields number, channel aspect ratio, and relative roughness number. The analysis is capable to capture the effects of grain size on azimuthal velocity perturbation, bed topography perturbation, bend amplification rate, and meandering propagation speed over a wide range of shear Reynolds numbers. The variations of resonant wavenumbers in different flow regimes with the Shields number, channel aspect ratio, and relative roughness number are addressed. For a specific flow regime, the upstream and downstream migrations of meandering channels are typically governed by the Shields number, channel aspect ratio, and relative roughness number.

Anti-parallel dimer and tetramer formation of propylene carbonate

Raman scattering and infrared (IR) absorption spectra of enantiopure (R)-propylene carbonate ((R)PC) and racemic propylene carbonate (PC) were recorded at room temperature, 25 °C, in benzene (Bz) solution and in the pure liquid state to investigate the presence of dimers and other higher order intermolecular associations. (R)PC and PC both demonstrated a strong C=O stretching vibrational band. The band exhibited changes in its shape and resonance wavenumber highly dependent on the concentrations of PCs, whereas a difference between the chirality of (R)PC and PC had little influence. In an extremely dilute condition, doubly split bands were observed at 1807 and 1820 cm-1 in both Raman and IR spectra, which are assigned to the characteristic bands of isolated monomeric PCs. An additional band appeared at 1795 cm-1 in a dilute to concentrated regime, and its magnitude strengthened with increasing concentrations accompanied with slight increasing in the magnitude of 1807 cm-1 band in Raman spectra, while an increase in the magnitude of 1807 cm-1 band was clearly greater than that of 1795 cm-1 band in IR spectra. The spectrum changes at 1795 and 1807 cm-1 were attributed to characteristics of anti-parallel dimer formation of PCs caused by strong dipole-dipole interactions between C=O groups. Moreover, another additional signal was clearly observed at 1780-1790 cm-1 in a concentrated regime, and became the primary signal in the pure liquid state with slight increasing in the intensity of 1795 cm-1 band in Raman spectra. On the other hand, in IR spectra the observed increasing of 1780-1790 cm-1 band was much less than that of 1795 cm-1 band. These newly found spectrum changes in the concentrated regime are attributed to the formation of anti-parallel tetramers of PCs based on the characteristics of band selection rule found in Raman and IR spectra. Equilibrium constants for the anti-parallel dimer (KD) and tetramer formation (KT) of PCs in Bz solution and in the pure liquid state were also determined using the Raman and IR data assuming chemical processes: 4PC ↔ 2(PC)2 ↔ (PC)4.

Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal

Shear to longitudinal mode conversion via second harmonic generation is studied theoretically and computationally for plane waves in a two-dimensional, adhesive, hexagonally close-packed microscale granular medium. The model includes translational and rotational degrees of freedom, as well as normal and shear contact interactions. We consider fundamental frequency plane waves in all three linear modes, which have infinite spatial extent and travel in one of the high-symmetry crystal directions. The generated second harmonic waves are longitudinal for all cases. For the lower transverse–rotational mode, an analytical expression for the second harmonic amplitude, which is derived using a successive approximations approach, reveals the presence of particular resonant and antiresonant wave numbers, the latter of which is prohibited if rotations are not included in the model. By simulating a lattice with adhesive contact force laws, we study the effectiveness of the theoretical analysis for non-resonant, resonant, and antiresonant cases. This work is suitable for the analysis of microscale and statically compressed macroscale granular media, and should inspire future studies on nonlinear two- and three-dimensional granular systems in which interparticle shear coupling and particle rotations play a significant role.

Justification of the 2D NLS equation for a fourth order nonlinear wave equation – quadratic resonances do not matter much in case of analytic initial conditions

The 2D Nonlinear Schrödinger (NLS) equation can be derived by multiple scaling analysis in order to describe slow temporal and spatial modulations of an oscillating wave packet in general dispersive wave systems. It is the purpose of this paper to establish estimates for the difference between the so-obtained approximation and true solutions of the original system in case that the original system possesses nontrivial quadratic resonances. We explain for a fourth order nonlinear wave equation that quadratic resonances do not matter much in case of analytic initial conditions for the NLS equation in the sense that independently of whether the resonance is stable or unstable an approximation property can be established on the natural time scale of the NLS equation if the set of resonant wave numbers is bounded away from the integer multiples of the basic wave number of the underlying wave packet. Finally, we explain that in case of an approximation by a so-called four wave interaction (FWI) system the same approach works and an even stronger result can be established.

Radio relics tracing the projected mass distribution in CIZA J2242.8+5301

Abstract We present a weak-lensing analysis for a merging galaxy cluster, CIZA J2242.8+5301, which hosts double radio relics, using three-band Subaru/Suprime-Cam imaging (Br′z′). Since the lifetime of dark matter halos colliding into clusters is longer than that of X-ray emitting gas halos, weak-lensing analysis is a powerful method to constrain merger dynamics. Two-dimensional shear fitting using a clean background catalog suggests that the cluster undergoes a merger with a mass ratio of about 2 : 1. The main halo is located around the gas core in the southern region, while no concentrated gas core is associated with the northern sub-halo. We find that the projected cluster mass distribution resulting from an unequal-mass merger is in excellent agreement with the curved shapes of the two radio relics and the overall X-ray morphology, except for the lack of the northern gas core. The lack of a prominent radio halo enables us to constrain an upper limit of the fractional energy of magnetohydrodynamic turbulence of $(\delta B/B)^2<\mathcal {O}(10^{-6})$ at a resonant wavenumber, by finding a balance between the acceleration time and the time after the core passage or the cooling time, with an assumption of resonant acceleration by a second-order Fermi process.