Lamb wave

The lamb wave (British: Lamb waves) is a kind of the elastic wave propagation mode to propagate the whole sheet of the elastic body which is homogeneous equal one. The speed dispersibility that phase velocity changes by frequency and innumerable propagation mode have the existing polymodal nature. Dispersion relations were led in 1917 by Horace Lamb [¹].

As for the lamb wave, the plane of vibration points to the thing which is perpendicular to the board surface in a guide wave propagating the whole board which is homogeneous equal one. In addition, a plane of vibration is called a board wave by the SH wave by the parallel thing for the board surface.

Table of contents

Basic equation

Dispersion expression of relations

I assume that the length of the perpendicular direction is infinite for a propagation direction. The dispersion relations (relations of angular frequency ω and cycle k) of the lamb wave that primary wave speed c_L, one of side wave speed c_T propagate the board of the one elastic body for thickness 2h put during a vacuum then are expressed as follows. When I assume the wavelength for the propagation direction of the lamb wave λ, cycle k is quantity expressed in k=λ/(2π) here.

${\displaystyle \Omega _{\rm {S}}(\omega ,k)\equiv \left(q^{2}-k^{2}\right)^{2}\left({\frac {\sin(qh)}{q}}\right)\cos(ph)+4k^{2}p\sin(ph)\cos(qh)=0,}$

${\displaystyle \Omega _{\rm {A}}(\omega ,k)\equiv \left(q^{2}-k^{2}\right)^{2}\left({\frac {\sin(ph)}{p}}\right)\cos(qh)+4k^{2}q\sin(qh)\cos(ph)=0.}$

This expression is called Reilly = lamb frequency equation in particular. But p,q in a ceremony expresses the cycle ingredient of the direction of a primary wave and the side wave out of the plane each,

${\displaystyle p={\sqrt {\left({\frac {\omega }{c_{\rm {L}}}}\right)^{2}-k^{2}}},}$

${\displaystyle q={\sqrt {\left({\frac {\omega }{c_{\rm {T}}}}\right)^{2}-k^{2}}},}$

I take であり, a real number or a purely imaginary number. The Reilly = lamb frequency equation has cycle k of a real number or the complex number in a solution for angular frequency ω of real number generally. In the case of a real number, k is called propagation mode by the lamb wave to be able to propagate a long distance. Because the amplitude damps according to propagation exponentially, in the case of complex number or a purely imaginary number, k becomes the non-propagation mode. It is all a lamb wave in the meaning to satisfy Reilly = lamb frequency equation, but may call lamb wave propagation mode with a lamb wave depending on context. The Reilly = lamb frequency equation includes a dispersion expression of relations of the Rayleigh wave as an infinite limit of the board thickness or the frequency. I show the property (speed dispersibility, the polymodal nature) as the lamb wave which board thickness in particular stated above for the wavelength of the elastic wave at the same level conspicuously.

Displacement distribution and symmetry, antisymmetric mode

I assume one point on the vertical plane the origin in a board, and the displacement place of the lamb wave is the following expressions, and a propagation direction is expressed using fixed number A, B, C, D when I assume x direction, a board thickness direction z.

${\displaystyle u_{x}=\operatorname {Re} \left\lbrack \left\lbrace kA\cos(pz)+qB\cos(qz)+kC\sin(pz)+qD\sin(qz)\right\rbrace \exp \left\lbrace i(kx-\omega t)\right\rbrace \right\rbrack ,}$

${\displaystyle u_{z}=\operatorname {Re} \left\lbrack i\left\lbrace pA\sin(pz)-kB\sin(qz)-pC\cos(pz)+kD\cos(qz)\right\rbrace \exp \left\lbrace i(kx-\omega t)\right\rbrace \right\rbrack ,}$

A,B,C,D is any fixed number to express the amplitude,

${\displaystyle {\begin{pmatrix}2kp\sin(ph)&(q^{2}-k^{2})\sin(qh)\\(q^{2}-k^{2})\cos(ph)&-2kq\cos(qh)\end{pmatrix}}{\begin{pmatrix}A\\B\end{pmatrix}}=0,}$

${\displaystyle {\begin{pmatrix}2kp\cos(ph)&(q^{2}-k^{2})\cos(qh)\\(q^{2}-k^{2})\sin(ph)&-2kq\sin(qh)\end{pmatrix}}{\begin{pmatrix}C\\D\end{pmatrix}}=0,}$

I satisfy を. When I satisfy Ω_S=0 in the upper dispersion expression of relations, I can take the solution except the own lucid explanation that A,B becomes 0 together (either or both sides can take non-zero A,B). The displacement ground is called a symmetry mode by the lamb wave propagation mode having such displacement ground for an aspect of z=0 in particular then to become symmetric.

Similarly, I can take the solution except the own lucid explanation that C,D becomes 0 together when I satisfy Ω_A=0 (either or both sides can take non-zero C,D). The displacement ground is called a particularly antisymmetric mode by the lamb wave propagation mode having such displacement ground for an aspect of z=0 then to become antisymmetric.

Phase velocity, group speed and dispersion curve

Phase velocity c_p of the lamb wave is referred to c_p=ω/k. For example, the speed that phase in the surface goes ahead for a propagation direction agrees in this phase velocity c_p when a lamb wave of the single frequency propagates a sheet. In addition, I can define c_g= ∂ ω/ ∂ k by considering angular frequency ω of the lamb wave to be function ω(k) of cycle k. This c_g is called the group speed of the lamb wave, and the wave packet of the lamb wave copes with speed to propagate.

A dispersion curve about the phase velocity. The blue or red line shows a symmetric and antisymmetric mode each. In addition, a solid line and the dotted line show two kinds of different dispersion curves of materials of Poisson ratio σ.

I take frequency on the cross axle, and call the figure which took phase velocity or the group speed in a vertical axis with a dispersion curve and play an important role in knowing the property of the lamb wave. The figure shows a dispersion curve about the phase velocity. A cross axle, the vertical axis are normalized by dividing angular frequency ω and board thickness d(=2h), phase velocity c_p (in a figure v) each in side wave speed c_T (in a figure v_s). The blue or red line shows a symmetric and antisymmetric mode each. In addition, a solid line and the dotted line show two kinds of different dispersion curves of materials of the Poisson ratio.

Property

When I activate an elastic wave from a trembler of the limited size, I spread in the direction except the propagation distance direction. Therefore, the elastic wave damps according to distance generally. Because the energy that the lamb wave is dynamic is shut in in a board, the decrement by this effect is lower than a primary wave and a side wave propagating an infinite medium. In addition, this is a general characteristic of the guide wave having a property to propagate along a border.

The lamb wave has strong speed dispersibility generally unless it is special. Therefore, a wave pattern changes according to propagation when I activate the wave which is broadband. In addition, plural modes can exist at the single frequency.

Extreme

The zeroth symmetric antisymmetric mode satisfies Reilly = lamb frequency equation at any frequency and becomes the propagation mode. The phase velocity and group of the zeroth modes speed accords at propagation speed of the Rayleigh wave in an infinite limit of frequency f or board thickness d(=2h). In addition, the phase velocity of the antisymmetric mode and the group speed agree at the propagation speed of the side wave in an infinite limit of frequency f or board thickness d(=2h) the primary above-mentioned symmetry.

The antisymmetric mode meets Reilly = lamb frequency equation only in higher than a certain frequency the primary above-mentioned symmetry. Therefore, among solutions satisfying Reilly = lamb frequency equation, I call the frequency in the limit of k → 0 of cycle k cut-off frequency [²]. Because the propagation mode of the lamb wave increases whenever frequency exceeds this cut-off frequency (quite), it is the frequency that is important in analysis in practical use. Mode symmetric at the limit of k → 0, angular frequency ω_cutoff of the antisymmetric mode each,

${\displaystyle \cos \left({\frac {\omega _{\rm {cutoff}}h}{c_{\rm {L}}}}\right)\sin \left({\frac {\omega _{\rm {cutoff}}h}{c_{\rm {T}}}}\right)=0,}$

${\displaystyle \sin \left({\frac {\omega _{\rm {cutoff}}h}{c_{\rm {L}}}}\right)\cos \left({\frac {\omega _{\rm {cutoff}}h}{c_{\rm {T}}}}\right)=0,}$

I satisfy を. In addition, cut-off frequency ω_cutoff may not become the smallest frequency in this lamb wave propagation mode closely because the cut-off frequency in a certain mode is a limit of k → 0 to the last. In other words, ω_cr <ωc_utoff and the case that it is exist when I assume the smallest frequency ω_cr in same lamb wave propagation mode.

References

1. ^ Lamb, H. "On Waves in an Elastic Plate." Proc. Roy. Soc. London, Ser. A 93, 114–128, 1917.
2. ^ Graff, K. F. "Wave Motion in Elastic Solids," "Chapter 8 Wave propagation in plates and rods," Dover, New York, 1975

• Viktorov, I. A. "Rayleigh and Lamb Waves: Physical Theory and Applications," Plenum Press, New York, 1967.
• Rose, J. L., "Ultrasonic Waves in Solid Media," Cambridge University Press, 1999.

It is acquired by "https://ja.wikipedia.org/w/index.php?title= lamb wave &oldid=58959766"

This article is taken from the Japanese Wikipedia Lamb wave

This article is distributed by cc-by-sa or GFDL license in accordance with the provisions of Wikipedia.

Wikipedia and Tranpedia does not guarantee the accuracy of this document. See our disclaimer for more information.

In addition, Tranpedia is simply not responsible for any show is only by translating the writings of foreign licenses that are compatible with CC-BY-SA license information.