Lamb waves: February 2012

Thursday, February 16, 2012

Lamb waves

Lamb after-effects bear in solid plates. They are adaptable after-effects whose atom motion lies in the even that contains the administration of beachcomber advancement and the bowl accustomed (the administration erect to the plate). In 1917, the English mathematician Horace Lamb appear his archetypal assay and description of acoustic after-effects of this type. Their backdrop angry out to be absolutely complex. An absolute average supports just two beachcomber modes traveling at different velocities; but plates abutment two absolute sets of Lamb beachcomber modes, whose velocities depend on the accord amid amicableness and bowl thickness.

Since the 1990s, the compassionate and appliance of Lamb after-effects has avant-garde greatly, acknowledgment to the accelerated access in the availability of accretion power. Lamb's abstract formulations accept begin abundant applied application, abnormally in the acreage of nondestructive testing.

The appellation Rayleigh–Lamb after-effects embraces the Rayleigh wave, a blazon of beachcomber that propagates forth a individual surface. Both Rayleigh and Lamb after-effects are accountable by the adaptable backdrop of the surface(s) that adviser them.

Lamb's characteristic equations

In general, adaptable after-effects in solid materials1 are guided by the boundaries of the media in which they propagate. An access to guided beachcomber propagation, broadly acclimated in concrete acoustics, is to seek sinusoidal solutions to the beachcomber blueprint for beeline adaptable after-effects accountable to abuttals altitude apery the structural geometry. This is a archetypal eigenvalue problem.

Waves in plates were a part of the aboriginal guided after-effects to be analyzed in this way. The assay was developed and appear in 19172 by Horace Lamb, a baton in the algebraic physics of his day.

Lamb's equations were acquired by ambience up ceremonial for a solid bowl accepting absolute admeasurement in the x and y directions, and array d in the z direction. Sinusoidal solutions to the beachcomber blueprint were postulated, accepting x- and z-displacements of the form

\xi = A_x f_x(z) e^{i(\omega t - kx)} \quad \quad (1)

\zeta = A_z f_z(z) e^{i(\omega t - k x)} \quad \quad (2)

This anatomy represents sinusoidal after-effects breeding in the x administration with amicableness 2π/k and abundance ω/2π. Displacement is a action of x, z, t only; there is no displacement in the y administration and no aberration of any concrete quantities in the y direction.

The concrete abuttals action for the chargeless surfaces of the bowl is that the basic of accent in the z administration at z = +/- d/2 is zero. Applying these two altitude to the above-formalized solutions to the beachcomber equation, a brace of appropriate equations can be found. These are:

\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {4 \alpha \beta k^2} {(k^2 - \beta^2)^2}\ \quad \quad \quad \quad (3)

and

\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {(k^2 - \beta^2)^2} {4 \alpha \beta k^2}\ \quad \quad \quad \quad (4)

where

\alpha^2 = \frac{\omega^2}{c_l^2} - k^2 \quad \quad \text{and}\quad\quad \beta^2 = \frac{\omega^2}{c_t^2} - k^2.

Inherent in these equations is a accord amid the angular abundance ω and the beachcomber amount k. Numerical methods are acclimated to acquisition the appearance acceleration cp = fλ = ω/k, and the accumulation acceleration cg = dω/dk, as functions of d/λ or fd. cl and ct are the longitudinal beachcomber and microburst beachcomber velocities respectively.

The band-aid of these equations aswell reveals the absolute anatomy of the atom motion, which equations (1) and (2) represent in all-encompassing anatomy only. It is begin that blueprint (3) gives acceleration to a ancestors of after-effects whose motion is balanced about the midplane of the bowl (the even z = 0), while blueprint (4) gives acceleration to a ancestors of after-effects whose motion is antisymmetric about the midplane. Figure 1 illustrates a affiliate of anniversary family.

Lamb’s appropriate equations were accustomed for after-effects breeding in an absolute bowl - a homogeneous, isotropic solid belted by two alongside planes aloft which no beachcomber activity can propagate. In formulating his problem, Lamb bedfast the apparatus of atom motion to the administration of the bowl accustomed (z-direction) and the administration of beachcomber advancement (x-direction). By definition, Lamb after-effects accept no atom motion in the y-direction. Motion in the y-direction in plates is begin in the alleged SH or shear-horizontal beachcomber modes. These accept no motion in the x- or z-directions, and are appropriately commutual to the Lamb beachcomber modes. These two are the alone beachcomber types which can bear with straight, absolute beachcomber fronts in a bowl as authentic above.

Velocity dispersion inherent in the characteristic equations

Lamb after-effects display acceleration dispersion; that is, their acceleration of advancement c depends on the abundance (or wavelength), as able-bodied as on the adaptable constants and body of the material. This abnormality is axial to the abstraction and compassionate of beachcomber behavior in plates. Physically, the key constant is the arrangement of bowl array d to amicableness λ. This arrangement determines the able acerbity of the bowl and appropriately the acceleration of the wave. In abstruse applications, a added applied constant readily acquired from this is used, namely the artefact of array and frequency:

f\cdot d = \frac {d\cdot c}{\lambda}, since for all after-effects c = fλ.

The accord amid acceleration and abundance (or wavelength) is inherent in the appropriate equations. In the case of the plate, these equations are not simple and their band-aid requires afterwards methods. This was an awkward botheration until the actualization of the agenda computer forty years afterwards Lamb's aboriginal work. The advertisement of computer-generated "dispersion curves" by Viktorov3 in the above Soviet Union, Firestone followed by Worlton in the United States, and eventually abounding others brought Lamb beachcomber approach into the branch of applied applicability. Experimental waveforms empiric in plates can be accepted by estimation with advertence to the burning curves.

Dispersion curves - graphs that actualization relationships amid beachcomber velocity, amicableness and abundance in dispersive systems - can be presented in assorted forms. The anatomy that gives the greatest acumen into the basal physics has ω (angular frequency) on the y-axis and k (wave number) on the x-axis. The anatomy acclimated by Viktorov, that brought Lamb after-effects into applied use, has beachcomber acceleration on the y-axis and d / λ, the thickness/wavelength ratio, on the x-axis. The a lot of applied anatomy of all, for which acclaim is due to J. and H. Krautkrämer as able-bodied as to Floyd Firestone (who, incidentally, coined the byword "Lamb waves") has beachcomber acceleration on the y-axis and fd, the frequency-thickness product, on the x-axis.

Lamb's appropriate equations announce the actuality of two absolute families of sinusoidal beachcomber modes in absolute plates of amplitude d. This stands in adverse with the bearings in great media area there are just two beachcomber modes, the longitudinal beachcomber and the axle or microburst wave. As in Rayleigh after-effects which bear forth individual chargeless surfaces, the atom motion in Lamb after-effects is egg-shaped with its x and z apparatus depending on the abyss aural the plate.4 In one ancestors of modes, the motion is balanced about the midthickness plane. In the added ancestors it is antisymmetric. The abnormality of acceleration burning leads to a affluent array of experimentally appreciable waveforms if acoustic after-effects bear in plates. It is the accumulation acceleration cg, not the above-mentioned actualization acceleration c or cp, that determines the modulations apparent in the empiric waveform. The actualization of the waveforms depends alarmingly on the abundance ambit called for observation. The flexural and analytic modes are almost simple to admit and this has been advocated as a address of nondestructive testing.

Velocity dispersion inherent in the characteristic equations

The zero-order modes

The balanced and antisymmetric zero-order modes deserve appropriate attention. These modes accept "nascent frequencies" of zero. Thus they are the alone modes that abide over the absolute abundance spectrum from aught to indefinitely top frequencies. In the low abundance ambit (i.e. if the amicableness is greater than the bowl thickness) these modes are generally alleged the “extensional mode” and the “flexural mode" respectively, agreement that call the attributes of the motion and the adaptable stiffnesses that administer the velocities of propagation. The egg-shaped atom motion is mainly in the even of the bowl for the symmetrical, analytic approach and erect to the even of the bowl for the antisymmetric, flexural mode. These characteristics change at college frequencies.

These two modes are the a lot of important because (a) they abide at all frequencies and (b) in a lot of applied situations they backpack added activity than the higher-order modes.

The zero-order balanced approach (designated s0) campaign at the "plate velocity" in the low-frequency administration area it is appropriately alleged the "extensional mode". In this administration the bowl stretches in the administration of advancement and affairs appropriately in the array direction. As the abundance increases and the amicableness becomes commensurable with the bowl thickness, arched of the bowl starts to accept a cogent access on its able stiffness. The appearance acceleration drops calmly while the accumulation acceleration drops somewhat precipitously appear a minimum. At college frequencies yet, both the appearance acceleration and the accumulation acceleration assemble appear the Rayleigh beachcomber acceleration - the appearance acceleration from above, and the accumulation acceleration from below.

In the low-frequency absolute for the analytic mode, the z- and x-components of the apparent displacement are in quadrature and the arrangement of their amplitudes is accustomed by:

\frac {a_z}{a_x} = \frac{\pi \nu}{(1 - \nu)} . \frac{d}{ \lambda}

where ν is Poisson's ratio.

The zero-order antisymmetric approach (designated a0) is awful dispersive in the low abundance administration area it is appropriately alleged the "flexural mode". For actual low frequencies (very attenuate plates) the appearance and accumulation velocities are both proportional to the aboveboard basis of the frequency; the accumulation acceleration is alert the appearance velocity. This simple accord is a aftereffect of the stiffness/thickness accord for attenuate plates in bending. At college frequencies area the amicableness is no best abundant greater than the bowl thickness, these relationships breach down. The appearance acceleration rises beneath and beneath bound and converges appear the Rayleigh beachcomber acceleration in the top abundance limit. The accumulation acceleration passes through a maximum, a little faster than the microburst beachcomber velocity, if the amicableness is about according to the bowl thickness. It again converges, from above, to the Rayleigh beachcomber acceleration in the top abundance limit.

In abstracts that acquiesce both analytic and flexural modes to be aflame and detected, the analytic approach generally appears as a higher-velocity, lower-amplitude forerunner to the flexural mode. The flexural approach is the added calmly aflame of the two, and generally carries a lot of of the energy.

The higher-order modes

As the abundance is raised, the higher-order beachcomber modes accomplish their actualization in accession to the zero-order modes. Anniversary higher-order approach is “born” at a beating abundance of the plate, and exists alone aloft that frequency. For example, in a ¾ inch (19mm) blubbery animate bowl at a abundance of 200 kHz, the aboriginal four Lamb beachcomber modes are present and at 300 kHz, the aboriginal six. The aboriginal few higher-order modes can be audibly empiric beneath favorable beginning conditions. Beneath beneath than favorable altitude they overlap and can not be distinguished.

The higher-order Lamb modes are characterized by nodal planes aural the plate, alongside to the bowl surfaces. Anniversary of these modes exists alone aloft a assertive abundance which can be alleged its "nascent frequency". There is no top abundance absolute for any of the modes. The beginning frequencies can be pictured as the beating frequencies for longitudinal or microburst after-effects breeding erect to the even of the plate, i.e.

d = \frac{n \lambda}{2} \quad \quad \text{or} \quad \quad f = \frac{nc}{2d}

where n is any absolute integer. Here c can be either the longitudinal beachcomber acceleration or the microburst beachcomber velocity, and for anniversary consistent set of resonances the agnate Lamb beachcomber modes are alternately balanced and antisymmetric. The coaction of these two sets after-effects in a arrangement of beginning frequencies that at aboriginal glance seems irregular. For example, in a 3/4 inch (19mm) blubbery animate bowl accepting longitudinal and microburst velocities of 5890 m/s and 3260 m/s respectively, the beginning frequencies of the antisymmetric modes a1, a2 and a3 are 86 kHz, 257 kHz and 310 kHz respectively, while the beginning frequencies of the symmetric modes s1, s2 and s3 are 155 kHz, 172 kHz and 343 kHz respectively.

At its beginning frequency, anniversary of these modes has an absolute appearance acceleration and a accumulation acceleration of zero. In the top abundance limit, the appearance and accumulation velocities of all these modes assemble to the microburst beachcomber velocity. Because of these convergences, the Rayleigh and microburst velocities (which are actual abutting to one another) are of above accent in blubbery plates. Simply declared in agreement of the actual of greatest engineering significance, a lot of of the high-frequency beachcomber activity that propagates continued distances in animate plates is traveling at 3000–3300 m/s.

Particle motion in the Lamb beachcomber modes is in accepted elliptical, accepting apparatus both erect to and alongside to the even of the plate. These apparatus are in quadrature, i.e. they accept a 90° appearance difference. The about consequence of the apparatus is a action of frequency. For assertive frequencies-thickness products, the amplitude of one basic passes through aught so that the motion is absolutely erect or alongside to the even of the plate. For particles on the bowl surface, these altitude action if the Lamb beachcomber appearance acceleration is √2ct or cl, respectively. These directionality considerations are important if because the radiation of acoustic activity from plates into adjoining fluids.

The atom motion is aswell absolutely erect or absolutely alongside to the even of the plate, at a mode's beginning frequency. Abutting to the beginning frequencies of modes agnate to longitudinal-wave resonances of the plate, their atom motion will be about absolutely erect to the even of the plate; and abreast the shear-wave resonances, parallel.

J. and H. Krautkrämer accept acicular out5 that Lamb after-effects can be conceived as a arrangement of longitudinal and microburst after-effects breeding at acceptable angles beyond and forth the plate. These after-effects reflect and mode-convert and amalgamate to aftermath a sustained, articular beachcomber pattern. For this articular beachcomber arrangement to be formed, the bowl array has to be just appropriate about to the angles of advancement and wavelengths of the basal longitudinal and microburst waves; this claim leads to the acceleration burning relationships.

Point sources and waves with cylindrical symmetry

While Lamb's assay affected a beeline wavefront, it has been shown* that the aforementioned appropriate equations administer to axisymmetric bowl after-effects (e.g. after-effects breeding with annular wavefronts from point sources, like ripples from a rock alone into a pond). The aberration is that admitting the "carrier" for the beeline wavefront is a sinusoid, the "carrier" for the axisymmetric beachcomber is a Bessel function. The Bessel action takes affliction of the aberancy at the source, again converges appear sinusoidal behavior at abundant distances.

Klaes, M, in Journées d'Etudes sur l'Emission Acoustique, INSA de Lyon (France), 1978.

This byword is absolutely generally encountered in non-destructive testing. "Guided Lamb Waves" can be authentic as Lamb-like after-effects that are guided by the bound ambit of absolute analysis objects. To add the prefix "guided" to the byword "Lamb wave" is appropriately to admit that Lamb's absolute bowl is, in reality, boilerplate to be found.

In absoluteness we accord with bound plates, or plates captivated into annular pipes or vessels, or plates cut into attenuate strips, etc. Lamb beachcomber approach generally gives a actual acceptable annual of abundant of the beachcomber behavior of such structures. It will not accord a absolute account, and that is why the byword "Guided Lamb Waves" is added actual than "Lamb Waves". One catechism is how the velocities and approach shapes of the Lamb-like after-effects will be afflicted by the absolute geometry of the part. For example, the acceleration of a Lamb-like beachcomber in a attenuate butt will depend hardly on the ambit of the butt and on whether the beachcomber is traveling forth the arbor or annular the circumference. Another catechism is what absolutely altered acoustical behaviors and beachcomber modes may be present in the absolute geometry of the part. For example, a annular aqueduct has flexural modes associated with actual movement of the accomplished pipe, absolutely altered from the Lamb-like flexural approach of the aqueduct wall.

Lamb waves in ultrasonic testing

The purpose of accelerated testing is usually to acquisition and characterize alone flaws in the article getting tested. Such flaws are detected if they reflect or besprinkle the abutting beachcomber and the reflected or broadcast beachcomber alcove the seek assemblage with acceptable amplitude.

Traditionally, accelerated testing has been conducted with after-effects whose amicableness is actual abundant beneath than the ambit of the allotment getting inspected. In this high-frequency-regime, the accelerated ambassador uses after-effects that almost to the infinite-medium longitudinal and microburst beachcomber modes, zig-zagging to and fro beyond the array of the plate. Although the lamb beachcomber antecedents formed on nondestructive testing applications and drew absorption to the theory, boundless use did not appear about until the 1990s if computer programs for artful burning curves and apropos them to experimentally appreciable signals became abundant added broadly available. These computational tools, forth with a added boundless compassionate of the attributes of Lamb waves, fabricated it accessible to devise techniques for nondestructive testing application wavelengths that are commensurable with or greater than the array of the plate. At these best wavelengths the abrasion of the beachcomber is less, so that flaws can be detected at greater distances.

A above claiming and accomplishment in the use of Lamb after-effects for accelerated testing is the bearing of specific modes at specific frequencies that will bear able-bodied and accord apple-pie acknowledgment "echoes". This requires accurate ascendancy of the excitation. Techniques for this cover the use of adjust transducers, wedges, after-effects from aqueous media and electro alluring acoustic transducers (EMAT's).

Acousto-ultrasonic testing differs from accelerated testing in that it was conceived as a agency of assessing accident (and added actual attributes) broadcast over abundant areas, rather than anecdotic flaws individually. Lamb after-effects are able-bodied ill-fitted to this concept, because they brighten the accomplished bowl array and bear abundant distances with constant patterns of motion.

Lamb waves in acoustic emission testing

Lamb waves

Acoustic discharge uses abundant lower frequencies than acceptable accelerated testing, and the sensor is about accepted to ascertain alive flaws at distances up to several meters. A ample atom of the structures commonly testing with acoustic discharge are bogus from animate bowl - tanks, burden vessels, pipes and so on. Lamb beachcomber approach is accordingly the prime approach for answer the arresting forms and advancement velocities that are empiric if administering acoustic discharge testing. Substantial improvements in the accurateness of AE antecedent area (a above techniques of AE testing) can be accomplished through acceptable compassionate and accomplished appliance of the Lamb beachcomber physique of knowledge.An approximate automated action activated to a bowl will accomplish a complication of Lamb after-effects accustomed activity beyond a ambit of frequencies. Such is the case for the acoustic discharge wave. In acoustic discharge testing, the claiming is to admit the assorted Lamb beachcomber apparatus in the accustomed waveform and to adapt them in agreement of antecedent motion. This contrasts with the bearings in accelerated testing, area the aboriginal claiming is to accomplish a single, well-controlled Lamb beachcomber approach at a individual frequency. But even in accelerated testing, approach about-face takes abode if the generated Lamb beachcomber interacts with flaws, so the estimation of reflected signals circuitous from assorted modes becomes a agency of blemish characterization.