Lamb waves

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.

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. \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 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: \frac{\tanh(\beta d / 2)} {\tanh(\alpha d / 2)} = \frac {4 \alpha \beta k^2} {(k^2 + \beta^2)^2}\ \quad \quad \quad \quad (3) for symmetric modes and \frac{\tanh(\beta d / 2)} {\tanh(\alpha d / 2)} = \frac {(k^2 + \beta^2)^2} {4 \alpha \beta k^2}\ \quad \quad \quad \quad (4) for antisymmetric modes, where \alpha^2 = k^2-\frac{\omega^2}{c_l^2} \quad \quad \text{and}\quad\quad \beta^2 = k^2-\frac{\omega^2}{c_t^2}. 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 c p = fλ = ω/k, and the group velocity c g = dω/dk, as functions of d/λ or fd. c l and c t 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. \frac {a_z}{a_x} = \frac{\pi \nu}{(1 - \nu)} . \frac{d}{ \lambda} where \nu is Poisson's ratio. The zero-order antisymmetric mode (designated A 0 ) 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. d = \frac{n \lambda}{2} \quad \quad \text{or} \quad \quad f = \frac{nc}{2d} where n is any positive integer. Here c can be either the longitudinal wave velocity or the shear wave velocity, and for each resulting set of resonances the corresponding Lamb wave modes are alternately symmetrical and antisymmetric. The interplay of these two sets results in a pattern of nascent frequencies that at first glance seems irregular. For example, in a 3/4 inch (19mm) thick steel plate having longitudinal and shear velocities of 5890 m/s and 3260 m/s respectively, the nascent frequencies of the antisymmetric modes A 1 and A 2 are 86 kHz and 310 kHz respectively, while the nascent frequencies of the symmetric modes S 1 , S 2 and S 3 are 155 kHz, 172 kHz and 343 kHz respectively. At its nascent frequency, each of these modes has an infinite phase velocity and a group velocity of zero. In the high frequency limit, the phase and group velocities of all these modes converge to the shear wave velocity. Because of these convergences, the Rayleigh and shear velocities (which are very close to one another) are of major importance in thick plates. 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. These components are in quadrature, i.e. they have a 90° phase difference. The relative magnitude of the components is a function of frequency. 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 c t or for symmetric modes only c l , respectively. These directionality considerations are important when considering the radiation of acoustic energy from plates into adjacent fluids. The particle motion is also entirely perpendicular or entirely parallel to the plane of the plate, at a mode's nascent frequency. Close to the nascent frequencies of modes corresponding to longitudinal-wave resonances of the plate, their particle motion will be almost entirely perpendicular to the plane of the plate; and near the shear-wave resonances, parallel. J. and H. Krautkrämer have pointed out that Lamb waves can be conceived as a system of longitudinal and shear waves propagating at suitable angles across and along the plate. These waves reflect and mode-convert and combine to produce a sustained, coherent wave pattern. For this coherent wave pattern to be formed, the plate thickness has to be just right relative to the angles of propagation and wavelengths of the underlying longitudinal and shear waves; this requirement leads to the velocity dispersion relationships.

See also

Acoustics Acoustic wave Wave equation Waveguide Waveguide (acoustics) Waveguide (electromagnetism)

References

Rose, J.L.; "Ultrasonic Waves in Solid Media," Cambridge University Press, 1999.

External links

Modes of Sound Wave Propagation at NDT Resource Center Lamb wave in Nondestructive Testing Encyclopedia Lamb Wave Analysis of Acousto-Ultrasonic Signals in Plate by Liu Zhenqing: an article which includes the complete Lamb wave equations.

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