1.Introduction

Materials, in general, have different mechanical properties, and consequently, body waves propagate differently inside of them. In the scientific community, there has always been an enormous interest in studying the propagation of body waves since they offer to determine in a non-invasive or destructive way, the mechanical properties of the medium through which they travel ^1-5. Its study has allowed obtaining information on the structure and composition of our planet’s various layers ⁶. But they have also been of great help in seismic prospecting to locate oil deposits on land, shallow and deep waters ⁷.

Among the mechanical waves, the most technologically attractive are those known as interfacial waves, which emerge at the interface of two media due to the coupling of both shear and compression waves ⁸. The characteristics of this type of waves, which in turn make them easy to detect, are: a) they rapidly decay in-depth, b) they propagate parallel to the interface between two media, c) they have the largest amplitude, and d) they decay more slowly than body waves. There are three types of interfacial waves ⁹: a) the Stoneley wave that emerges at the interface of two solids, b) the Scholte wave that appears at a solid-fluid interface, and c) the Rayleigh wave or (surface wave) that arises in the free surface of a solid exposed to a vacuum. Although interface waves are commonly used to detect defects in the surface of materials ^10,11, they can also be used to develop biosensors, temperature sensors, pressure sensors, and humidity sensors, among others ^12,13. Due to the feasibility of the physical conditions under which Scholte waves can be excited and measured, so its characterization is very attractive for the development of technological devices with practical applications ^14,15.

Theoretically, the zeros of Scholte’s secular equation determine the speed with which a wave moves at the solid/fluid interface ¹⁶. Before discussing the complexity of solving this equation, let us consider a particular case: the Rayleigh’s secular equation. This last equation was formulated for the first time by Lord Rayleigh ¹⁷, and it arises from posing the problem of the propagation of a wave (with speed C) along the free surface of an elastic and isotropic solid and is described as

Here A(C) and B(C) are in terms of α and β (the compressional and shear velocities, respectively) as

To find a distinct solution from the trivial one (C = 0) in Eq. (1), Lord Rayleigh proposed to find solutions to A(C)2=B(C)2 (known as the “rationalization method”), which will lead us to a third order polynomial. The counterpart of this procedure can be obtained by introducing the following new function

Note that the roots of Eq. (3) do not correspond to the problem of the Rayleigh wave propagation, but the equality

leads to the same polynomial as the one obtained by Rayleigh. Thus, it is evident that this procedure introduces spurious roots that come from f(C). Lord Rayleigh (based on physical arguments) showed for some particular cases, that Eq. (1) has only one physically acceptable root, which should be real and with speed C<β. Later on, using the same argument as Rayleigh, Knopoff et al.¹⁸ numerically extended the solution to the entire range of physically allowable Poisson radii. However, for the case of Rayleigh’s wave propagation at the interface of a viscoelastic medium, the non trivial solution to Eq. (1) is not restricted to real solutions ¹⁹. Thus, in this last case the discrimination of solutions which come from F(C) and f(C) is no longer straightforward.

As discussed above, the rationalization method restates the solution to Eq. (1) in terms of a third-order polynomial, the roots of which allow a piecewise solution to be constructed ^20-24. The availability of analytical expressions for the respective complex roots may incorrectly suggest additional solutions ²⁵ to Eq. (1). Nevertheless, due to the simplicity of the method, it has been used to investigate analytically the solutions of both Scholte and Stoneley secular Eqs.(26). However, because these equations have a larger number of square roots terms, the polynomial order increases, and consequently, the number of spurious roots introduced also increases. For these cases, even the numerical evaluation of the roots is not an easy task, and in the case of viscoelastic media, it is practically unthinkable. Nkemzi ²⁷ and Romeo (19), in terms of a method based on Cauchy integrals, considered Eq. (1) for both the elastic and viscoelastic cases, respectively, and demonstrated the existence of only one physical solution. Later Antúnez-García would consider the same method to solve the Scholte equation for a specific range of fluid velocities, which would be extended by Vinh (29), both showing that it also has only one physical solution.

As previously shown, the Scholte secular equation’s parametrization in terms of the slowness drives directly to a unique physical solution of this expression ²⁸ without any additional assumptions; then, we propose extending those results in the full range of possible fluid velocities. Unlike the work presented by Vinh ²⁹, the current one makes a more detailed analysis of the solution’s behavior for the different speed ranges with which a wave can propagate in a fluid. So the relevance of the present work falls in obtaining a simple analytical expression to describe the slowness of the Scholte wave for all the range of possible elastic and isotropic media. We also show that the Rayleigh’s wave is a singular solution (at the free surface limit) that rises specifically at the connection point of two fluid speed ranges.

The article is organized as follows: Sec. 2 is devoted to give a brief introduction of the Scholte’s secular equation and the requirements for the existence of that solution. In Sec. 3, the quotient of discontinuities associated to the Scholte’s secular equation are analyzed, and the continuous continuation of them for different speed ranges was established. In Sec. 4 the expression for the unique physical root of the Scholte’s secular equation is obtained and particularly, it is used to recover the Rayleigh wave solution. Additionally, numerical calculations were included to test the solution of the Scholte’s equation. Finally, the conclusions are presented.

2.Scholte’s secular equation

In general the existence of a wave propagating through the interface of two homogeneous and isotropic semi-infinite media (exponentially decaying) was predicted by Stoneley . A particular case of a Stoneley wave arises when one of those semi-infinite media becomes fluid, which is known as a Scholte wave. The speed propagation of this wave (C) is a zero of the Scholte secular equation ^26,31-33

under the following physical restrictions

which ensures solutions with an exponentially decaying behavior. Here α and β are the compressional and shear velocities of elastic waves in a solid, C_F is the speed of a sound wave in a fluid, p and ρ_F are the densities of a solid and fluid medium, respectively. Note that the terms inside of the square brackets in Eq. (5) correspond to the Rayleigh characteristic Eq. (1). By convenience we rewrite Eqs. (5) and (6) in therms of the dimensionless variable z=β2/C2 and parameter γ=α2/β2, γ†=CF2/β2 as

subject to the restrictions

For any elastic and isotropic medium α, β, and C_F are real and positive quantities. In particular, the ratio of speeds can be described in terms of Poisson’s (v) ratio as

Thus, there are three cases for the study of the characteristic Scholte equation (7): (a) CF>α>β, (b) α>CF>β and (c) α>β≥CF, which will be discussed below.

3.Discontinuities of Scholte’s secular equation

In general the branch points of F(z) are localized along the real axes z=1,1/γ† and 1/γ. For each of the cases mentioned in the previous section, these branch points define two main branch cuts (see Fig. 1). For the follow discussion, we will make reference to the Rayleigh branch cut ΓR, as the one defined by

Figure 1 Branch cuts for the function F(z) for cases in which: a) C_F > α > β, b) α > C_F > β and c) α > β ¸ C_F . 

which describes the discontinuities presented by the Rayleigh secular equation ²⁸.

3.2.Case (b): α>CF>β

For a fluid under this range of velocities, the discontinuities (see Fig. 1b)) are described by

G3(t)=F+(t)F-(t)|Γ3=1+iH3(t)1-iH3(t),G4(t)=F+(t)F-(t)|Γ4=1-iH4(t)1+iH4(t) (16)

with the auxiliary functions

H3(t)=γ(2t-1)24t1-tγt-1+γ†ρFργt-1γ†t-1,H4(t)=4t1-tγt-1γ(2t-1)2+γ†ρFργt-11-γ†t, (17)

which are defined along the arcs

Γ3(t)=t, ∀ t ∈ 1/γ<t≤1/γ†,Γ4(t)=t, ∀ t ∈ 1/γ†≤t≤1. (18)

Here the continuous continuation and unicity of solutions is granted by

limt→1/γ†G2(t)Γ3=limt→1/γ†G3(t)Γ4=1,

while for the limit with ρF→0 satisfies

limρF→0H3(t)Γ3=0,  limρF→0H4(t)Γ4=1-iHR(t)1+iHR(t) (19)

as in the previous case. Particularly for this case, note that the Rayleigh branch cut ΓR moves from partially screened to unscreened as the presence of a fluid vanishes (see Fig. 2a) and b)). Here HR(t) is described by Eq. (14). As in a previous case, it could be considered a fluid medium with density ρF=ρ″≠ρ' in the range 0<ρ″≪1 such that CF=α+ϵ, with 0<ϵ≪1. At this limit we will have

limCF→α+ϵH3(t)|Γ1≠limCF→α-ϵH1(t)|Γ1≠0,andlimCF→α+ϵH4(t)|Γ2≠limCF→α-ϵH2(t)|Γ2≠HR(t). (20)

Thus the connection of discontinuities on both cases (a) and (b) is granted trough the free-surface limit.

Contrarily to the unscreened Rayleigh branch cut limit, the full-screening of ΓR occurs when γ†→1 (see Fig. 2c)) where

limγ†→1H4(t)Γ4=0,

and

limγ†→1H3(t)Γ3=γ(2t-1)24t1-tγt-1+ρFργt-11-t. (21)

then, the extreme values of 𝐺 3 (𝑡), particularly corresponds to well-defined values

limγ†→1G3t=1γ=-1,limγ†→1G3t=1=1.

Figure 2 Branch cuts for the function F(z) in the case that α > C _F > β for: a) the general case, b) p _F → 0 and c) 1= y^† → 1. The branch cut colored in blue represents the fluid discontinuities’ contribution to the three different cases. The cases in which Γ _R is totally unscreened and totally screened correspond to Figures b) and c) respectively. 

4.The Scholte solution

Since we have guaranteed the continuous continuation of the Scholte discontinuities for the entire range of fluid velocities, we are in a position to determine the polynomial containing the zeros of Eq. (7). According to the construction 5 for the canonical solution of the Riemann problem, the polynomial with only zeros of the Scholte characteristic Eq. (7) is

which was obtained through the Bourniston method . This method involves the expansion of Eq. (A.3) in terms of Laurent series. It should be noted that unlike previous works ^19,27,29, the parametrization in terms of slowness conduces directly to a simple polynomial (26), and additional considerations are not necessary to solve it. In fact, directly from this polynomial is possible to have one and only one physical solution for the Scholte slowness, that is

Here δ is a correction parameter introduced to enhance the numerical accuracy of the Scholte root, which is a consequence of the truncation process during the Laurent series expansion (Eq. (7)). The Ia, Ib Ic terms represent the contribution of the discontinuities for the different speed ranges and are expressed in terms of Eqs. (12), (17) and (24) as

Ib=1π∫Γ3arctanH3t dt-∫Γ4arctanH4t dt, (28b)

Ic=1π∫Γ5arctanH5t dt+∫Γ6arctanH6t dt. (28c)

4.1.The free surface limit

Contributions associated exclusively with the ΓR branch cut, occur only at the point at which the continuous continuation of the speed range between cases (a) and (b) occurs; i.e., the pole contribution. Thus, the free surface limit in terms of Eqs. (27), (28) and (28b) corresponds to

limρF⟶0⁡zSc=zR=2γ2+γ-124γ(γ-1)+1π∫1/γ1arctan⁡HR(t)dt  (29)

Figure 3 Sholte (z _Sc ) versus Rayleigh (z _R ) slowness as function of the Poisson’s ratio. 

or explicitly to

zR=(βCR)2=2γ2+(γ-1)24γ(γ-1)+1π∫1/γ1arctan⁡(4t1-tγt-1γ(2t-1)2)dt, (30)

which, according to ²⁸, describes the exact slowness of the Rayleigh’s wave propagation. Note that the δ term is not included in this expression.

5.Conclusions

In terms of Cauchy integrals, we have obtained a robust analytic formula to predict the existence of a unique physical solution for the Scholte slowness in all range of possible elastic and isotropic media.The appropriated analysis of the discontinuities associated to the Scholte secular equation allows us to identify the free-surface limit as a pole contribution.

Unlike previous results ²⁹, due to the secular Scholte equation’s fractional-order, our results show that the truncation process involved in obtaining the rational polynomial does not allow getting an exact solution. However, our numerical results show that this solution presents a reasonable approximation to the exact value. Additionally, since slowness is the inverse of speed, the results show that a Scholte wave’s propagation speed is less than or equal to that of a Rayleigh wave.