Lamb waves

In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type.

An infinite medium supports just two wave modes traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, whose velocities depend on the relationship between wavelength and plate thickness.

Since the 1990s, the understanding and utilization of Lamb waves have advanced greatly, thanks to the rapid increase in the availability of computing power.

Lamb's theoretical formulations have found substantial practical application, especially in the field of non-destructive testing.

In general, elastic waves in solid materials[2] are guided by the boundaries of the media in which they propagate.

These are: for symmetric modes and for asymmetric modes, where Inherent in these equations is a relationship between the angular frequency ω and the wave number k. Numerical methods are used to find the phase velocity cp = fλ = ω/k, and the group velocity cg = dω/dk, as functions of d/λ or fd.

This was an intractable problem until the advent of the digital computer forty years after Lamb's original work.

The publication of computer-generated "dispersion curves" by Viktorov[4] in the former Soviet Union, Firestone followed by Worlton in the United States, and eventually many others brought Lamb wave theory into the realm of practical applicability.

Lamb's characteristic equations indicate the existence of two entire families of sinusoidal wave modes in infinite plates of width

The phenomenon of velocity dispersion leads to a rich variety of experimentally observable waveforms when acoustic waves propagate in plates.

The flexural and extensional modes are relatively easy to recognize and this has been advocated as a technique of nondestructive testing.

In the low-frequency limit for the extensional mode, the z- and x-components of the surface displacement are in quadrature and the ratio of their amplitudes is given by:

Simply stated in terms of the material of greatest engineering significance, most of the high-frequency wave energy that propagates long distances in steel plates is traveling at 3000–3300 m/s.

Particle motion in the Lamb wave modes is in general elliptical, having components both perpendicular to and parallel to the plane of the plate.

For certain frequencies-thickness products, the amplitude of one component passes through zero so that the motion is entirely perpendicular or parallel to the plane of the plate.

For particles on the plate surface, these conditions occur when the Lamb wave phase velocity is √2ct or for symmetric modes only cl, respectively.

The Bessel function takes care of the singularity at the source, then converges towards sinusoidal behavior at great distances.

Thus a plate's response to a point disturbance can be expressed as a combination of Lamb waves, plus evanescent terms in the near field.

The overall result can be loosely visualized as a pattern of circular wavefronts, like ripples from a stone dropped into a pond but changing more profoundly in form as they progress outwards.

One question is how the velocities and mode shapes of the Lamb-like waves will be influenced by the real geometry of the part.

Another question is what completely different acoustical behaviors and wave modes may be present in the real geometry of the part.

Traditionally, ultrasonic testing has been conducted with waves whose wavelength is very much shorter than the dimension of the part being inspected.

Although the lamb wave pioneers worked on non-destructive testing applications and drew attention to the theory, widespread use did not come about until the 1990s when computer programs for calculating dispersion curves and relating them to experimentally observable signals became much more widely available.

These computational tools, along with a more widespread understanding of the nature of Lamb waves, made it possible to devise techniques for nondestructive testing using wavelengths that are comparable with or greater than the thickness of the plate.

Lamb waves are well suited to this concept, because they irradiate the whole plate thickness and propagate substantial distances with consistent patterns of motion.

Acoustic emission uses much lower frequencies than traditional ultrasonic testing, and the sensor is typically expected to detect active flaws at distances up to several meters.

A large fraction of the structures customarily testing with acoustic emission are fabricated from steel plate - tanks, pressure vessels, pipes and so on.

Substantial improvements in the accuracy of source location (a major technique of AE testing) can be achieved through good understanding and skillful utilization of the Lamb wave body of knowledge.

An arbitrary mechanical excitation applied to a plate will generate a multiplicity of Lamb waves carrying energy across a range of frequencies.

In acoustic emission testing, the challenge is to recognize the multiple Lamb wave components in the received waveform and to interpret them in terms of source motion.