Normal Mode Solutions Research Articles

Normal Mode Solutions of Target Strengths of Solid-filled Spherical Shells and Discussion of Influence Parameters

The normal mode solution for the form function and target strength (TS) of a solid-filled spherical shell is derived. The calculation results of the spherical shell’s acoustic TS are in good agreement with the results of the finite element method (FEM). Based on these normal mode solutions, the influences of parameters such as the material, radius, and thickness of the inner and outer shells on the TS of a solid-filled spherical shell are analyzed. An underwater spherical shell scatterer is designed, which uses room temperature vulcanized (RTV) silicone rubber as a solid filling material and does not contain a suspension structure inside. The scatterer has a good TS enhancement effect.

Boson star normal modes

Boson stars are gravitationally bound objects that arise in ultralight dark matter models and form in the centers of galactic halos or axion miniclusters. We systematically study the excitations of a boson star, taking into account the mixing between positive and negative frequencies introduced by gravity. We show that the spectrum contains zero-energy modes in the monopole and dipole sectors resulting from spontaneous symmetry breaking by the boson star background. We analyze the general properties of the eigenmodes and derive their orthogonality and completeness conditions which have non-standard form due to the positive-negative frequency mixing. The eigenvalue problem is solved numerically for the first few energy levels in different multipole sectors and the results are compared to the solutions of the Schrödinger equation in fixed boson star gravitational potential. The two solutions differ significantly for the lowest modes, but get close for higher levels. We further confirm the normal mode spectrum in 3D wave simulations where we inject perturbations with different multipoles. As an application of the normal mode solutions, we compute the matrix element entering the evaporation rate of a boson star immersed in a hot axion gas. The computation combines the use of exact wavefunctions for the low-lying bound states and of the Schrödinger approximation for the high-energy excitations.

Mechanism of sound emission produced by enclosed cavity upon a sphere entering the water

A mathematical model for the acoustic pressure field induced by an enclosed cavity upon a solid sphere impacting into water is established. The relation between sound emission, cavity ripples, and the falling sphere is elaborated analytically. To describe the water domain above the sphere, physical and co-moving cylindrical coordinate systems are introduced. The water domain is analyzed by the perturbation method and potential flow theory. The air inside the cavity is assumed to be uniform and adiabatic. In the physical coordinates, the zeroth-order cavity surface is governed by the Rayleigh–Plesset Equation and is cylinder-like. In the co-moving coordinates, the sphere is stationary and subjected to an upward free stream. The slender cavity geometry permits linear surface waves with a vertical normal-mode solution, which corresponds to the first-order cavity ripples. The formula of ripples suggests that the cavity nose is a falling, pulsating, flat, cylindrical acoustic source in water. The induced acoustic wave is formulated in another spherical coordinate system. The sound frequency equals the vibration frequency of the cavity nose and also equals the formation/propagation frequency of the ripples. Moreover, the natural frequency of the cavity nose and the natural normal mode of the ripples are formulated. A water entry experiment is implemented and validates the theory. The normal mode of the cavity ripples, the criteria of the sound emission, and the initiation of the ripples and sound emission are discussed.

Study of primary and secondary instabilities arising due to a chemical reaction in a two-component Rayleigh–Bénard system

Manifestation of stationary and oscillatory convection and secondary instabilities due to a chemical reaction in a two-component convective fluid system is reported in the paper by considering idealistic as well as physically realistic boundaries. Using a normal mode solution, analytical expression of the critical Rayleigh number for a stationary and oscillatory disturbances, and the natural frequency are reported. The range of parameters is identified where oscillatory motion happens. Further, the parameters’ range for existence of oscillatory regime is found to be larger for rigid boundaries compared to that of free boundaries. Furthermore, for both the boundaries, parameters’ range for this regime increases when the chemical reaction rate increases, leading to the conclusion that the oscillatory motion emerges as the most preferred mode in the two-component system due to the presence of a chemical reaction and the size of this domain is directly proportional to the chemical reaction rate. The marginal stability plots depict that the oscillatory and stationary regimes respectively correspond to Hopf and direct pitchfork bifurcations. The critical Rayleigh number and the wave number where codimension two bifurcation exists are documented in the paper for fixed values of parameters. It is shown that the codimension two bifurcation that arose in the problem is not a Takens–Bogdanov bifurcation. In a stationary regime, the domain for secondary instabilities of Eckhaus and zigzag is obtained using the spatio-temporal Newell–Whitehead–Segel equation. These instabilities grow with increasing chemical reaction rate. In the oscillatory regime, the complex Ginzburg–Landau equation is used to predict the appearance of the Benjamin–Feir instability.

Quasinormal Mode Theory of Chiral Power Flow from Linearly Polarized Dipole Emitters Coupled to Index-Modulated Microring Resonators Close to an Exceptional Point

Chiral emission can be achieved from a circularly polarized dipole emitter in a nanophotonic structure that possess special polarization properties such as a polarization singularity, namely with right or left circularly polarization (C-points). Recently, Chen et al. [Nature Physics 16, 571 (2020)] demonstrated the surprising result of chiral radiation from a linearly-polarized (LP) dipole emitter, and argued that this effect is caused by a decoupling with the underlying eigenmodes of a non-Hermitian system, working at an exceptional point (EP). Here we present a quasinormal mode (QNM) approach to model a similar index-modulated ring resonator working near an EP and show the same unusual chiral power flow properties from LP emitters, in direct agreement with the experimental results. We explain these results quantitatively without invoking the interpretation of a missing dimension (the Jordan vector) and a decoupling from the cavity eigenmodes, since the correct eigenmodes are the QNMs which explain the chiral emission using only two cavity modes. By coupling a LP emitter with the dominant two QNMs of the ring resonator, we show how the chiral emission depend on the position and orientation of the emitter, which is also verified by the excellent agreement with respect to the power flow between the QNM theory and full numerical dipole solutions. We also show how a normal mode solution will fail to capture the correct chirality since it does not take into account the essential QNM phase. Moreover, we demonstrate how one can achieve frequency-dependent chiral emission, and replace lossy materials with gain materials in the index modulation to reverse the chirality.

Electric and viscous correction for viscous potential flow analysis of electrohydrodynamic instability of an electrified leaky-dielectric jet

The electric and viscous correction of viscous potential flow (EVCVPF) is developed for analyzing the electrohydrodynamic instability of an electrified leaky-dielectric viscous jet. The EVCVPF model is based on the viscous potential flow (VPF) and the viscous correction of VPF (VCVPF), proposed by Joseph and Wang [“The dissipation approximation and viscous potential flow,” J. Fluid Mech. 505, 365–377 (2004)]. The purpose is to resolve the discrepancy between the non-zero irrotational viscous and the electric tangent stresses. The power of the pressure correction is introduced to compensate the neglected viscous dissipation in the flow bulk in VPF, which is equal to the average power of the irrotational viscous and the electric tangent stresses. The model has been validated by comparing it to the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF). The energy budget is also performed to assist in understanding underlying mechanisms. Results show that EVCVPF is accurate for charged jets with low and moderate viscosities, i.e., the Ohnesorge number approximately Oh ≤ 0.1. The inaccuracy for highly viscous jets are the limitations of VPF itself. The electric field has less influence compared to the fluid viscosity. To achieve more accurate approximations, VCVPF and VPF are chosen for axisymmetric and non-axisymmetric modes under weak electric fields. EVCVPF is in remarkably good agreement with FVF under moderate and strong electric fields. In general, as VCVPF extends the applicability in fluid viscosity of VPF, EVCVPF further improves the adequacy when studying the electrohydrodynamic instability.

Linear and non-linear magneto-convection in couple stress fluid with non-classical heat conduction law

A theoretical examination of the thermal convection for a couple stress fluid which is electrically conductive and possessing significant thermal relaxation time with an externally applied magnetic field is carried out. Fourier’s law fails when fluids are subjected to rapid heating or when it is confined and in the case of nano-devices. A frame invariant constitutive equation for heat flux is considered. The linear analysis is carried out implementing a normal mode solution and the non-linear stability of the system is analyzed using a double Fourier series. The analysis of the transfer of heat is determined in terms of the Nusselt number.

Developing efficient finite element techniques for atmospheric sound propagation

Applying the finite element method to problems in atmospheric sound propagation is challenging because the use of a standard approach, such as meshing the entire domain, quickly delivers a prohibitive problem size. The finite element method does, however, offer many advantages such as the ability to accommodate continuous variations in fluid properties, as well as scattering from obstacles of complex geometry. This means the method is ideally placed to provide benchmark solutions for more popular approaches. Accordingly, ideas for developing more efficient versions of the finite element method suitable for atmospheric sound propagation are explored here for two dimensional problems. These are based on the use of one dimensional normal mode expansions, and then mapping these on to two dimensional solutions for non-uniform obstacles. The normal mode solution for a range independent inhomogenous moving fluid is presented, and numerical mode matching techniques are introduced to demonstrate the inclusion of a point source. The extension of these mode matching methods to accommodate scattering from obstacles is then discussed, and the conditions necessary to deliver efficient finite element solutions are reviewed.

Forced guided waves in linearly elastic plates (II) – A modified normal-mode expansion method

The classical normal-mode expansion method developed by Auld and Kino in 1973 has been used widely by numerous researchers and practitioners to study forced guided waves in plates and layered media, including both elastic and electromagnetic waves. However, it was shown in Part I of this study that, although this method gives the correct solution when applied to horizontally polarized shear waves and electromagnetic waves, it does not yield the correct elastodynamic solution when applied to Lamb waves. To address this shortcoming, we develop in this paper a modified normal-mode expansion method that yields the correct elastodynamic solution in that the solution satisfies all the elastodynamic governing equations and the boundary conditions for forced Lamb waves in a plate. The efficacy of the modified normal-mode expansion method is further confirmed by comparing its solution with the finite element solution. Further, it is showed that when applied to horizontally polarized shear waves, the modified normal-mode expansion method yields numerically the same result as that of the classical normal-mode expansion method. However, the modified normal-mode expansion solution converges much faster.

Updating Geometrically Nonlinear Reduced-Order Models Using Nonlinear Modes and Harmonic Balance

Nonlinear reduced-order models are being investigated for use as digital twins for advanced aircraft and spacecraft. In concept, the digital twin would be tuned to accurately describe the behavior of the vehicle, and then it would be updated as the vehicle is modified or ages. Unfortunately, the linear model correlation and validation techniques that are commonly used in the aerospace industry are not valid for nonlinear models, so a new set of tools is needed. This work presents an algorithm to update geometrically nonlinear reduced-order models using nonlinear normal modes as a correlation metric. The nonlinear normal modes serve as a strong metric to correlate the numerical models because they can be extracted from experiments, and they describe the dynamics of the nonlinear system over a range of amplitudes and are independent of the loading applied to the system. This paper presents a novel method of computing analytical gradients of nonlinear normal mode solutions with respect to system parameters using the multiharmonic balance method. The procedure is first tested using numerical simulations and then applied to tune the reduced-order model of a curved beam based on nominal blueprints and idealized boundary conditions to match a nonlinear normal mode that has been measured experimentally from a three-dimensional printed test specimen.

Formulation of a Triaxial Three‐Layered Earth Rotation: Theory and Rotational Normal Mode Solutions

AbstractIn this study, we formulated a triaxial three‐layered anelastic Earth rotation theory considering various core‐mantle couplings, including the pressure and gravitational couplings acting on the elastic inner core by the outer core and mantle, and the viscoelectromagnetic couplings between the outer core and mantle and between the outer and inner cores. With this formulation, using the eigenvalue method, we solved four rotational normal modes, including the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN), and the Inner Core Wobble (ICW). The triaxiality of the actual Earth has notable effects on rotational normal modes, that is, 0.005, 0.0, 0.071, and 1.174 d (mean solar days), respectively, for the CW, FCN, FICN, and ICW. In the case that the triaxiality is bigger, for instance, if the Earth's equatorial dynamical ellipticity were , which is 1 order of magnitude larger than the presently determined ellipticity ( ), the CW and ICW periods will be lengthened by 3.579 and 94.912 d, and the FICN period will be decreased by 0.591 d. Based on our present knowledge, the numerical solutions of the four normal modes under the new rotation frame suggest that the periods of CW, FCN, FICN, and ICW are, respectively, 432.2, 429.9, 934.0, and 2,718.7 d. Our theory for Earth can be extended to other planets in our solar system.

Nonlinear normal modes in the [formula omitted]-Fermi-Pasta–Ulam-Tsingou chain

Nonlinear normal mode solutions of the β-FPUT chain with fixed boundaries are presented in terms of the Jacobi sn function. Exact solutions for the two particle chain are found for arbitrary linear and nonlinear coupling strengths. Solutions for the N-body chain are found for the case of purely nonlinear couplings. Three distinct solution types are presented: a linear analogue, a chaotic amplitude mapping, and a localized nonlinear mode. The relaxation of perturbed modes are also explored using l1-regularized least squares regression to estimate the free energy functional near the nonlinear normal mode solution. The perturbed modes are observed to decay sigmoidally towards a quasi-equilibrium state and a logarithmic relationship between the perturbation strength and mode lifetime is found.

Linear Modes for Channels of Constant Cross-Section and Approximate Dirichlet–Neumann Operators

We study normal modes for the linear water wave problem in infinite straight channels of bounded constant cross-section. Our goal is to compare semi-analytic normal mode solutions known in the literature for special triangular cross-sections, namely isosceles triangles of equal angle of \(45^{\circ }\) and \(60^{\circ }\), see Lamb (Hydrodynamics. Cambridge University Press, Cambridge, 1932) , Macdonald (Proc Lond Math Soc 1:101–113, 1893), Greenhill (Am J Math 97–112, 1887), Packham (Q J Mech Appl Math 33:179–187, 1980), and Groves (Q J Mech Appl Math 47:367–404, 1994), to numerical solutions obtained using approximations of the non-local Dirichlet–Neumann operator for linear waves, specifically an ad-hoc approximation proposed in Vargas-Magana and Panayotaros (Wave Motion 65:156–174, 2016), and a first-order truncation of the systematic depth expansion by Craig et al. (Proc R Soc Lond A: Math, Phys Eng Sci 46:839–873, 2005). We consider cases of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with sinusoidal dependence in the longitudinal direction. The triangular geometries considered have slopping beach boundaries that should in principle limit the applicability of the approximate Dirichlet–Neumann operators. We nevertheless see that the approximate operators give remarkably close results for transverse even modes, while for odd transverse modes we have some discrepancies near the boundary. In the case of longitudinal modes, where the theory only yields even modes, the different approximate operators show more discrepancies for the first two longitudinal modes and better agreement for higher modes. The ad-hoc approximation is generally closer to exact modes away from the boundary.

Normal mode analysis of 3D incompressible viscous fluid flow models

In this paper, we study the normal mode solutions of 3D incompressible viscous fluid flow models. The obtained theoretical results are then applied to analyze several time-stepping schemes for the numerical solutions of the 3D incompressible fluid flow models.

Numerical solutions of rotational normal modes of a triaxial two-layered anelastic Earth

The Earth's rotational normal modes depend on Earth model used, including the layer structures, principal inertia moments of different layers and the compliances. This study focuses on providing numerical solution of the rotational normal modes of a triaxial two-layered anelastic Earth model without external forces but with considering the complex forms of compliances and the electromagnetic coupling between the core and mantle. Based on the present knowledge of the Chandler wobble (CW) and Free Core Nutation (FCN), we provide a set of complete compliances which could be used for reference in further investigations. There are eight rotational normal mode solutions, four of which might exist in nature. However, in reality only two of these four solutions correspond to the present motion status of the prograde CW and the retrograde FCN. On one hand, our numerical calculations show that the periods and quality factors (Qs) of CW and FCN are respectively 434.90 and 429.86 mean solar days (d) and 76.56 and 23988.47 under frequency-dependent assumption, and the triaxiality prolongs CW about 0.01 d and has hardly effect on FCN. On the other hand, we analyze the sensibility of compliances and electromagnetic coupling parameter on the periods and Qs of CW and FCN and find the sensitive parameters with respect to them.

Temporal Coherence for Sound Transmission Through Fluctuating Deep Water with Adiabatic Perturbed Modes

Adiabatic approximation (AP) combined with perturbation theory gives a fast normal-mode solution of temporal coherence for sound field in a two-dimensional deep water with time-varying random internal waves. Internal waves induced mode changes are deduced using the first-order perturbation theory [C. T. Tindle, L. M. O’Driscoll and C. J. Higham, Coupled mode perturbation theory of range dependence, J. Acoust. Soc. Am. 108(1) (2000) 76–83]. And mode perturbations in amplitude are neglected by the adiabatic method with wavenumber perturbations in phase merely considered. The AP expression of temporal coherence function is theoretically identical to the adiabatic transport equation theory [J. A. Colosi, T. K. Chandrayadula, A. G. Voronovich and V. E. Ostashev, Coupled mode transport theory for sound transmission through an ocean with random sound speed perturbations: Coherence in deep water environments, J. Acoust. Soc. Am. 134(4) (2013) 3119–3133]. Numerical results of the adiabatic temporal coherence function for several low frequencies and ranges up to 1000[Formula: see text]km are calculated. Then the coherence time scales obtained from the calculations are examined by a one-way coupled theory considering forward scattering [A. G. Voronovich, V. E. Ostashev and J. A. Colosi, Temporal coherence of acoustic signals in a fluctuating ocean, J. Acoust. Soc. Am. 129(6) (2011) 3590–3597]. Comparisons demonstrate that the range and frequency dependence of coherence time for both methods are quite close. And this shows good agreement with the well-known inverse frequency and inverse square root range laws. In addition, the internal wave energy dependence of coherence time is also studied.

Passive manipulation of low resonance scattering frequency associated with subsonic Rayleigh waves on submerged polymer spheres

Observation of the backscattering form function for a solid polymethylmeth-acrylate(PMMA) sphere reveals a significant enhancement at low frequencies, which is associated with the subsonic Rayleigh waves resonance. The presence of these resonances suggests that solid PMMA spheres may be used as passive sonar targets for certain applications.The slightly deformed PMMA spheres with regular corrugation are constructed for studying and manipulating the low-frequency resonances of subsonic Rayleigh waves. For the regular-corrugated PMMA spheres, the first-order approximate of Rayleigh normal mode solution is obtained with the perturbation method and the backscattering form function is analyzed in each order. It can be found that coupling among multiple resonance modes is introduced by the boundary deformation and leads to the shift of resonance frequency. An approximate formula for the frequency shift with respect to the parameters of corrugated boundary is deduced. Based on this formula, it is possible to evaluate resonance frequency of subsonic Rayleigh waves quickly and then manipulate acoustic scattering filed of the slightly deformed PMMA spheres in water passively.

An improved Gaussian beam caustic correction for Bellhop at low frequencies

Gaussian beams are commonly used in ray-tracing to mitigate the effects of caustics and shadow zones. The problem is how to determine the width of these “fuzzy” beams. Porter and Bucker [J. Acoust. Soc. Am. 82, 1349–1359, 1987] proposed a method that expressed the beamwidth and curvature in terms of p and q of the dynamic ray equations. We call q the beamwidth factor. In the Gaussian beam implementation caustics are not caused by the crossing of two rays; rather they occur when the beamwidth factor, which appears in the denominator of the amplitude, becomes small. In practice, this is generally not a problem at high frequencies, but as the frequency gets lower the problem gets more severe. The widely used Bellhop model has a procedure, which “caps” the beamwidth when q becomes too small, but the procedure eventually breaks down at low frequencies. Here, we propose a different cap based on a cylindrical wave front converging to the focal point of a caustic. The various caps are compared with the “exact” normal mode solution for a shallow-water upward refracting environment, illustrating how the new cap provides better reduction of the caustic anomalies.

Normal mode coupling through horizontally variable sound speed

In conjunction with construction of a normal mode model specifically designed to compute acoustic propagation in shallow water environments, the coupling associated with horizontally variable sound speed is predicted using a closed form solution in a modified Pekeris wave guide with a hard bottom. The code enables analysis of the correspondence between mode summation and the corresponding wave propagation as represented by ray propagation. The closed form solution is being used to provide insights into horizontal mode coupling and as a benchmark for verification of coupling in the shallow water normal mode solution.In conjunction with construction of a normal mode model specifically designed to compute acoustic propagation in shallow water environments, the coupling associated with horizontally variable sound speed is predicted using a closed form solution in a modified Pekeris wave guide with a hard bottom. The code enables analysis of the correspondence between mode summation and the corresponding wave propagation as represented by ray propagation. The closed form solution is being used to provide insights into horizontal mode coupling and as a benchmark for verification of coupling in the shallow water normal mode solution.

Acoustic characterization of a Ukelin

An acoustical characterization was performed on a ukelin, a wooden hybrid stringed instrument designed to combine features of the ukulele and the violin. This instrument consists of two resonant cavities at opposite ends of the instrument, one supporting sixteen strings that are individually bowed, and the other supporting 16 strings arranged in groups of four-note diatonic chords that are simultaneously plucked with the free hand, allowing an individual player to produce a melody/accompaniment combination. The ukelin design was patented in the early 20th century and sold door-to door as a novelty instrument. Modal Analysis results of a sample instrument are presented and compared with an FEA normal modes solution using ANSYS. Preliminary results show good agreement of the mode shapes for several modes between the modal results and the ANSYS predictions, but a significant difference in mode frequencies for a given mode shape. The differences in frequency are attributed primarily to unknown material properties of the wood of the sample instrument. Implications of the use of a finite-element model and of the Dynamic Modification feature of modal analysis are discussed.