Normal Mode Expansion Research Articles

Application of Normal Mode Expansion to Accoustic Emission Waves in Finite Plates

A normal mode solution to the classical plate bending equatiion was investigted of its applicability to AE. The same source time function chosen by Weavr and Pao is considered. However, arbitrary source and receiver position are chosen relative to the boundaries of the plate

Mode control of ultrasonic guided waves in thick cylinders for crack detection.

Experiments on ultrasonic propagation in thick annular structures show that one can simplify the modal structure of the received signals by carefully controlling the wedge angles of the generation and detection transducers. This is important if one is to detect cracks along the path between source and receiver. In particular, one should excite and detect only those modes whose energy is concentrated where the cracks are expected to form. To do so, the following methodology is used: First, experimental signals are obtained for various source–receiver configurations with a standard pulser/receiver; a wavelet transform is then applied to the signals to localize precisely in time the arrival of a given mode packet at a given frequency and to determine the corresponding group velocity. These modes are then identified by comparing the results with predictions from theoretical dispersion curves. One can then predict the energy distribution within the structure by using the method of normal mode expansion for transients. [Work supported by the ONR, Code 332.]

A boundary element solution for a mode conversion study on the edge reflection of Lamb waves

The boundary element method, well known for bulk wave scattering, is extended to study the mode conversion phenomena of Lamb waves from a free edge. The elastodynamic interior boundary value problem is formulated as a hybrid boundary integral equation in conjunction with the normal mode expansion technique based on the Lamb wave dispersion equation. The present approach has the potential of easily handling the geometrical complexity of general guided wave scattering with improved computational efficiency due to the advantage of the boundary-type integral method. To check the accuracy of the boundary element program, vertical shear wave diffraction, due to a circular hole, is solved and compared with previous analytical solutions. Edge reflection factors for the multibackscattered modes in a steel plate are satisfied quite well with the principle of energy conservation. In the cases of A0, A1, and S0 incidence, the variations of the multireflection factors show similar tendencies to the existing results for glass. It is observed that the reflection of an incident wave becomes close to zero over a certain frequency range seen through energy interaction with other reflected modes, and increases again beyond this minimum point due to reverse mode conversion. The reflections of the higher symmetric incident modes, S1 and S2, are also investigated. It turns out that S1 mode is an unusual mode which is nearly unaffected by the mode conversion in the Lamb wave edge reflection.

Excitations of thermoelastic waves in plates by a pulsed laser

The method of the eigenfunction expansion, also known as the expansion in normal modes, is employed to study numerically the axisymmetric excitation of the thermoelastic waves in plates by a pulsed laser. This method gives a systematic treatment and allows one to investigate not only the quasistatic and dynamic thermoelastic responses of pulsed photothermal deformation on the time scale of 1 μs, but also the thermoelastic generation of longitudinal, transverse, and surface acoustic waves in thick materials, as well as the excitations of the Rayleigh-Lamb wave modes in thin plates. The formalism is particularly suitable for waveform analyses of the excitations of transient Lamb waves in thin plates because one needs only to calculate the contributions of several lower eigenmodes. The numerical technique provides a quantitative tool for the experimental determination of material properties, especially the mechanical and elastic properties of free-standing films and thicker sheet materials by thermoelastic detection.

Formulation and its energy balance verification for ultrasonic non-destructive characterization of a single fibre composite interphase

It is widely acknowledged that the structural performance of fibre-reinforced composites is critically affected by the interphase, the finite-thickness layer located between the fibres and matrix. A need to characterize the interphase thus arises. A promising ultrasonic non-destructive evaluation (NDE) technique for the characterization of the interphase is considered. For the successful design and implementation of the technique, an understanding of elastic wave propagation and scattering involved in this technique is required. As a fundamental approach, a two-dimensional plane-strain model is formulated for elastic wave scattering of time-harmonic plane incident waves from a single fibre having a homogeneous interphase embedded in an elastic matrix. Two modes of incident waves, longitudinal and in-plane shear, are considered. The formal solution to the formulated problem is found via the normal mode expansion method. A computer program is developed for the calculation of the formal solution and its validity is confirmed by two methods: limiting case analysis and energy balance analysis.

Method of images for the harmonic response of beams and rectangular plates

The feasibility of calculating the narrow band harmonic response of thin plates and beams by using the method of images or the ray tracing technique is examined. The fundamental solution of an infinite plate is employed, in conjunction with appropriately placed images, to obtain the dynamic compliance spectra of simply supported rectangular plates over a wide range of frequencies. Predictions yielded by the method of images are in excellent agreement with the normal mode expansion technique. Simply supported beams are analyzed first by considering the fundamental solution of an infinite beam, and second as a case of a narrow plate with simply supported and roller boundary conditions. Both approaches match well with the closed form solutions and with measurements conducted on two beams. Several issues including convergence characteristics and high frequency behavior are discussed.

Molecule-radiation and molecule-molecule processes in condensed media: a microscopic QED theory

A microscopic QED theory has been developed for the description of molecule-radiation and molecule-molecule processes in condensed molecular media. Adopting a polariton model, an explicit normal-mode expansion has been derived for the operator of the local displacement field. (It is the local displacement operator that describes the coupling of the individual molecules with the field.) The expansion manifestly embodies refractive influences, including, inter alia, local field contributions. To this end, the local field effects intrinsically emerge within the present formalism that systematically treats Umklapp processes; in addition, the theory accomoddates an arbitrary number of molecular levels contributing to the refractive index. The mode expansion is applicable for the investigation of a variety of molecule-radiation and molecule-molecule processes in condensed phases. In the present study, medium-induced modifications are analysed for spontaneous emission and linear absorption; application to the two-centre (bimolecular) processes are also discussed.

The instability of internal gravity waves to localised disturbances

Abstract. The instability of an internal gravity wave due to nonlinear wave-wave interaction is studied theoretically and numerically. Three different aspects of this phenomenon are examined. 1. The influence of dissipation on both the resonant and the nonresonant interactions is analysed using a normal mode expansion of the basic equations. In particular, the modifications induced in the interaction domain are calculated and as a result some modes are shown to be destabilised by dissipation. 2. The evolution of an initial unstable disturbance of finite vertical extent is described as the growth of two secondary wave packets travelling at the same group velocity. A quasi-linear correction to the basic primary wave is calculated, corresponding to a localised amplitude decrease due to the disturbance growth. 3. Numerical experiments are carried out to study the effect of a basic shear on wave instability. It appears that the growing secondary waves can have a frequency larger than that of the primary wave, provided that the shear is sufficient. The instability of waves with large amplitude and long period, such as tides or planetary waves, could therefore be invoked as a possible mechanism for the generation of gravity waves with shorter period in the middle atmosphere.

Dispersion of wind-induced inertial waves by a barotropic jet

This note attempts to reinterpret previous results on the dispersion of wind-induced inertial waves by a geostrophic barotropic jet in the ocean. The approach is to consider the jet vorticity influence on the different baroclinic modes using a vertical normal mode expansion. Numerical and analytical analysis of the linear equations shows that vorticity effects on a single baroclinic mode strongly depend on the ratio of its Rossby radius and the length scale of the geostrophic vorticity: trapping of the near-inertial energy occurs when this ratio is small. When this ratio is of order one, inertial waves are almost unaffected by the geostrophic vorticity because dispersion efficiently overcomes the jet vorticity effects. A 2-D primitive-equation model is used to examine the scattering of wind-induced inertial waves in realistic situations. Results indicate that contribution of the lowest baroclinic modes, unaffected by the jet vorticity, explain some striking features reported in previous studies as the downward phase propagation of near-inertial waves in the positive vorticity region

Guided-mode interactions in thin films with surface corrugation

The guided modes in a thin-film planar dielectric waveguide sandwiched between a cover and a substrate (two different dielectrics) are considered. The interface between the cover and the film has a smooth corrugation in the longitudinal direction. For weak corrugations, the guided-mode interactions are investigated using the expansion in terms of ideal normal modes. A corresponding treatment is given for the not-so-weak corrugations using the expansion in terms of local normal modes. The coupling coefficients are evaluated and reduced to simple forms. The theories are specialized for the treatment of contradirectional coupling between two guided modes taking place selectively in the neighborhood of the Bragg frequency. The coupled-mode equations governing the contradirectional interaction obtained from the local normal mode expansion procedure, in the limit of weak periodic corrugations, are identical to those deduced directly using the ideal normal mode expansion technique. The treatments for both the transverse electric and the transverse magnetic modes are included.

Chiral current at high temperatures.

The BP-FTW effective action is a good laboratory to study the nature of chiral symmetry at high $T$. It is invariant under a new Noether charge, $\Qbeta$. By explicit quantization, performed to order $\Tprime$ we show that the thermal vacuum is annihilated by $\Qbeta$, so that the new chiral symmetry is not broken spontaneously at high $T$. This high $T$ Noether charge is, however, different from the usual zero temperature chirality, $\Q$. Our quantization also shows that the thermal vacuum is not annihilated by $\Q$, so that the old chiral symmetry remains broken at high $T$. The thermal vacuum is thus every bit as `structured' and `complex' as the Nambu-Jona-Lasinio vacuum at zero temperature. We exhibit the relation between the normal mode expansion for $\Qbeta$ versus the usual expansion for $\Q$. In an appendix, we point out the analyticity properties of the fermion propagator in real time formalism, and verify the canonical nature of the thermal Green function.

Horizontal wave propagation in the equatorial waveguide

The dispersive (i.e. non-Kelvin) linear wave field on the equatorial β-plane, in a single vertical mode, is fully described by a single potential φ. Long Rossby waves, which are weakly dispersive, are represented in this field. This description is free from the problem of the ‘spurious solution’ encountered when working with an evolution equation for the meridional velocity; addition of this unwanted solution represents a gauge transformation that leaves the physical fields unaltered.The general solution of the ray equations is found, including trajectories, and the amplitudes and phase fields. This solution is asymptotically valid for either high or low frequencies. The ray paths are identical in both limits, but the phase field is not, reflecting the isotropy of Poincaré waves, in one case, and the zonal anisotropy of Rossby waves, in the other.Two examples are studied by ray theory: meridional normal modes and wave radiation from a point source in the equator. In the first case, the exact dispersion relation is obtained. In the second one, northern and southern caustics bend towards the equator, meeting there at focal points. The full solution is the superposition of many leaves and has a structure that would be hard to find in a normal modes expansion.

Excitation of Guided Waves in Generally Anisotropic Layers Using Finite Sources

The excitation of guided wave modes in generally anisotropic layers by finite sized strip sources placed on the surfaces of the layer is examined. The general problem of arbitrarily applied harmonic surface tractions is first solved using the normal mode expansion technique in conjunction with the complex reciprocity relation of elastodynamics. This general solution is then specialized to loading situations modelling those commonly used to excite guided waves in layers for use in nondestructive evaluation. The amplitudes of the generated modes are written as the product of an “excitation function” which depends only on the distribution of the applied tractions and an “excitability function” which depends only on the properties of the specific mode(s) being excited and which determines how receptive the modes are to the applied tractions. Expressions are obtained for the −9 dB wave number and phase velocity bandwidths (σβ and σν respectively) which determine the widths of the wavenumber or phase velocity excitation spectra at the −9 dB generation point. Finally, the problem of transient loading is addressed by superimposing time harmonic solutions via an integration over the dispersion curves of the layer.

Exact Finite Elements for Wave Propagation in Range-Independent Fluid-Solid Media

A finite element technique (FEM) is proposed for solving the two-point ordinary differential equation (ODE) boundary value problem for the wavefield in a layered horizontally homogeneous fluid-solid medium at fixed horizontal wavenumber k. The medium may consist of an arbitrary mixture of fluid and solid layers, bounded at both ends by homogeneous half-spaces. In homogeneous layers the FEM basis functions are chosen to be exact local solutions of the governing source-free ODE's, and in a medium composed of homogeneous layers only the overall solution, at fixed horizontal wavenumber k , is exact without subdivision of the layers. In layers with non-constant material parameters, standard second order accurate linear elements are used. The FEM stiffness matrix is symmetric with dimension and bandwidth essentially half of those obtained with a parallel shooting global matrix technique. The stiffness matrix remains bounded and well conditioned as the lengths of exact elements grow, and it can, in general, be decomposed into triangular factors without scaling or pivoting, which contributes significantly to computational efficiency. The use of the finite element technique for wavefield synthesis by Hankel transform integration or by normal mode expansion is discussed. In both cases, general purpose numerical quadrature and zero-finding techniques can be used if proper care is taken of the poles of the stiffness matrix in the complex k plane. A pole-free dispersion function, efficiently computable from the FEM stiffness matrix, is derived. A simple but useful general result in the theory of non-linear eigenvalue problems for matrices is pointed out and used for computing the excitation coefficients of the modes of the discretized system. The usefulness of the method is illustrated by presenting numerical results from a computationally intensive application example.

Numerical Simulation of Baroclinic Jovian Vortices

This thesis consists of two papers on the dynamics of Jovian planet atmospheres. The first paper discusses the uses of a normal-mode expansion in the vertical for modeling the dynamics of Jupiter's atmosphere. The second paper uses a non-linear numerical model based on the normal-mode expansion of the first paper to study the dynamics of baroclinic vortices. The abstracts for the two papers are reproduced below. Paper 1: We propose a non-linear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the non-linear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the barotropic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the normal-mode model for Jupiter using two simple problems: forced, vertically propagating Rossby waves, using two and three baroclinic modes, and baroclinic instability, using two baroclinic modes. We find that the normal-mode model provides qualitatively correct results, even with only a very limited number of vertical degrees of freedom. Paper 2: We examine the evolution of baroclinic vortices in a time dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and first two baroclinic modes (Achterberg and Ingersoll 1989). Our results for the stability of baroclinic vortices on an ƒ-plane in the absence of a mean zonal flow are consistent with previous results in the literature, although the presence of the deep fluid interior on the Jovian planets appears to shift the stability boundaries to smaller length scales. The presence of a mean zonal shear flow acts to stabilize vortices against instability, significantly modifies the finite amplitude form of baroclinic instabilities, and combined with internal barotropic instability (Gent and McWilliams 1986) produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable elliptical vortices in a zonal shear flow are observed in this baroclinic model, as in simpler two-dimensional models (Kida 1981). The meridional propagation and decay of vortices on a β-plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model assumption for the flow in the deep interior of Jupiter.

Acoustic diffraction spectrum of a SiC fiber in a solid elastic medium

An analytical model has been developed to compute the diffraction spectrum and phase of a plane compression wave incident on a long fiber embedded in a solid elastic matrix. The spectrum is computed as an expansion of normal modes, thereby permitting insight into the resonant conditions set up in the fiber. The technique was used to model a reinforcing fiber, consisting of a graphite core surrounded by a vapor-deposited SiC layer, all consolidated within a titanium alloy matrix. Reasonable agreement was noted between experiment and theoretical prediction of the resonant frequencies, although bandwidth limitations of the experiments severely restricted the number of resonance lines that could be included in the comparison. A series of numerical studies was conducted to assess the characteristics of each observed resonance. The technique has potential application to characterization of the interfacial layer that links reinforcing fibers to their matrix.

The structure of small clusters: Multiple normal-modes model

A simple model for the structural fluctuations, isomerizations, and phase transitions of small rare-gas clusters is defined (multinormal-model) which studies the statistical equilibrium of several isomers making use of the normal mode expansion of the free energy. It is evaluated classically and quantum mechanically and its results are compared to those of extensive (path integral) simulation calculations.

A method for analyzing properties of rectangular waveguides of steeply tapered width

AbstractA method is described for calculations of the characteristics of the tapered waveguide by means of the electromagnetic field method (normal mode expansion method) in which many higher‐order modes are taken into account. The tapered waveguide to be treated is assumed to be straight along the axis and its thermal loss is assumed negligible. The shape is such that the generated higher‐order mode sequence contains only one parameter. (As an example, either m or n is constant in the TEmn modes.) In the eigenmode analysis of such a tampered waveguide, it is possible to derive solutions which agree with the rigorous solutions taking into account the infinitely many higher‐order modes with sufficient accuracy in practice by means of an application of the extrapolation with relatively few higher‐order modes. The validity of the results is confirmed by experiments. The present method can be applied to a structure in which the slope of the taper is almost 90° (or a step shape) so that it has a wide application range.

Vortex dynamics in a type-II superconducting film and complex linear-response functions.

The dynamics of interacting vortices in a type-II superconducting film responding to a distribution of currents parallel to the film is investigated. The nonlocality of vortex interactions is taken into account by use of an expansion in normal modes of the flux-line lattice and a wave-vector-dependent dynamical matrix. The theory is applied to obtain linear response functions including the complex mutual inductance and film impedance. For a particular geometry of driving current and pickup coil, the complex mutual inductance is calculated and it is shown that only longitudinal modes contribute to the measurable change in flux when pinning is absent or when it is represented as a linear restoring force acting the same way on each vortex. The configuration so described provides a sensitive means of studying vortex dynamics.

Excitation of guided elastic wave modes in hollow cylinders by applied surface tractions

The problem of the generation of guided elastic wave modes in infinite hollow circular cylinders by loading applied to the boundaries of the cylinder, apart from being an interesting problem in itself, has important applications in the field of nondestructive evaluation. Hollow cylinders are commonly used in the nuclear industry in heat exchangers and hence must be inspected for defects regularly to ensure continued safe operation. An exact, closed-form solution is given for the amplitude with which any propagating waveguide mode is generated due to the application of prescribed (time harmonic) surface tractions. Instead of using classical integral transform techniques to solve the problem, the normal mode expansion technique is used. In this technique, analogous to the expansion of a function in terms of an orthogonal (and complete) set of functions, the fields generated in the cylinder due to the application of the surface loading are expanded in terms of the normal modes of the cylinder. The general results are then specialized to loading configurations of practical importance in nondestructive testing.