The uniqueness problem of Rayleigh wave in Kelvin viscoelastic
half-space and possible method to solve the problem
Cheng Hua
Department of Aeronautics and Astronautics, Fudan University, China
huacheng@fudan.edu.cn
Abstract: Waves that propagate in elastic solid can be divided into two maincategories: body waves and surface waves. Two and only two types of body waves inan unbounded solid, namely longitudinal wave (P-wave) and shear waves (SV-waveand SH-wave), can be propagated independently. Rayleigh wave is one kind of surface
waves which propagates along the free boundary and decays exponentially away fromthe boundary. Rayleigh wave is essentially the formation of interference on the surfaceof the medium of P-wave and SV-wave. Therefore, it was firstly introduced as asolution of the free boundary problem for an elastic half space by Lord Rayleigh (1885)[1], who summed it up as the simultaneous solution of the equations of P-wave andSV-wave. Yet to this day, the method of the characteristic equation of surface wave
velocity he deduced is still the important way to seek for Rayleigh wave [2].
It is important to realize that since a solution (Rayleigh wave) can be found using
Elastic theory and Helmholtz decomposition, it can also be seen as the superposition of
two separate components: one P-wave and the other SV-wave. So, we think it isnecessary to review in detail the mechanical process and mathematical method of
deducing Rayleigh wave so far. In this report another main attention is a new andsimplified characteristic equation about Rayleigh wave, which is the result recently
deduced with special aim at future purposes in viscoelastic media.
There remains a dispute about the number of Rayleigh waves in viscoelastic medium
until now. Rayleigh wave in viscoelastic media was early and deeply studied by
Scholte [3], who proofed that Rayleigh wave also existed on the surface of half-spaceviscoelastic media, but he was not sure whether the Rayleigh wave he calculated was
the only one valid. In 1960, from the mathematical aspect of the effective characteristic
root, Bland [4] raised the question of the existence and uniqueness of Rayleigh wave in
viscoelastic media “It has not yet been shown that for any viscoelastic material there isone and only one such root” [4, p.75]. In response to this question, after a series of studies
based on linear viscoelastic models, Currie et al. [5,6,7] came to a conclusion in 1977
that there was more than one possible Rayleigh wave in the viscoelastic half-space
surface. In most cases, there existed two while in some special material there would
even be three possible Rayleigh waves [6]. What’s more, they predicted that thevelocity of viscoelastic surface wave would sometimes be higher than that of the bodywave, and there would even be retrograde wave [5,6].
After that, Bland’s question seemed to be solved. But, on one hand, there is no reportabout the practical examples that the velocity of surface wave will surpass the bodywave velocity nor the retrograde propagation phenomenon. On the other hand, in 2001
Romeo [8,9] promoted Nkemzi’s elastic Rayleigh wave equation [10] to linear
viscoelastic situation, thus put forward an opposite viewpoint. He took advantage ofcomplex modulus to analyze the root number of the characteristic equation ofviscoelastic half-space Rayleigh wave. Although his derivation process contained
multiple variable substitutions, which appeared complicated and obscure, he deemedthat there is only one complex root valid, while the other two are invalid. This impliedthat there is only one truly valid Rayleigh wave in viscoelastic half-space surface.
Obviously, Romeo’s opinion denied the conclusion by Currie et al., but it hasn’t been
widely accepted yet. Except that Ivanov and Savova’s [11] study supports Romeo’s
opinion only in the sense of wavelength fixed condition, studies in the common senseof frequency fixed condition [12,13] and more researchers’ studies [14,15,16] are in favor of
the conclusion of Currie et al. The possibility of the existence of a retrograde Rayleighsurface wave is even suggested [16], and there maybe third types of Rayleigh waves in
the case of some special combinations of material parameters [16]. What is worth notingis that the study in 2014 by Chirita et al [16]. is carried out based on Kelvin viscoelastic
model, which is simple but very commonly adopted in practical engineering. It isinteresting but a little unbelievable that the process of derivation, settings of variable
and multiple variable substitutions by Chirita et al. in the study [16] were similar to
Romeo’s method, but in the conclusion, Romeo's opinion is completely negated. The
obscure of mathematical derivation process in Chirita et al.’s article brings somedifficulty to understanding for us, but the authors’ work arouses our interest and makes
us want to find out the truth.
This report is to propose a brief way of handling this essential problem within
half-space Kelvin viscoelastic medium. Starting from the dynamic equations oftransverse wave and longitudinal wave based on Kelvin viscoelastic model, we derive
the characteristic equation of Rayleigh surface wave in Kelvin viscoelastic half-spacein a much simpler way and in terms of complex wavenumber. We then propose a
relatively convenient and concise method which is able to directly analyze the validityof characteristic roots in a physical sense. Finally, we reach the conclusion that there is
only one Rayleigh wave in Kelvin viscoelastic half-space surface, confirming Romeo’s[8,9]conclusion under the assumption of Kelvin model. Meanwhile, we point out theerror of Chirita et al [16] have made in handling the result, negating their viewpoint that
there is not only one Rayleigh wave in Kelvin viscoelastic half-space surface.
Keywords: Rayleigh wave; viscoelastic; wave equation; characteristic equation; Kelvin model

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