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Geofísica internacional

versión On-line ISSN 2954-436Xversión impresa ISSN 0016-7169

Geofís. Intl vol.59 no.2 Ciudad de México abr./jun. 2020  Epub 02-Oct-2020

 

Original papers

Short notes

May Rayleigh waves propagate with group- and phase-velocities of opposite sign in the valley of Mexico city?

Peter G. Malischewsky1  * 

Thomas Forbriger2 

1Institute of Geosciences Friedrich-Schiller University Jena

2Karlsruhe Instiute of Technology 76187 Karlsruhe, Germany


Abstract

On the basis of the group-velocity formula the occurrence of negative group velocities of Rayleigh waves is theoretically discussed for simplified models of Mexico City's basin by using results of a preceding article of the authors.

Key words: Rayleigh waves; negative group velocities; Valley of Mexico City

Resumen

Con base a la fórmula que determina la velocidad de grupo de las ondas sísmicas y utilizando resultados previamente publicados por los autores, se discute en forma teórica la ocurrencia de velocidades de grupo negativas de ondas de Rayleigh en modelos simplificados de la cuenca del Valle de México.

Palabras clave: Ondas de Rayleigh; velocidades del grupo negativas; Valle de México

Introduction

This short note is a complement of the article about unusual, equivocal Rayleigh dispersion curves by Malischewsky et al. (2017) in the Special Volume on the 30TH Anniversary of the 19 September 1985 Earthquake of Geofisica Internacional. It was not demonstrated, that the strange dispersion curves presented there lead to negative group velocities, but this is indeed the case as shown below. So this short comment in some way serves the popularization and understanding of negative group velocities, well-known in many fields of physics and techniques [see e. g. Nedopasov et al. (2017), Tamm et al.. (2017), Gérardin et al.. (2019)], in seismology as well [see Lysmer (1970), Forbriger (2017)] . Let us point out that in literature on normal mode theory solutions to the boundary value problem usually are discussed without an explicit source. The sign of propagation velocity then is not defined with respect to a source of wave energy. The term "negative group velocity" thus actually means that "group velocity has the opposite sign of phase velocity" and is often coupled with the conception of "backward waves" in literature. It is not the intention of this short comment to shed light onto this connection. We only note, that with Shevchenko (2007) there are three definitions for forward and backward waves, where the classical one is that a wave is considered a forward (backward) wave if the directions of its phase and group velocities are the same (opposite).

Group velocities

The formula for the group velocity of waves [see e. g. Malischewsky (2018).] is

UF=C2C-FdC/dF (1)

with dimensionless phase velocity C and dimensionless frequency F, which are defined depending on the model under consideration. Already Lamb (1904) has pointed out that it is possible for the group velocity to be of opposite sign with respect to phase velocity. He discusses the solution of the wave equation in the presence of a source of energy and points out that wave groups always propagate away from the source, while the complete wave train is actually a superposition of harmonic waves with negative phase velocity. Having a look on formula (1) U(F) and C are of opposite sign only for anomalous dispersion, i. e.

dCdF>0 (2)

with the additional condition

dCdF>CF (3)

which is a differential inequality.

The dispersion curves in Fig. 1 of Malischewsky et al. (2017) concern the first higher Rayleigh mode in the simple LFB (Layer with Fixed Bottom) - model for the Poisson ratios ν = 0.37 and 0.44, respectively. It can be demonstrated that both conditions (2) and (3) are fulfilled for specific frequency ranges, and the occurrence of negative group velocities is obvious. By using eq. (1) and numerical differentiation, the group velocity U is obtained and is presented together with the former phase velocity curves C in Fig. 1. The negative branches of group velocity occur in the frequency ranges

F Є (0.53, 0.55) for v = 0.37 and

F Є (0.64, 0.75) for v = 0.44 .

Figure 1 Phase (thin) and group (thick) velocity curves for Poisson ratios v=0.37 and v=0.44, respectively; the ZGV-points with lower frequencies are marked with black points 

These pictures are very similar to those ones obtained by Negishi (1987) in Lamb-wave context.

Obviously, a necessary precondition for negative group velocities is the existence of so-called ZGV - points (Zero Group Velocity points, see Fig. 1). The derivative d C / d F becomes infinite for the phase-velocity curve at these points. A marginal note: Condition (2) can be easily fulfilled for the fundamental Rayleigh mode of the LOH-model (Layer Over Halfspace) by using the results of Tuan et al. (2008) and specifying the parameters for which the dispersion curve starts with d C/dF > 0 for zero frequency. However, in absence of a ZGV-point, condition (3) is not fulfilled, and a negative group velocity does not occur.

Following Forbriger et al. (2020) these effects described here for the first higher mode occur for the fundamental mode in certain more complex models as well.

It is yet unclear, what the role of these special waves could be in the seismic regime of Mexico City's valley, but their pure possible existence is scientifically interesting enough to be mentioned and to position it into a larger context.

Let us come back to the title question by affirming it: If any facts are theoretically possible, the nature usually realizes them. Successful predictions is what we expect from a consistent theory. The most prominent example of recent years is the confirmation of the existence of gravitational waves after Einstein's prediction from 1916. For the phenomenon discussed here, the seismological observation has not been made yet. However, on laboratory scale the phenomenon is confirmed by experiment [Wolf et al. (1988), Cès et al. (2011)].

Acknowledgements

PM devotes this comment to the memory of his good friend and former colleague Dr. Dieter Freund (1943-2018), who taught him to look cheerfully at the life in good and bad times.

References

Cès, M., Clorennec, D. Royer, D. Prada, C., 2011, Thin layer thickness measurements by zero group velocity Lamb mode resonances, Rev. Sci. Instrum., 82, 114902, doi: 10.1063/1.3660182. [ Links ]

Forbriger, T., 2017, Surface waves - benefits and pitfalls for seismic exploration (in German), In: DGG-Kolloquium. Neue Entwicklungen in der Angewandten Seismik. Ed.: Deutsche Geophysikalische Gesellschaft. Mitteilungen der Deutschen Geophysikalischen Gesellschaft, Sonderband I/2017, ISSN 0947-1944, 3-19. http://dx.doi.org/10.S44S/IR/1000068706 [ Links ]

Forbriger, T., Gao, L., Malischewsky, P., Ohrnberger, M., Pan, Y., 2020, A single Rayleigh mode may exist with multiple values of phase-velocity at one frequency, Geophys. J. Int., ggaa123. (doi: 10.1093/gji/ggaa123). [ Links ]

Gérardin, B., Laurent, J., Legrand, F., Prada, C. , Aubry, A., 2019, Negative reflection of elastic guided waves in chaotic and random scattering media, Scientific Reports, 9, 1-8. [ Links ]

Lamb, H., 1904, On group-velocity, Proc. London Math. Soc., s2-1.1, 473-479. [ Links ]

Lysmer, J., 1970, Lumped mass method for Rayleigh waves, Bull. Seism. Soc. Am., 60, 89-104. [ Links ]

Malischewsky, P. G., Forbriger, T. , Lomnitz, C., 2017, Unusual equivocal Rayleigh-dispersion curves for simple models taking into account the special propagation conditions in the valley of Mexico City (CDMX) - Preliminary results, Geofísica Internacional, 56, 7-12. [ Links ]

Malischewsky, P. G. , 2018, Surface waves, In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg, 1-9. [ Links ]

Nedopasov, I. A., Mozhaev, V. G., Kuznetsova, I. E., 2017, Unusual energy properties of leaky backward Lamb waves in a submerged plate, Ultrasonics, 77, 95-99. [ Links ]

Negishi, K., 1987, Existence of negative group velocities in Lamb waves, Jpn. J. Appl. Phys., 26-1, 171-173. [ Links ]

Shevchenko, V. V., 2007, Forward and backward waves: three definitions and applicability, Physics-Uspekhi, 50, 287 292. [ Links ]

Tamm, K., Peets, T., Engelbrecht, J., Kartoffelev, D., 2017, Negative group velocity in solids, Wave Motion, 71, 127 138. [ Links ]

Tran Thanh Tuan, Malischewsky, P. G., Scherbaum, F., Ohrnberger, M. , 2008, Dispersion of zero-frequency Rayleigh waves in an isotropic model 'Layer over half-space', Geophys. J. Int. , 175, 537-540. [ Links ]

Wolf, J., Ngoc, T. D. K., Kille, R., Mayer, W. G., 1988, Investigation of Lamb waves having a negative group velocity. J. Acoust. Soc. Am., 83, 122-126. [ Links ]

Received: April 16, 2019; Accepted: January 14, 2020; Published: April 01, 2020

*Corresponding authorp.mali@ uni-jena.de

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