Anomalous Dispersion of the 𝑆₁ Lamb Mode

Abstract

The 𝑆₁ 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.

1. Introduction

The dispersion relation for symmetric Lamb modes propagating in an infinite isotropic plate of thickness 2ℎ is given by the well known Rayleigh-Lamb equation [1]: tan𝑝ℎtan𝑞ℎ=−4𝑝𝑞𝑘²𝑝²−𝑘²², (1) where 𝑝=𝜔²𝑐²_𝑇−𝑘²,𝑞=𝜔²𝑐²_𝐿−𝑘². (2)

In (1), 𝑐_𝑇 and 𝑐_𝐿, 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 𝜔𝑐=𝑘. (3)

If 𝑐 is plotted as a function of the frequency, the spectrum appears as in Figure 1, which depicts the spectrum for a steel plate with 𝑐_𝑇=3.24 km/s and 𝑐_𝐿=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 Figure 1 is the shape of the 𝑆₁ mode which has a turning point at 𝑘_𝑇=2.686, and the phase velocity becomes double valued for 𝑘_𝑇ℎ 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 of 𝑆₁ 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 with 𝜈=1/3 that is 𝑐_𝐿=2𝑐_𝑇, each of the modes 𝑆_3𝑛+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 the 𝑆₁ 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

Let 𝑘 and 𝑐, respectively, denote the wave number and phase speed of the mode. Define the dimensionless speed 𝑐/𝑐_𝑇 by 𝑦 and the dimensionless frequency 𝜔ℎ/𝑐_𝑇 by 𝑢.

Then 𝑝ℎ=ℎ𝜔²𝑐²_𝑇−𝑘²=ℎ𝑘𝑦²−1=ℎ𝑘𝑐/𝑐_𝑇𝑐/𝑐_𝑇𝑦²𝑢−1=𝑦𝑦²−1,𝑢𝑞ℎ=𝑦𝑦²𝜅²−1. (4)(5)

With respect to the variables 𝑢 and 𝑦, (1) becomes 𝑓=𝑦𝑢,𝑦₂−2²sin𝑦²𝑢−1𝑦cos𝑦²𝜅²𝑢−1𝑦+4𝑦²−1𝑦²𝜅²−1sin𝑦²𝜅²𝑢−1𝑦×cos𝑦²𝑢−1𝑦=0. (6)

In this section, we will consider 𝜅≠2. We will calculate the derivative 𝑑𝑦/𝑑𝑢 at two positions of the spectrum. We write 𝐷_𝑛,𝜅=𝑑𝑦𝑑𝑢_{𝑢=𝑢𝑛,𝑦=𝜅},𝐷_𝑛,𝐿=𝑑𝑦𝑑𝑢_{𝑢=𝑢𝑛,𝑦=𝐿≫𝜅}, (7) where the subscript 𝑛 refers to the mode under consideration. For 2/√3<𝜅<3.39 (−1<𝜈<0.452), we will show that 𝐷_𝑛,𝜅<0 for all modes. On the other hand, 𝐷_1,𝐿>0 for the 𝑆₁ mode while 𝐷_𝑛,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 Appendix A, we give expressions for 𝜕𝑓/𝜕𝑢 and 𝜕𝑓/𝜕𝑦. The derivative 𝑑𝑦/𝑑𝑢 is found as 𝑑𝑦𝑑𝑢=−𝜕𝑓/𝜕𝑢.𝜕𝑓/𝜕𝑦 (8)

When 𝑦=𝜅, (6) becomes 𝑢√sin𝜅²−1𝜅=0. (9) Therefore, 𝑢_𝑛=𝑛𝜋𝜅√𝜅².−1 (10)

Equation (10) shows that the line 𝑦=𝜅 intersects the modes at infinitely many points. At 𝑦=𝜅 and 𝑢=𝑢_𝑛, partial derivatives (A.1) become 𝜕𝑓𝜕𝑢_{𝑢=𝑢𝑛,𝑦=𝜅}=𝜅²−2²√𝜅²−1𝜅,𝜕𝑓𝜕𝑦_{𝑢=𝑢𝑛,𝑦=𝜅}=𝜅²−2²𝜅²√𝜅²√−1+4𝜅²−1𝜅²𝑢_𝑛. (11) Hence, 𝐷_𝑛,𝜅𝜅=−²−1^3/2𝜅²−2²𝜅²−2²+4𝜅².−1𝑛𝜋 (12)

It is clear that 𝐷_𝑛,𝜅<0 for all modes. Also |𝐷_𝑛,𝜅| becomes progressively smaller as 𝑛 increases and 𝐷_𝑛,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 find 𝐷_𝑛,𝐿. For 𝑦≫𝜅, the expressions for 𝜕𝑓/𝜕𝑢 and 𝜕𝑓/𝜕𝑦 are given by 𝜕𝑓𝜕𝑢≃𝑦⁴𝑢cos𝑢cos𝜅−1𝜅𝑢sin𝑢sin𝜅𝜕𝑓𝑢𝜕𝑦≃𝑢𝑦cos𝑢cos𝜅+4𝑦³𝑢cos𝜅+sin𝑢8𝑦𝜅𝑢cos𝑢sin𝜅𝑢−𝜅𝑢𝑦sin𝑢sin𝜅. (13)(14)

Thus, 𝑑𝑦𝑦𝑑𝑢≃−⁴−𝑦⁴/𝜅tan𝑢tan(𝑢/𝜅)𝑢𝑦+4𝑦³()().tan𝑢+8𝑦/𝜅tan𝑢/𝜅−𝜅𝑢𝑦tan𝑢tan𝑢/𝜅 (15)

For 𝑦≫𝜅, (6) gives 𝑦²()≃−4tan𝑢/𝜅.𝜅tan𝑢 (16)

Also, from (15) and (16), we have 𝐷_𝑛,𝐿𝑦≃−³+𝑦⁵/4tan²𝑢_𝑛𝑢_𝑛+𝑦²tan𝑢_𝑛2+𝜅²/4𝑢_𝑛tan𝑢_𝑛, (17) or equivalently 𝐷_𝑛,𝐿𝑦≃−³+𝑦⁵/4tan²𝑢_𝑛𝑢_𝑛−(4/𝜅)tan𝑢_𝑛/𝜅2+𝜅²/4𝑢_𝑛tan𝑢_𝑛. (18)

We have replaced 𝑢 by 𝑢_𝑛 in (17) and (18), since, for a fixed 𝑦, (16) yields infinitely many roots 𝑢_𝑛, 𝑛=1,2,3….

To fix ideas, we consider the case of a steel plate for which 𝜅=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 mode 𝑆₁, we have 𝐷_1,𝐿𝑦≃−³+𝑦⁵/4tan²(𝜅𝜋/2)(𝜅𝜋/2)+𝑦²tan(𝜅𝜋/2)2+𝜅²⋅()(./4𝜅𝜋/2tan𝜅𝜋/2) (19)

For 𝜅=1.83 and for large 𝑦,𝐷_1,𝐿>0.

For 2/√3<𝜅<2 and large𝑦, approximate values of 𝑢_𝑛 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 of 𝑢_𝑛 when 𝑦=20 for the first five modes.

We have seen previously that 𝐷_1,𝐿>0 for the 𝑆₁ mode. Now for the 𝑆₂ mode 𝑢₂≃𝜋 and from (18) we get 𝐷_2,𝐿𝑦≃−³()().𝜋−8/𝜅tan𝜋/𝜅 (20)

Here, 𝜅=1.83 and tan(𝜋/𝜅)<0; thus 𝐷_2,𝐿<0. In a similar fashion by successively using (17) and (18), we can show that 𝐷_𝑛,𝐿<0 for all 𝑛≥2.

From the previous, we conclude that for steel, the 𝑆₁ 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.

If 𝜅>2, the 𝑆₁ mode occurs, when 𝑢₁≃𝜋, and from (18), the 𝑆₁ mode will have positive slope as long as 8𝜋−𝜅𝜋tan𝜅<0, (21) or 2<𝜅<3.39 which corresponds to 1/3<𝜈<0.452.

Thus, we have established that the 𝑆₁ 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 Case 𝜅=2

The case 𝜅=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 pairs 𝑆₁-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.

With 𝜅=2, (6) becomes, for 𝑦≫1, 𝑦⁴𝑢cos2sin𝑢+𝑦²𝑢sin2cos𝑢=0, (22) which is satisfied for 𝑢_𝑛=2𝑛𝜋, 𝑛=1,2,3…. It is shown in Appendix B that a more accurate solution of (6) is 𝑢_𝑛1=𝑛𝜋2+𝑦². (23)

In addition to (23), (16) has roots given by −12𝑦²≃tan(𝑢/2)=1tan𝑢21−tan²𝑢2. (24)

Ignoring 1 in comparison with 𝑦², we get tan²𝑢2≃𝑦², (25) or cot²𝑢2≃1𝑦². (26) Let 𝑢2=(2𝑛+1)𝜋2+𝜖. (27) Equation (26) becomes cot²𝜋2≃1+𝜖𝑦², (28) or tan²1𝜖≃𝑦², (29) which leads to 1𝜖≃±𝑦. (30)

Thus, for large 𝑦, 𝑢₁=𝜋−2𝜖, 𝑢₂=𝜋+2𝜖, 𝑢₄=3𝜋−2𝜖, 𝑢₅=3𝜋+2𝜖, and so forth.

Since 𝜖→0 as 𝑦→∞, the modes 𝑢₁,𝑢₂;𝑢₄,𝑢₅;𝑢₇,𝑢₈;… appear to coalesce for large 𝑦. Now, with 𝜅=2, (18) gives 𝐷_1,𝐿>0, 𝐷_4,𝐿>0, and 𝐷_7,𝐿>0 while the slopes for all other modes are negative. This argument establishes the anomalous dispersion of the modes 𝑆_3𝑛+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 small 𝑦. 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.

Appendices

A. The Partial Derivatives

Expressions for the partial derivatives are as follows: 𝜕𝑓=𝑦𝜕𝑢₂−2²𝑦²𝑢−1cos𝑦²−1𝑦𝑢×cos𝑦²/𝜅²−1𝑦𝑦⁻¹+4𝑦²𝜅²−1𝑦²𝑢−1cos𝑦²−1𝑦𝑢×cos𝑦²/𝜅²−1𝑦𝑦⁻¹−𝑦²−2²𝑦²𝜅²𝑢−1sin𝑦²−1𝑦𝑢×sin𝑦²/𝜅²−1𝑦𝑦⁻¹−4𝑦²−1𝑦²𝜅²𝑢−1sin𝑦²−1𝑦𝑢×sin𝑦²/𝜅²−1𝑦𝑦⁻¹,𝜕𝑓=𝑦𝜕𝑦₂−2²𝑢𝑦²−𝑢−1𝑦²−1𝑦²𝑢×cos𝑦²−1𝑦𝑢cos𝑦²/𝜅²−1𝑦+4𝑦²−1𝑦²/𝜅²×𝑢−1𝜅²𝑦²/𝜅²−𝑢−1𝑦²/𝜅²−1𝑦²𝑢×cos𝑦²−1𝑦𝑢cos𝑦²/𝜅²−1𝑦+4𝑦𝑦²𝑢−2sin𝑦²−1𝑦𝑢cos𝑦²/𝜅²−1𝑦+4𝑦𝑦²𝑢−1cos𝑦²−1𝑦𝑢×sin𝑦²/𝜅²−1𝑦𝜅²𝑦²/𝜅²−1⁻¹+4𝑦𝑦²/𝜅²𝑢−1cos𝑦²−1𝑦𝑢×sin𝑦²/𝜅²−1𝑦𝑦²−1⁻¹−4𝑦²−1𝑦²/𝜅²𝑢−1𝑦²−𝑢−1𝑦²−1𝑦²𝑢×sin𝑦²−1𝑦𝑢sin𝑦²/𝜅²−1𝑦−𝑦²−2²𝑢𝜅²𝑦²/𝜅²−𝑢−1𝑦²/𝜅²−1𝑦²𝑢×sin𝑦²−1𝑦𝑢sin𝑦²/𝜅²−1𝑦. (A.1)

B. The Mode 𝑆_3𝑛 for 𝜅=2

For 𝑦≫1, 𝑦²−1/𝑦≃1−1/(2𝑦²) and (𝑦²/4)−1/𝑦≃1/2−1/𝑦².

Let 𝑢_𝑛≃2𝑛𝜋+(𝛼_𝑛/𝑦²) and ignore the terms of order 1/𝑦⁴ or higher. Then, cos𝑦²−1𝑦𝑢_𝑛≃1,cos𝑦²/4−1𝑦𝑢_𝑛≃−1,sin𝑦²−1𝑦𝑢_𝑛𝛼≃sin^𝑛−𝑛𝜋𝑦²≃𝛼_𝑛−𝑛𝜋𝑦²,sin𝑦²/4−1𝑦𝑢_𝑛𝛼≃∓sin^𝑛−2𝑛𝜋𝑦²𝛼≃∓^𝑛−2𝑛𝜋𝑦². (B.1) Putting these expressions in (6), we have −𝑦⁴𝛼_𝑛−𝑛𝜋𝑦²∓2𝛼_𝑛−2𝑛𝜋=0, (B.2) or 𝑦²±2𝛼_𝑛=𝑛𝜋𝑦²∓4𝑛𝜋, (B.3) or 𝛼_𝑛=𝑛𝜋𝑦²∓4𝑛𝜋𝑦²±2≃𝑛𝜋, (B.4) for large 𝑦.

This result shows that the modes 𝑆_3𝑛, 𝑛=1,2,3,…, intersect the line 𝑦=𝑦₀, for large 𝑦₀, at points slightly to the right of 𝑢_𝑛=2𝑛𝜋, 𝑛=1,2,3,….

Acknowledgment

Faiz Ahmad is grateful to the Higher Education Commission of Pakistan for financial support.