TABLE OF CONTENTS

Abstract Introduction Theoretical Background Rayleigh Wave Equation Once Again 3-D Visualization of Rayleigh Function The RTSS waves behavier in plane half-space solid Computing, Modeling, and Interpreting Amplitude Distribution of the RTSS waves Recent Development Conclusions Discussion References

Abstract

Rayleigh surface waves were discovered theoretically as a solution of wave equation. Their physical existence was confirmed later thanks to numerous experimental investigations and practical applications in acoustics, geophysics, electronics, etc. Conditions of Rayleigh surface waves propagation are defined by one of the real roots of Rayleigh equation. The other real roots of the equation were assumed to be extraneous and were not taken into account. This paper is considering possibility to use additional (extraneous) real roots of Rayleigh equation for explanation of existence of a special type of leaky waves, which propagate on the border of half-space solid and vacuum. Conditionally these waves could be referred to as Rayleigh transformed sub-surface waves as they propagate along the surface, their main energy concentrates in a local zone under the surface, and below this zone they transform into bulk waves taking away the wave energy. Some computer simulations and calculations are presented as a theoretical explanation of conditions of Rayleigh transformed sub-surface waves propagation.
Keywords: acoustics, Rayleigh equation, surface waves, ultrasonics, phase velocity.

Introduction

Physical Acoustics as fundamental science was born about two hundred years ago thanks to monumental works of great scholars: Rayleigh, Lamb, and Love [1, 2, 3]. The second part of the 20^th century demonstrated high performance acoustic applications in several fields of science and technology. Ultrasonics as a branch of acoustics is one of the main consumers of physical acoustics fundamentals. The latest advances of electrical and electronic engineering gave opportunity to build a wide class of ultrasonic equipment allowing verification of basic acoustic concepts. Some recent experimental results cannot be explained unambiguously by existing basic theories especially in the field of acoustic surface waves classification. Depending on classification of the wave type testing results may be interpreted differently. In spite of the fact that a lot of basic and experimental works are published on the problem of analysis and classification of Rayleigh type surface waves, distance between theoretical models and real materials is hard to come by.

Theoretical Background

Transformed surface (TS) waves are a recently discovered [4,5] special class of elastic waves in solids. This class of waves is intermediate between bulk and surface waves, which propagate along the border of solid medium and transform acoustic energy from surface co-boundary space or into the solid depth. Their characteristic behavior may be described as follows: they appear in a local area near the energy source, propagate further with splitting into bulk and surface wave components, have limited zone of propagation because of fast energy transformation from surface to bulk component and absorption and dissipation of energy in solid.

It is possible to distinguish a sub-class of TS waves, which have got a name of "leaky surface waves" (LSW). At present two types of the LSW have been investigated: waves on the border of two solid half-spaces with different physical properties and on the border of solid half-space and liquid. The first type of the LSW could be presented by Stoneley waves [6]. In the case of Stoneley waves acoustic energy is leaking from one solid half-space to the other, and the wave propagates on the border between them. The second type of the LSW could be presented by Rayleigh type of waves: on the border between solid half-space and liquid [7]; and Lamb type of waves in plates immersed into liquid [8]. In this case acoustic energy is leaking from half-space solid or plate to liquid [9].

The other sub-class of the TS waves includes Rayleigh sub-surface transformed (RSST) waves, which were discovered in 80's [10,11]. The RSST waves could propagate on the border of solid half-space and vacuum (or gas) where acoustic energy radiates into the same half-space solid transforming to other types of waves. This type of waves is more complex and interesting for practical application than other TS wave types [12]. The present paper is considering only conditions of RSST wave propagation in plane half-space solid. The accepted model can be useful for description of one-side ultrasonic testing of "thick" (more than 3-5 wavelengths) structures.

Rayleigh Wave Equation Once Again

For better understanding of conditions of Rayleigh transformed surface waves propagation it is proposed to return "back to basics" and consider classical elaboration of Rayleigh problem once again [13]. The general motion equation for isotropic homogeneous and ideally elastic medium has the following form (1)

(1)

Where U- displacement component; r- medium density; l,m- Lame constants; D- Laplace operator. Displacement component vector can be presented as

U=Ul + Ut

(2)

where U_l = gradj; U_t = roty ; r, y- scalar and vector potentials accordingly. By substituting (2) into (1) reduce this equation to two independent formulae (3) and (4)

	(3)
	(4)

The first one (3) describes propagation of longitudinal waves, and the second (4) - transversal waves. After that the classical Rayleigh wave propagating in positive direction along the border of half-space solid and vacuum represented by axis x is taken for consideration (Fig. 1).

Fig 1: Scheme for Rayleigh problem calculations

In this case the wave motion does not depend on coordinate y and vector potential y and has other than zero values only along axis y. The motion equations for plane harmonic wave are satisfied if potentials j, y are solutions of the two wave equations (5) and (6)

	(5)
	(6)

Where - wave numbers corresponding to longitudinal and transversal waves accordingly. Searching for solutions of (5) and (6) corresponding to surface waves we assume that . Substituting these expressions into (5) and (6) receive two linear differential equations for functions F(z) and G(z) (7):

(7)

The following functions represent two linear independent solutions for each of the written equations: and . It is possible to assume a priori that k² > k_t² > k_l². Then solutions with positive radicals in exponential correspond to increasing with depth motion, i.e. in-depth penetration, and solutions with negative radicals correspond to along-surface motion, i.e. surface propagation wave mode. Thus, j,y have the following form (8)

(8)



In accordance with (2) displacement of medium components along axes x and z are expressed through potentials j,y in the form (9)

(9)

Using linear dependence between strain and stress tensors (Hooke's law) in elastic solid as described in [14] and relations (9) it is possible to present stress tensors components through j,y (10)

(10)

On the border of half-space and vacuum (z = 0) stress tensors T_zz,T_xz vanish. Substituting expressions of j,y for these conditions a system of homogeneous functions relative to arbitrary constants A,B is received (11)

(11)

Function determinant F_(k) = 0 is the necessary condition for nontrivial solution of the system. It gives the following characteristic equation for finding the wave number k(12)

(12)

This equation is called Rayleigh equation. Often the equation (12) is written in polynomial form (13) [15]

(13)

where and k,k_l, k_t are wave numbers for phase velocities of surface, longitudinal, and transversal waves respectively. As it is clear from (13) there are several roots in the equation. One of them describes pure Rayleigh surface wave h_R. In a number of papers, for example [16, 17] an approximation of this root depending on Poisson ratio n is given using Bergmann's [18] formula (14)

(14)

This work is an attempt to explain physical nature of additional roots. The equation is considered here as a polynomial function Â(v)

(15)

The search for values of the roots is made under condition that Â(v) = 0 [19].

3-D Visualization of Rayleigh Function

It is proposed to take a look at a 3-D representation of Rayleigh function. Two different side-views of the function are given in the Fig. 2 (a, b).

Fig 2a:

Fig 2b:

Fig 2: Full-scale two-side views of Rayleigh function depending on Poisson ratio v and x parameter as a relation of wave numbers of longitudinal kl and transversal kt waves.Elaboration with MATHCAD PLUS 6.0: a - rotation: 98° and tilt: 45°; b - rotation: 280° and tilt: 50°.

This original representation of Rayleigh function in 3-D gives the opportunity to better understand the physical nature of the surface wave behavior in solids with different elastic characteristics. It is possible to see that the function crosses the zero plane in several places corresponding to different groups of real roots. These real roots include the root describing Rayleigh surface wave h_R as well as the other two roots presumably corresponding to the so-called Rayleigh transformed sub-surface (RTSS) waves h_RTSS.

The RTSS waves behavier in plane half-space solid

The 3-D representation for solution of Rayleigh equation for condition h(x , n) £ 0 demonstrates characteristic behavior of h_R and h_RTSS. In the Fig. 3 it is possible to see two places showing searched roots: almost linear inflection corresponding to Rayleigh surface wave (roots h_R) and further ellipse-shape concavity corresponding to the RTSS waves (roots h_RTSS).

Fig 3: Rayleigh function depending on Poisson ratio v and x parameter as relations of wave numbers of longitudinal kl and transversal kt waves cut by zero plane h(x,v)£ 0 for visualization of different groups of real roots

Computing, Modeling, and Interpreting

Values of Rayleigh function real roots were approximated using nonlinear power polynomials. This approach has given a possibility to tie functionally values of Rayleigh wave velocities V_R(v) depending on transversal wave velocities V_t and Poisson ratio v (19):

(19)

The roots describing the RTSS waves have a parabolic form and are uniquely defined over Poisson ratio. That is why velocity values are approximated separately for the upper and lower branches of this function. Then velocity values of the RTSS waves V_{RTSS_1}(v) corresponding to the lower branch of the function are described by (20)

(20)

and velocity values of the RTSS V_{RTSS_2}(v) corresponding to the upper branch are described by (21)

(21)

In the Figure 4 schematic representation of relative values of velocities V_{RTSS_1}(v),V_{RTSS_2}(v) for the RTSS and transversal waves V_t(v) are given.

Fig 4: Schematic representation of relative values of velocities VRTSS_1(v),VRTSS_2(v) for the RTSS and transversal waves Vt(v) depending on Poisson ratio v   in the range of 0 to 0.26345.

Amplitude Distribution of the RTSS waves

Knowing the phase velocity of the RTSS waves and values of A and B constants from (11) it is possible to get values of the potentials j,y (16):

(16)

They show that the wave consists of two different components with amplitudes that attenuate with depth according to the rule for longitudinal and transversal modes. Then after substitution of these expressions into (11) receive shift components of amplitude distribution along co-boundary half-surface U_x (17):

(17)

and in depth of solid U_x (18):

(18)

It is possible to assume that wave number k_RTSS for the RTSS waves could be free of limitations k_RTSS ³ k_l or k_RTSS£ k_l . Then amplitude distribution could be defined using calculated values of velocities of the RTSS waves in (20), (21) and placing k_RTSS = k into expressions (17), (18). Results of modeling of amplitude distribution along co-boundary half-surface U_x are presented in the Figure 5.

Fig 5: Results of modeling of horizontal shift component Ux distribution against relative distance x / l (where l is the wave length) for Poisson ratio a - 0.18, b - 0.22, c - 0.26.

Fig 6: Results of modeling of vertical shift component Ux distribution against relative distance z / l (where l is the wave length) for Poisson ratio: a - 0.18, b - 0.22, c - 0.26.

Results of modeling of amplitude distribution in depth of solid U_z are presented in the Figure 6.

Recent Development

Recently leaky Lamb waves (LLW) have been investigated for nondestructive testing applications in aerospace industry [20]. In spite of considerable achievements in this area, applications of these waves are limited to laboratory testing of materials and elements of structures due to requirements of emersion techniques. It is clear that LLW cannot be properly used for nondestructive evaluation of materials in-situ or structures in service where application of emersion techniques is not appropriate.
Investigation of the RTSS waves presumably propagating on the border of solid and vacuum or gas could give a new approach to nondestructive testing in-situ. The RTSS waves could simplify testing procedure because they, unlike the LLW, do not have velocity dispersion.
The theoretical assumption of existence of the RTSS waves is at the stage of experimental verification on the materials with Poisson ratio less than 0.26, for example ceramics, concrete, and some types of metals.

Conclusions

Theoretical explanation of the conditions of the RTSS waves propagation is based on the same mathematical model as Rayleigh wave propagation. But the other group of real roots of Rayleigh equation, which was not considered by Rayleigh and other authors, is taken and investigated. Interpretation of these additional real roots allows assuming existence of a special type of sub-surface waves with phase velocity that could be more, equal, or less than phase velocities of bulk waves in solids.
Computer simulation allows to visualize existence of different groups of real roots of Rayleigh equation, which correspond to different types of surface waves: Rayleigh surface wave and Rayleigh transformed sub-surface waves. Highly accurate approximations of data for the roots of the both groups are obtained as polynomial functions for various values of Poisson ratio.
Calculations of horizontal amplitude distribution of the RTSS waves based on the polynomial approximations of the velocity values show that these waves attenuate at a distance of 2-3 wavelengths. Calculations of vertical amplitude distribution of the RTSS waves show that the waves concentrate at a depth of 1-2 wavelengths and below this level they transform into bulk waves.

Discussion

Specialists interested in this topic are invited for discussion through NDT.net or directly with the author.

References

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