NDT.net - September 2000, Vol. 5 No. 09 |
TABLE OF CONTENTS |
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.
(1) |
Where U- displacement component; r- medium density; l,m- Lame constants; D- Laplace operator. Displacement component vector can be presented as
U=U_{l} + U_{t} | (2) |
(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 k^{2} > k_{t}^{2} > k_{l}^{2}. 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].
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 k_{l} and transversal k_{t} 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}.
Fig 3: Rayleigh function depending on Poisson ratio v and x parameter as relations of wave numbers of longitudinal k_{l} and transversal k_{t} waves cut by zero plane h(x,v)£ 0 for visualization of different groups of real roots |
(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 V_{RTSS_1}(v),V_{RTSS_2}(v) for the RTSS and transversal waves V_{t}(v) depending on Poisson ratio v in the range of 0 to 0.26345. |
(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 U_{x} 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 U_{x} 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.
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