全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Anomalous Dispersion of the Lamb Mode

DOI: 10.1155/2013/903934

Full-Text   Cite this paper   Add to My Lib

Abstract:

The 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 is given by the well known Rayleigh-Lamb equation [1]: where In (1), and , respectively, denote the phase speeds of the transverse and longitudinal bulk waves in the material. Also, and , respectively, denote the frequency and the wave number of the mode. The phase velocity, , of a mode is given by If is plotted as a function of the frequency, the spectrum appears as in Figure 1, which depicts the spectrum for a steel plate with ?km/s and ?km/s. Figure 1: Symmetric Lamb modes on a steel plate ( ) showing phase velocity as a function of normalized frequency. The most striking feature in Figure 1 is the shape of the mode which has a turning point at , and the phase velocity becomes double valued for in . 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 mode was first noticed by Tolstoy and Usdin [9] in . In all isotropic materials with , only the mode has this “anomalous behavior” and other modes behave normally. We will call it the anomaly. An explanation of this peculiar shape of mode has posed a challenge since its discovery in . For the special case of a material with that is , each of the modes , , 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 , 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 mode and the first harmonic 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

References

[1]  J. D. Achenbach, Wave Propagation in Elastic Solids, chapter 6, North-Holland, Amsterdam, The Netherlands, 1980.
[2]  A. H. Meitzler, “Backward wave tranmission stress pulses in elstic cylinders and plates,” Journal of the Acoustical Society of America, vol. 38, pp. 835–842, 1965.
[3]  C. Prada, D. Clorennec, and D. Royer, “Local vibration of an elastic plate and zero-group velocity Lamb modes,” Journal of the Acoustical Society of America, vol. 124, no. 1, pp. 203–212, 2008.
[4]  K. Nishimiya, K. Mizutani, N. Wakatsuki, and K. Yamamoto, “Determination of 12 condition for fastest NGV of Lamb-Type waves under each density ratio of solid and liquid layers,” Acoustics 08 Paris, pp. 3613–3618.
[5]  J. Wolf, T. D. K. Nook, R. Kille, and W. G. Mayer, “Investigation of lamb waves having negative group velocity,” Journal of the Acoustical Society of America, vol. 83, pp. 122–126, 1988.
[6]  O. Balogun, T. W. Murray, and C. Prada, “Simulation and measurement of the optical excitation of the S1 zero group velocity Lamb wave resonance in plates,” Journal of Applied Physics, vol. 102, no. 6, Article ID 064914, 2007.
[7]  K. Negishi, “Negative group velocities of Lamb waves,” Journal of the Acoustical Society of America, vol. 64, no. 1, p. S63, 1978.
[8]  M. Germano, A. Alippi, A. Bettucci, and G. Mancuso, “Anomalous and negative reflection of Lamb waves in mode conversion,” Physical Review B, vol. 85, no. 1, Article ID 012102, 2012.
[9]  I. Tolstoy and E. Usdin, “Wave propagation in elastic plates, low and high mode dispersion,” Journal of the Acoustical Society of America, vol. 29, pp. 37–42, 1957.
[10]  M. F. Werby and H. überall, “The analysis and interpretation of some special properties of higher order symmetric Lamb waves: the case for plates,” Journal of the Acoustical Society of America, vol. 111, no. 6, pp. 2686–2691, 2002.
[11]  N. A. Whitaker Jr. and H. A. Haus, “Backward wave effect in acoustic scattering measurements,” IEEE Ultrasonics Symposium, pp. 891–894, 1983.
[12]  H. Uberall, B. Hosten, M. Deschamps, and A. Gerard, “Repulsion of phase-velocity dispersion curves and the nature of plate vibrations,” Journal of the Acoustical Society of America, vol. 96, no. 2, pp. 908–917, 1994.
[13]  T. Hussain and F. Ahmad, “Lamb modes with multiple zero-group velocity points in an orthotropic plate,” Journal of the Acoustical Society of America, vol. 32, pp. 641–645, 2012.
[14]  R. D. Mindlin, An Introduction To the Mathematical Theory of Vibrations of Elastic Plates, Monograph. Sec. 2.11, U.S. Army Signal Corps Eng. Lab., Ft Monmouth, NJ, USA, 1995, Edited by, J. Yang, World Scientific, Singapore, 2006.

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133