Scientific and technical journal "Bulletin Caratovskogo State Technical University», №04 (43), 2009.
On the question of the degree of attenuation of Lamb waves in the plate in contact with liquid
VG Tokarev, OM Kachanov, AI Kurenkov
Normal Lamb waves are now widely used in ultrasonic technology. Basic information parameter that is most commonly used in various processes of diagnosis and control, with an amplitude of the waves. Method based on the analysis of amplitude characteristics, timegated individual sections Signals Transmitted and reflected in the plates in their surface layer, showed sufficient stability and reliability of the results of such monitoring. However, in case the contact surfaces of the plates with the liquid medium on the amplitude signal characteristics of Lamb waves is affected by several additional factors which reduce the stability of these characteristics. The authors studied experimentally the ratio of energy characteristics of signals Lamb waves propagated in plates of different thicknesses in contact with a liquid medium. Experiments were conducted on the steel plates contacting with a thick layer of water is significantly superior ultrasonic wave length. As a transmitter and receiver pair of identical piezoelectric transducers used Xcut, perpendicular surface of the plates. Excitation of piezoelectric transducers carried out leading edge of periodic pulses (step exposure to a predetermined amplitude) that are longer than the time of propagation of Rayleigh waves between the transmitter and the receiver. As is well known [1], this method of excitation leads to the simultaneous occurrence in the plates mostly symmetric and antisymmetric Lamb waves of zero order. It is also known that wave Lamb represents a heterogeneous group of nonlinear configuration of the normal modes, which are described by the following equations:
where U,W components of general longitudinal and transverse displacements, respectively; U_{s}, U_{a}  longitudinal displacement components related to the symmetric and antisymmetric wave, respectively; W_{s}, W_{a}  transverse displacement components related to the symmetric and antisymmetric wave, respectively; A, В  arbitrary constants determined by the initial conditions of excitation; z  the lateral position of a point relative to the median plane of the plate; d  the thickness of the plate; x  distance along the plate; ω  angular frequency of oscillation; t  time; к,q,s  wave numbers associated relations:
where k_{(s,a)}  wave numbers corresponding to the symmetric and antisymmetric waves, k_{l} , k_{t}  wave numbers corresponding to the longitudinal and transverse waves, respectively. Analysis of the equations shows that most of the energy for the antisymmetric wave is concentrated in the transverse vibrations, whereas for symmetric, on the contrary  in the longitudinal. Analytical solution of the problem of determining the degree of absorption of different types of Lamb waves for specific conditions rather complex. However, the nature of this relationship is already visible on the ratios of the damping coefficients of Rayleigh waves, which are most similar to the zero modes of Lamb waves degenerating into them at d≫λ_{R}, where λ_{R}  Rayleigh wavelength. Shown [1], that the greatest contribution to the damping coefficient of Rayleigh waves made by the transverse wave. Analytical dependence for the attenuation coefficient is expressed by the following formula::
where α  attenuation coefficient of the Rayleigh wave longitudinal, β  attenuation coefficient of transverse Rayleigh wave, A  нa certain quantity depending only on the properties of the material, expressed in terms of Poisson's ratio v. Steel, for example, the above expression takes the following form γ= 0,11 α + 0,89β. The latter indicates that the attenuation of these waves is determined to a greater extent transverse component.
This circumstance must be different impact on the degree of absorption of the fluid contact layer symmetric and antisymmetric Lamb waves, similar to Rayleigh. Experiments performed to assess the extent of absorption of zero symmetric and antisymmetric modes of Lamb waves water contact layer on one side of plates showed the following results, which are presented in the table. The following table presents the results for energy characteristics of signals related to the zero modes of Lamb waves for the free surfaces of the plates (in the air) and one surface in contact with water. To evaluate the characteristics of the signal energy ratio used ratio of sums of squares of samples of the quantized signals in time relating to the symmetric and antisymmetric waves. The table shows the calculated ratio is following:
where N_{C},N_{B}  sample sizes for signals antisymmetrical and symmetrical waves, respectively; (X_{i})_{(C,s)}  the signal values of the samples belonging to the symmetric Lamb wave plate for free; (X^{i})^{(C,a)}   the values of samples of signals relating to the antisymmetric Lamb wave plate for free; ; (X_{i})_{(B,s)}  the signal values of the samples belonging to the symmetric Lamb wave plate for contacting one surface of the water; (X_{i})_{(B,a)}  the values of samples of signals relating to the antisymmetric Lamb wave plate for contacting one surface with water.
The relative values of the absorption of symmetric and antisymmetric Lamb waves of zero order
A distance between emitter and receiver, mm
25 50 100

ТPlate thickness mm
12 16 50
2,28 2,04 1,92
2,54 2,32 2,18
3,02 2,87 2,47

The separation of these signals related to the symmetric and antisymmetric waves carried over time, as the propagation velocity of these waves are different. Quantization of the signal over time for the sampling carried out with a frequency of 10.0 MHz.
10,0 МГц.
Conclusions:
1. The ratio of the degree of absorption of energy of the antisymmetric and symmetric Lamb waves, zeroth order, when in contact with a water surface plate boundaries are proportions of from
3: 1 to
2: 1, and this characteristic is stable for the chosen experimental conditions..
2. Stability characteristics detected experiment can be used for creating efficient industrial devices noninvasive monitoring liquid levels in the tanks.
Literature
1. Victoria IA The physical basis for the use of ultrasonic Rayleigh and Lamb waves in the art. M .: Nauka, 1966.
Received 25.09.09, accepted for publication 25.11.09