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Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal (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. Their properties turned out to be quite complex. 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 has 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 nondestructive 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 materials〔Achenbach, 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 1917〔Lamb, 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: : and : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal (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. Their properties turned out to be quite complex. 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 has 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 nondestructive testing.The term Rayleigh–Lamb wavesRayleigh–Lamb waves redirects here --> 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:\xi = A_x f_x(z) e^ \quad \quad (1) :\zeta = A_z f_z(z) e^ \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 = - \frac\ \quad \quad \quad \quad (3)and :\frac = - \frac\ \quad \quad \quad \quad (4)where: \alpha^2 = \frac - k^2\quad \quad \text\quad\quad \beta^2 = \frac - k^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 ''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.」の詳細全文を読む スポンサード リンク
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