In solid mechanics, Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate. 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, longitudinal and shear, traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, symmetric and antisymmetric, 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.
The term Rayleigh–Lamb waves embraces the Rayleigh wave, a type of wave that propagates along a single surface. Both Rayleigh and Lamb waves are constrained by the elastic properties of the surface(s) that guide them.
Lamb's characteristic equations
In general, elastic waves in solid materialsAchenbach, J. D. “Wave Propagation in Elastic Solids”. New York: Elsevier, 1984. are guided by the boundaries of the media in which they propagate. An approach to guided wave propagation, widely used in physical acoustics, is to seek sinusoidal solutions to the wave equation for linear elastic waves subject to boundary conditions representing the structural geometry. This is a classic eigenvalue problem.
Waves in plates were among the first guided waves to be analyzed in this way. The analysis was developed and published in 1917Lamb, H. "On Waves in an Elastic Plate." Proc. Roy. Soc. London, Ser. A 93, 114–128, 1917. by Horace Lamb, a leader in the mathematical physics of his day.
Lamb's equations were derived by setting up formalism for a solid plate having infinite extent in the x and y directions, and thickness d in the z direction. Sinusoidal solutions to the wave equation were postulated, having x- and z-displacements of the form
This form represents sinusoidal waves propagating in the x direction with wavelength 2π/k and frequency ω/2π. Displacement is a function of x, z, t only; there is no displacement in the y direction and no variation of any physical quantities in the y direction.
The physical boundary condition for the free surfaces of the plate is that the component of stress in the z direction at z = ±d/2 is zero. Applying these two conditions to the above-formalized solutions to the wave equation, a pair of characteristic equations can be found. These are:
for symmetric modes and
for antisymmetric 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. cl and ct are the longitudinal wave and shear wave velocities respectively.
The solution of these equations also reveals the precise form of the particle motion, which equations (1) and (2) represent in generic form only. It is found that equation (3) gives rise to a family of waves whose motion is symmetrical about the midplane of the plate (the plane z = 0), while equation (4) gives rise to a family of waves whose motion is antisymmetric about the midplane. Figure 1 illustrates a member of each family.
Lamb’s characteristic equations were established for waves propagating in an infinite plate – a homogeneous, isotropic solid bounded by two parallel planes beyond which no wave energy can propagate. In formulating his problem, Lamb confined the components of particle motion to the direction of the plate normal (z-direction) and the direction of wave propagation (x-direction). By definition, Lamb waves have no particle motion in the y-direction. Motion in the y-direction in plates is found in the so-called SH or shear-horizontal wave modes. These have no motion in the x- or z-directions, and are thus complementary to the Lamb wave modes. These two are the only wave types which can propagate with straight, infinite wave fronts in a plate as defined above.
Velocity dispersion
Lamb waves exhibit velocity dispersion; that is, their velocity of propagation c depends on the frequency (or wavelength), as well as on the elastic constants and density of the material. This phenomenon is central to the study and understanding of wave behavior in plates. Physically, the key parameter is the ratio of plate thickness d to wavelength . This ratio determines the effective stiffness of the plate and hence the velocity of the wave. In technological applications, a more practical parameter readily derived from this is used, namely the product of thickness and frequency: since for all waves
The relationship between velocity and frequency (or wavelength) is inherent in the characteristic equations. In the case of the plate, these equations are not simple and their solution requires numerical methods. 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 ViktorovViktorov, I. A. “Rayleigh and Lamb Waves: Physical Theory and Applications”, Plenum Press, New York, 1967. 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. The free "Dispersion Calculator" (DC) software allows computation of dispersion diagrams for isotropic plates and multilayered anisotropic specimens. Experimental waveforms observed in plates can be understood by interpretation with reference to the dispersion curves.
Dispersion curves - graphs that show relationships between wave velocity, wavelength and frequency in dispersive systems – can be presented in various forms. The form that gives the greatest insight into the underlying physics has (angular frequency) on the y-axis and k (wave number) on the x-axis. The form used by Viktorov, that brought Lamb waves into practical use, has wave velocity on the y-axis and , the thickness/wavelength ratio, on the x-axis. The most practical form of all, for which credit is due to J. and H. Krautkrämer as well as to Floyd Firestone (who, incidentally, coined the phrase "Lamb waves") has wave velocity on the y-axis and fd, the frequency-thickness product, on the x-axis.
Lamb's characteristic equations indicate the existence of two entire families of sinusoidal wave modes in infinite plates of width . This stands in contrast with the situation in unbounded media where there are just two wave modes, the longitudinal wave and the transverse or shear wave. As in Rayleigh waves which propagate along single free surfaces, the particle motion in Lamb waves is elliptical with its x and z components depending on the depth within the plate.This link shows a video of the particle motion. In one family of modes, the motion is symmetrical about the midthickness plane. In the other family it is antisymmetric. The phenomenon of velocity dispersion leads to a rich variety of experimentally observable waveforms when acoustic waves propagate in plates. It is the group velocity cg, not the above-mentioned phase velocity c or cp, that determines the modulations seen in the observed waveform. The appearance of the waveforms depends critically on the frequency range selected for observation. The flexural and extensional modes are relatively easy to recognize and this has been advocated as a technique of nondestructive testing.
Zero-order modes
The symmetrical and antisymmetric zero-order modes deserve special attention. These modes have "nascent frequencies" of zero. Thus they are the only modes that exist over the entire frequency spectrum from zero to indefinitely high frequencies. In the low frequency range (i.e. when the wavelength is greater than the plate thickness) these modes are often called the “extensional mode” and the “flexural mode" respectively, terms that describe the nature of the motion and the elastic stiffnesses that govern the velocities of propagation. The elliptical particle motion is mainly in the plane of the plate for the symmetrical, extensional mode and perpendicular to the plane of the plate for the antisymmetric, flexural mode. These characteristics change at higher frequencies.
These two modes are the most important because (a) they exist at all frequencies and (b) in most practical situations they carry more energy than the higher-order modes.
The zero-order symmetrical mode (designated S0) travels at the "plate velocity" in the low-frequency regime where it is properly called the "extensional mode". In this regime, the plate stretches in the direction of propagation and contracts correspondingly in the thickness direction. As the frequency increases and the wavelength becomes comparable with the plate thickness, curving of the plate starts to have a significant influence on its effective stiffness. The phase velocity drops smoothly while the group velocity drops somewhat precipitously towards a minimum. At higher frequencies yet, both the phase velocity and the group velocity converge towards the Rayleigh wave velocity – the phase velocity from above, and the group velocity from below.
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:
where is Poisson's ratio.
The zero-order antisymmetric mode (designated A0) is highly dispersive in the low frequency regime where it is properly called the "flexural mode" or the "bending mode". For very low frequencies (very thin plates) the phase and group velocities are both proportional to the square root of the frequency; the group velocity is twice the phase velocity. This simple relationship is a consequence of the stiffness/thickness relationship for thin plates in bending. At higher frequencies where the wavelength is no longer much greater than the plate thickness, these relationships break down. The phase velocity rises less and less quickly and converges towards the Rayleigh wave velocity in the high frequency limit. The group velocity passes through a maximum, a little faster than the shear wave velocity, when the wavelength is approximately equal to the plate thickness. It then converges, from above, to the Rayleigh wave velocity in the high frequency limit.
In experiments that allow both extensional and flexural modes to be excited and detected, the extensional mode often appears as a higher-velocity, lower-amplitude precursor to the flexural mode. The flexural mode is the more easily excited of the two and often carries most of the energy.
Higher-order modes
As the frequency is raised, the higher-order wave modes make their appearance in addition to the zero-order modes. Each higher-order mode is “born” at a resonant frequency of the plate, and exists only above that frequency. For example, in a inch (19mm) thick steel plate at a frequency of 200 kHz, the first four Lamb wave modes are present, and at 300 kHz, the first six. The first few higher-order modes can be distinctly observed under favorable experimental conditions. Under less than favorable conditions they overlap and can not be distinguished.
The higher-order Lamb modes are characterized by nodal planes within the plate, parallel to the plate surfaces. Each of these modes exists only above a certain frequency which can be called its "nascent frequency". There is no upper frequency limit for any of the modes. The nascent frequencies can be pictured as the resonant frequencies for longitudinal or shear waves propagating perpendicular to the plane of the plate, i.e.