# Anomalous Dispersion of theS1 Lamb Mode

Authors: Faiz Ahmad; Takasar Hussain
Journal: Advances in Acoustics and Vibration
 (2013)

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## Abstract

TheS1 mode of the Lamb spectrum of an isotropic plate exhibits negative group velocity in a narrow frequency domain. This anomalous behavior is explained analytically by examining the slope of each mode first in its initial state and then near its turning points.


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## Body

## 1. Introduction

The dispersion relation for symmetric Lamb modes propagating in an infinite isotropic plate of thickness2h is given by the well known Rayleigh-Lamb equation [1]:
(1)tan(ph)tan(qh)=-4pqk2(p2-k2)2,
where
(2)p=ω2cT2-k2,q=ω2cL2-k2.In (1), cT and cL, respectively, denote the phase speeds of the transverse and longitudinal bulk waves in the material. Also, ω and k, respectively, denote the frequency and the wave number of the mode. The phase velocity, c, of a mode is given by
(3)c=ωk.Ifc is plotted as a function of the frequency, the spectrum appears as in Figure 1, which depicts the spectrum for a steel plate with cT=3.24 km/s and cL=5.95 km/s.Figure 1
Symmetric Lamb modes on a steel plate (κ=1.83) showing phase velocity as a function of normalized frequency.The most striking feature in Figure1 is the shape of the S1 mode which has a turning point at kT=2.686, and the phase velocity becomes double valued for kTh in [2.686,2.873]. This phenomenon of negative group velocity is of great technical significance and has been observed in a large number of experiments [2–8].The afore mentioned feature ofS1 mode was first noticed by Tolstoy and Usdin [9] in 1957. In all isotropic materials with κ≠2(ν≠1/3), only the S1 mode has this “anomalous behavior” and other modes behave normally. We will call it the S1 anomaly. An explanation of this peculiar shape of mode S1 has posed a challenge since its discovery in 1957.For the special case of a material withν=1/3 that is cL=2cT, each of the modes S3n+1, n=0,1,2,3,…, exhibits anomalous behavior [10]. Anomalous pairs of modes may also occur for certain special values of the Poisson ratio. We will call it the pair anomaly.Although (1) governs the behavior of all modes, anomalous or otherwise, no simple theory seems to exist which should provide a satisfactory explanation of why certain modes in the spectrum should possess a bulge while others proceed in a normal manner. However, certain physical explanations of this phenomenon exist. In 1983, Whitaker and Haus [11] noted the fact that “propagation of waves with dispersion of this sort has been experimentally verified [2] but the reason for their appearance is not well understood.” They used the coupled mode theory to argue that, when the fundamental P mode and the first harmonic S mode are nearly degenerate at cutoff, a coupling effect can occur at the boundaries. Uberall et al. [12] hypothesize that “one observes a repulsion phenomenon between neighboring dispersion curves similar to that encountered in atomic physics for quasidegenerate energy levels of atoms when combining into molecules.” Prada et al. [3] express the same view in the words, “this phenomenon leads to a strong repulsion between the dispersion curves of the neighbouring modes.”It is clear that all of these authors focused on the pair anomaly only because, in theS1 anomaly, the S1 mode remains distinct and it does not coincide with any other mode at cutoff. Mode repulsion cannot explain S1 anomaly. To the best of our knowledge, the S1 anomaly still remains an unsolved mystery. Recently Hussain and Ahmad [13] considered ZGV points in the spectrum of Lamb modes in compressible orthotropic plate. It was found that, in addition to modes with a single ZGV point, a large number of modes exist with multiple such points.In this paper we will examine Rayleigh-Lamb spectrum for the symmetric modes of an isotropic material. We will analyze (1) and derive mathematical expressions which will explain both types of anomalies.


## 2. The Mode Spectrum

Letk and c, respectively, denote the wave number and phase speed of the mode. Define the dimensionless speed c/cT by y and the dimensionless frequency ωh/cT by u.Then(4)ph=hω2cT2-k2=hky2-1=hkc/cTc/cTy2-1=uyy2-1,(5)qh=uyy2κ2-1.With respect to the variablesu and y, (1) becomes
(6)f(u,y)=(y2-2)2sin(y2-1uy)cos(y2κ2-1uy)+4y2-1y2κ2-1sin(y2κ2-1uy)×cos(y2-1uy)=0.In this section, we will considerκ≠2. We will calculate the derivative dy/du at two positions of the spectrum. We write
(7)Dn,κ=dydu|u=un,y=κ,Dn,L=dydu|u=un,y=L≫κ,
where the subscript n refers to the mode under consideration. For 2/3<κ<3.39 (-1<ν<0.452), we will show that Dn,κ<0 for all modes. On the other hand, D1,L>0 for the S1 mode while Dn,L<0 for all other modes. Since the derivative for the S1 mode changes from positive to negative, it must exhibit a bulge before y=κ. No other mode undergoes a reversal of the slope; hence, all other modes continue their downward journey until they asymptotically approach the line y=1.In AppendixA, we give expressions for ∂f/∂u and ∂f/∂y. The derivative dy/du is found as
(8)dydu=-∂f/∂u∂f/∂y.Wheny=κ, (6) becomes
(9)sin(uκ2-1κ)=0.
Therefore,
(10)un=nπκκ2-1.Equation (10) shows that the line y=κ intersects the modes at infinitely many points. At y=κ and u=un, partial derivatives (A.1) become
(11)∂f∂u|u=un,y=κ=(κ2-2)2κ2-1κ,∂f∂y|u=un,y=κ=[(κ2-2)2κ2κ2-1+4κ2-1κ2]un.
Hence,
(12)Dn,κ=-(κ2-1)3/2(κ2-2)2[(κ2-2)2+4(κ2-1)]nπ.It is clear thatDn,κ<0 for all modes. Also |Dn,κ| becomes progressively smaller as n increases and Dn,κ→0 as n→∞. This means that, for y=κ, there is a plateau region and this plateau is flatter for higher modes.Next, we findDn,L. For y≫κ, the expressions for ∂f/∂u and ∂f/∂y are given by
(13)∂f∂u≃y4[cosucosuκ-1κsinusinuκ](14)∂f∂y≃uycosucosuκ+4y3cosuκsinu+8yκcosusinuκ-κuysinusinuκ.Thus,(15)dydu≃-y4-(y4/κ)tanutan(u/κ)uy+4y3tanu+(8y/κ)tan(u/κ)-κuytanutan(u/κ).Fory≫κ, (6) gives
(16)y2≃-4tan(u/κ)κtanu.Also, from (15) and (16), we have
(17)Dn,L≃-y3+(y5/4)tan2unun+y2tanun(2+(κ2/4)untanun),
or equivalently
(18)Dn,L≃-y3+(y5/4)tan2unun-(4/κ)tan(un/κ)(2+(κ2/4)untanun).We have replacedu by un in (17) and (18), since, for a fixed y, (16) yields infinitely many roots un, n=1,2,3….To fix ideas, we consider the case of a steel plate for whichκ=1.83. The general case follows on similar lines.From (16), we see that, for the S1 mode, u1/κ should be slightly less than π/2, so that u1/κ is in the first quadrant and the corresponding u1 is in the second quadrant to yield a large positive y2.Hence, for the modeS1, we have
(19)D1,L≃-y3+(y5/4)tan2(κπ/2)(κπ/2)+y2tan(κπ/2)(2+(κ2/4)·(κπ/2)tan(κπ/2)).Forκ=1.83 and for large y,D1,L>0.For2/3<κ<2 and largey, approximate values of un from (16) for the first few modes are κπ/2, π, 3κπ/2, 2π, and 5κπ/2. For the steel plate, these values are compared in Table 1 with the exact values found from (6) when y=20.Table 1
Approximate and exact values ofun when y=20 for the first five modes.
Modes
Approximateun
Exact value aty=20
S
1
2.875
2.856
S
2
3.142
3.180
S
3
6.283
6.289
S
4
8.624
8.650
S
5
9.425
9.449We have seen previously thatD1,L>0 for the S1 mode. Now for the S2 mode u2≃π and from (18) we get
(20)D2,L≃-y3π-(8/κ)tan(π/κ).Here,κ=1.83 and tan(π/κ)<0; thus D2,L<0. In a similar fashion by successively using (17) and (18), we can show that Dn,L<0 for all n≥2.From the previous, we conclude that for steel, theS1 is the only mode which reverses its slope while going from large values of y to y=κ. The previous analysis applies to all materials with κ<2.Ifκ>2, the S1 mode occurs, when u1≃π, and from (18), the S1 mode will have positive slope as long as
(21)π-8κtanπκ<0,
or 2<κ<3.39 which corresponds to 1/3<ν<0.452.Thus, we have established that theS1 mode will be anomalous for all materials falling in the range 2/3<κ<3.39. Since the slope of S1 becomes negative for all y when κ>3.39, the mode will lose its anomalous character beyond κ=3.39 which corresponds to ν=0.452.


## 3. The Exceptional Caseκ=2

The caseκ=2 merits special treatment. The spectrum of symmetric modes appears as in Figure 2.Figure 2
Symmetric Lamb modes on a nickel plate (κ=2).The pairsS1-S2, S4-S5, S7-S8… appear to merge for large y and then bifurcate as they descend to lower values of the phase speed. On the other hand, S3,S6,S9,… appear to behave normally.This phenomenon was first reported by Mindlin [14]. Each of the modes S1,S4,S7,… shows anomalous dispersion.Withκ=2, (6) becomes, for y≫1,
(22)y4cosu2sinu+y2sinu2cosu=0,
which is satisfied for un=2nπ, n=1,2,3…. It is shown in Appendix B that a more accurate solution of (6) is
(23)un=nπ(2+1y2).In addition to (23), (16) has roots given by
(24)-12y2≃tan(u/2)tanu=12(1-tan2u2).Ignoring1 in comparison with y2, we get
(25)tan2u2≃y2,
or
(26)cot2u2≃1y2.
Let
(27)u2=(2n+1)π2+ϵ.
Equation (26) becomes
(28)cot2(π2+ϵ)≃1y2,
or
(29)tan2ϵ≃1y2,
which leads to
(30)ϵ≃±1y.Thus, for largey, u1=π-2ϵ, u2=π+2ϵ, u4=3π-2ϵ, u5=3π+2ϵ, and so forth.Sinceϵ→0 as y→∞, the modes u1,u2;u4,u5;u7,u8;… appear to coalesce for large y. Now, with κ=2, (18) gives D1,L>0, D4,L>0, and D7,L>0 while the slopes for all other modes are negative. This argument establishes the anomalous dispersion of the modes S3n+1, n=0,1,2,3… As u3,u6,u9,… occur slightly above 2π,4π,6π,…, (17) gives a negative value for the slope of each of these modes. Hence, these modes behave in a normal manner.


## 4. Conclusion

We have explained analytically the anomalous behavior of Lamb modes for an isotropic material by looking at the slope of each mode for large as well as smally. For small y, slope is found at y=κ. This simple technique explains, in an analytic manner, theoretical results given by several authors about the anomalous dispersion of the S1 mode.



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*Source: 903934-2013-08-06.xml*