Rayleigh-Lamb Frequency Equation Research Articles

Study of the sound speed of components in a medical ultrasonic one-dimensional array transducer

In this work the sound speed of components in a one-dimensional array transducer used to acquire ultrasonic cross-sectional images for medical diagnosis was studied. The transducer is composed of several channels having a similar width and thickness, so the sound speed of these components depends on their dimensions. First, the components of the transducer were divided into active and passive components, and then the sound speed according to dimensions was derived using coupled-mode theory and the Rayleigh–Lamb frequency equation. After classifying the aspect ratios of the components according to the range of change, different wave velocities were proposed according to this range. For transducers made of different materials with various frequencies, the sound speed of the components was derived by applying the devised method. After making a prototype of each transducer, its performance was measured and the feasibility of the devised method presented.

Non-contact and non-destructive Lamb wave generation using laser-induced plasma shock wave

This paper proposes a non-contact and non-destructive method to generate Lamb waves against a target structure using the impulse excitation force generated by laser-induced plasma (LIP). When a high power pulse laser is irradiated in air and its laser fluence exceeds 1015 W/m2, a plasma is formed. While the plasma in air expands in high speed, shock waves on the spherical surface are generated and these shock waves become the impulse excitation force against a target structure, resulting in the non-contact and non-destructive approach. A 2024 aluminum alloy plate is used as the test piece in the experiment, and the dynamic characteristics of the Lamb waves generated from LIP shock waves are measured. Phase velocity and group velocity of generated Lamb waves were compared to the calculated values from Rayleigh-Lamb frequency equations and we found that maximum error was 5% and its frequency component included at least 400 kHz. Further, we investigated the relationship between the distance from the LIP shock wave–generating location to a test piece and the dynamic characteristics of the generated Lamb waves. This method can control the amplitude and the frequency components of generated Lamb waves by changing this distance.

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.

Measurement of wavenumber of wall-thinned plate using spatial local wavenumber filtering

The surface of a shell-type structure can generate cracks or wall-thinning due to corrosion, etc. Those can eventually lead to the fracture of the structure, which can trigger enormous fatality and property loss. Thereby, a laser imaging technology on such structures as thin plate structureor piping whose thickness is relatively thin in comparison to the area, has been steadily studied for the past 10 years. The most typical among the laser imaging technology is the pulse laser imaging. By using the same, a new technology for inspecting and imaging a desired area within a relatively short period of time was developed, so as to scan various structures including the thin-plate structure and piping. However, this method builds image by measuring waves reflected from defects, and has a time delay of a few milliseconds at each scanning point. Moreover, complexityof the systems is so high due to additional components such as laser focusing parts. This paper proposes laser imaging method with increased scanning speed based on excitation and measurement of standing waves in structures. It is shown that defects in a structure can be visualized bygenerating standing waves with single frequency and scanning the waves at each point by the laser scanning system suggested in this work. To quantitatively evaluate thickness of a plate, wavenumber is calculated by acoustic wavenumber spectroscopy, and relationship between wavenumber and thickness of the plate is established by Rayleigh-Lamb frequency equation. The proposed technique is validated by wall-thinned plates that have constant wall loss and a linear thickness variation wall loss.

Optical coherence elastography assessment of corneal viscoelasticity with a modified Rayleigh-Lamb wave model

The biomechanical properties of the cornea play a critical role in forming vision. Diseases such as keratoconus can structurally degenerate the cornea causing a pathological loss in visual acuity. UV-A/riboflavin corneal collagen crosslinking (CXL) is a clinically available treatment to stiffen the cornea and restore its healthy shape and function. However, current CXL techniques do not account for pre-existing biomechanical properties of the cornea nor the effects of the CXL treatment itself. In addition to the inherent corneal structure, the intraocular pressure (IOP) can also dramatically affect the measured biomechanical properties of the cornea. In this work, we present the details and development of a modified Rayleigh-Lamb frequency equation model for quantifying corneal biomechanical properties. After comparison with finite element modeling, the model was utilized to quantify the viscoelasticity of in situ porcine corneas in the whole eye-globe configuration before and after CXL based on noncontact optical coherence elastography measurements. Moreover, the viscoelasticity of the untreated and CXL-treated eyes was quantified at various IOPs. The results showed that the stiffness of the cornea increased after CXL and that corneal stiffness is close to linear as a function of IOP. These results show that the modified Rayleigh-Lamb wave model can provide an accurate assessment of corneal viscoelasticity, which could be used for customized CXL therapies.

Evaluating the Effects of Riboflavin/UV-A and Rose-Bengal/Green Light Cross-Linking of the Rabbit Cornea by Noncontact Optical Coherence Elastography.

PurposeThe purpose of this study was to use noncontact optical coherence elastography (OCE) to evaluate and compare changes in biomechanical properties that occurred in rabbit cornea in situ after corneal collagen cross-linking by either of two techniques: ultraviolet-A (UV-A)/riboflavin or rose-Bengal/green light.MethodsLow-amplitude (≤10 μm) elastic waves were induced in mature rabbit corneas by a focused air pulse. Elastic wave propagation was imaged by a phase-stabilized swept source OCE (PhS-SSOCE) system. Corneas were then cross-linked by either of two methods: UV-A/riboflavin (UV-CXL) or rose-Bengal/green light (RGX). Phase velocities of the elastic waves were fitted to a previously developed modified Rayleigh-Lamb frequency equation to obtain the viscoelasticity of the corneas before and after the cross-linking treatments. Micro-scale depth-resolved phase velocity distribution revealed the depth-wise heterogeneity of both cross-linking techniques.ResultsUnder standard treatment settings, UV-CXL significantly increased the stiffness of the corneas by ∼47% (P < 0.05), but RGX did not produce statistically significant increases. The shear viscosities were unaffected by either cross-linking technique. The depth-wise phase velocities showed that UV-CXL affected the anterior ∼34% of the corneas, whereas RGX affected only the anterior ∼16% of the corneas.ConclusionsUV-CXL significantly strengthens the cornea, whereas RGX does not, and the effects of cross-linking by UV-CXL reach deeper into the cornea than cross-linking effects of RGX under similar conditions.

Quantitative methods for reconstructing tissue biomechanical properties in optical coherence elastography: a comparison study

We present a systematic analysis of the accuracy of five different methods for extracting the biomechanical properties of soft samples using optical coherence elastography (OCE). OCE is an emerging noninvasive technique, which allows assessment of biomechanical properties of tissues with micrometer spatial resolution. However, in order to accurately extract biomechanical properties from OCE measurements, application of a proper mechanical model is required. In this study, we utilize tissue-mimicking phantoms with controlled elastic properties and investigate the feasibilities of four available methods for reconstructing elasticity (Young’s modulus) based on OCE measurements of an air-pulse induced elastic wave. The approaches are based on the shear wave equation (SWE), the surface wave equation (SuWE), Rayleigh-Lamb frequency equation (RLFE), and finite element method (FEM), Elasticity values were compared with uniaxial mechanical testing. The results show that the RLFE and the FEM are more robust in quantitatively assessing elasticity than the other simplified models. This study provides a foundation and reference for reconstructing the biomechanical properties of tissues from OCE data, which is important for the further development of noninvasive elastography methods.

Quantitative assessment of corneal viscoelasticity using optical coherence elastography and a modified Rayleigh-Lamb equation.

We demonstrate the use of a modified Rayleigh–Lamb frequency equation in conjunction with noncontact optical coherence elastography to quantify the viscoelastic properties of the cornea. Phase velocities of air-pulse-induced elastic waves were extracted by spectral analysis and used for calculating the Young’s moduli of the samples using the Rayleigh–Lamb frequency equation (RLFE). Validation experiments were performed on 2% agar phantoms (n ¼ 3) and then applied to porcine corneas (n ¼ 3) in situ. The Young’s moduli of the porcine corneas were estimated to be ∼60 kPa with a shear viscosity ∼0.33 Pa · s. The results demonstrate that the RLFE is a promising method for noninvasive quantification of the corneal biomechanical properties and may potentially be useful for clinical ophthalmological applications.

Extensional and Transversal Wave Motion in Transversely Isotropic Thermoelastic Plates by Using Asymptotic Method

The present investigation is concerned with the study of extensional and transversal wave motions in an infinite homogenous transversely isotropic, thermoelastic plate by using asymptotic method in the context of coupled thermoelasticity, Lord and Shulman (1967, “The Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids, 15, pp. 299–309), and Green and Lindsay (1972, “Thermoelasticity,” J. Elast., 2, pp. 1–7) theories of generalized thermoelasticity. The governing equations for extensional, transversal, and flexural motions have been derived from the system of three-dimensional dynamical equations of linear thermoelasticity. The asymptotic operator plate model for extensional motion in a homogeneous transversely isotropic thermoelastic plate leads to sixth degree polynomial secular equation that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion. The Rayleigh–Lamb frequency equation for the plate is expanded in power series in order to obtain polynomial frequency equation and velocity dispersion relations. Their validation has been established with that of asymptotic method. The special cases of short and long wavelength waves are also discussed. The expressions for group velocity of extensional and transversal modes have been derived. Finally, the numerical solution is carried out for homogeneous transversely isotropic plate of single crystal of zinc material. The dispersion curves of phase velocity and attenuation coefficient are presented graphically.

Phase and group velocity matching for cumulative harmonic generation in Lamb waves

Owing to the enhanced sensitivity of nonlinear acoustic methods to material damage, the nonlinear Lamb wave propagation is pertinent to the nondestructive evaluation of platelike structures, and it is typically manifested as generation of higher harmonics. For dispersive waves such as Lamb waves, however, the cumulative growth of harmonics requires that the primary mode and the generated higher harmonic modes possess identical phase and group velocities. In this paper, this issue of the phase and group velocity matching in Lamb waves is explored based on a systematic analysis of the Rayleigh-Lamb frequency equations. The analysis shows that for certain values of the phase velocity, the Rayleigh-Lamb frequency equations are satisfied at equi-spaced frequencies which are multiples of the smallest. Such frequencies, together with the corresponding phase velocities and the Lamb modes, are determined analytically. Four such types of Lamb modes are identified: (i) Lamé modes, (ii) symmetric modes with dominant longitudinal displacements, (iii) intersections of symmetric and antisymmetric modes and (iv) extra Rayleigh modes. For the first three types, it is also established that the primary and the harmonic modes have the same group velocity, and that the surface motion of the plate is featured with vanishing vertical or horizontal displacements. In contrast to these three types, the fourth type only exists for a special range of the transverse to longitudinal wave speeds of the solid. This type is not featured with a common group velocity, and neither of the vertical or horizontal displacement vanishes on the plate surfaces. The obtained results are summarized as tables, and demonstrated graphically on the dispersion curves for aluminum as well as iron plates.

Generalized thermoelastic extensional and flexural wave motions in homogenous isotropic plate by using asymptotic method

In this paper the asymptotic method has been applied to investigate propagation of generalized thermoelastic waves in an infinite homogenous isotropic plate. The governing equations for the extensional, transversal and flexural motions are derived from the system of three-dimensional dynamical equations of linear theories of generalized thermoelasticity. The asymptotic operator plate model for extensional and flexural free vibrations in a homogenous thermoelastic plate leads to sixth and fifth degree polynomial secular equations, respectively. These secular equations govern frequency and phase velocity of various possible modes of wave propagation at all wavelengths. The velocity dispersion equations for extensional and flexural wave motion are deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate. The approximation for long and short waves along with expression for group velocity has also been obtained. The Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations and its equivalence established with that of asymptotic method. The numeric values for phase velocity, group velocity and attenuation coefficients has also been obtained using MATHCAD software and are shown graphically for extensional and flexural waves in generalized theories of thermoelastic plate for solid helium material.

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.

S0302-1-3 ラム波における累積的高調波発生に関する基礎的検討([S0302-1]超音波非破壊評価の高精度化(1))

Despite its importance in nonlinear ultrasonic measurements, the harmonic generation in Lamb wave propagation is a difficult issue due to the dispersive nature and the multiple mode existence. Based on a systematic analysis of Rayleigh-Lamb frequency equations, the frequencies of Lamb waves which may cause cumulative harmonic generation have been derived in this study. The derived formulas giving angular frequencies, wave numbers, phase velocity and group velocity of the cumulative Lamb modes are given in terms of longitudinal wave velocity, transverse wave velocity, plate thickness and two positive integer numbers. The phase velocity of the fundamental and cumulative harmonic modes is dependent on the ratio of two integer numbers, and there are infinite numbers of such phase velocities at which cumulative harmonic generation is possible. These conclusions can be applied to any elastic material which is isotropic and homogenous.

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.

Asymptotics of wave motion in transversely isotropic plates

The present investigation is concerned with elastic wave motion in infinite transversely isotropic plate by asymptotic method. The differential equations for the flexural and extensional motions are derived from the system of three-dimensional dynamical equations of linear elasticity. All coefficients of the differential operator are presented as explicit functions of the material parameter γ= c s 2/ c l 2, the ratio of the squared velocities of flexural (shear) and extensional (longitudinal) waves. The velocity dispersion equations for the flexural and extensional wave motions are deduced analytically from the three-dimensional analog of Rayleigh–Lamb frequency equation for plates. The approximations for long and short waves are also obtained. The dispersion curves for phase velocity and group velocity spectrum are shown graphically for flexural and extensional wave motions of the plate. The results for isotropic materials have been deduced as special cases.

Determination of Lamb mode eigenvalues.

An original method is presented to determine the complex Lamb wave spectrum by using a numerical spectral method applied to the elasticity equations. This method presents the advantage to directly determine complex wave numbers for a given frequency via a classical matricial eigenvalue problem, and allows the wave numbers to be determined at relatively high frequencies (i.e., corresponding to many propagating modes). It does not need initial guess values for the wave numbers, contrary to the usual method of root finding of the Rayleigh-Lamb frequency equations (dispersion relation) in the complex plane. Results are presented and the method is discussed.

On the Equivalence of Dispersion Relations Resulting from Rayleigh-Lamb Frequency Equation and the Operator Plate Model

The Rayleigh-Lamb frequency equations for the free vibrations of an infinite isotropic elastic plate are expanded into the infinite power series and reduced to the polynomial frequency and velocity dispersion relations. The latter are compared to those of the operator plate model developed in [Losin, N. A., 1997, “Asymptotics of Flexural Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 64, No. 2, pp. 336–342; Losin, N. A., 1998, “Asymptotics of Extensional Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 65, No. 4, pp. 1042–1047] for both symmetric and antisymmetric vibrations. As a result of comparative analysis, the equivalence of the corresponding dispersion polynomials is established. The frequency spectra, generated by Rayleigh-Lamb equations, are illustrated graphically and briefly discussed with reference to those published in [Potter, D. S., and Leedham, C. D., 1967, “Normalized Numerical Solution for Rayleigh’s Frequency Equation,” J. Acoust. Soc. Am., 41, No. 1, pp. 148–153].

Eigenvalue problems for the beam and an upper and lower bound for each branch of solutions of Rayleigh-Lamb frequency equation

The Rayleigh-Lamb frequency equation is derived in studying eigenvalues and eigenfunctions for the beam problem in a domain (−a,a) x (−d, d). It is shown that there is an upper bound and a lower bound for each branch of the frequency spectrum which consists of all roots of the frequency equation.

Transient SV-Wave Motion in Plane Strain of an Elastic Plate.

In this study, a plate subjected to a suddenly applied surface force which travels constant velocity on the surface is analyzed. The transformed solutions in the Laplace and Fourier domains include the function of Rayleigh-Lamb frequency equation. Their solutions are expanded into infinite series form, and then the inverse transforms are carried out for the each term of the series using the Cagniard-deHoop method. The exact transient closed-form solutions in the time domain are also expressed in infinite series form and their each term represents physical transient wave. It is shown that the amplitude ratios of reflection of P wave and SV wave are same for the cases of harmonic waves. The numerical results of velocity fields are shown by graphical representation.

Asymptotics of Flexural Waves in Isotropic Elastic Plates

The long and short-wave asymptotics of order O(k6h6) for the flexural vibrations of an infinite, isotropic, elastic plate are studied. The differential equation for the flexural motion is derived from the system of three-dimensional dynamic equations of linear elasticity. All coefficients of the differential operator are presented as explicit functions of the material parameter γ = cs2/cL2, the ratio of the velocities squared of the flexural (shear) and extensional (longitudinal) waves. Relatively simple frequency and velocity dispersion equations for the flexural waves are deduced in analytical form from the three-dimensional analog of Rayleigh-Lamb frequency equation for plates. The explicit formulas for the group velocity are also presented. Variations of the velocity and frequency spectrums depending on Poisson’s ratio are illustrated graphically. The results are discussed and compared to those obtained and summarized by R. D. Mindlin (1951, 1960), Tolstoy and Usdin (1953, 1957), and Achenbach (1973).