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Dispersion Curves in Guided Wave Testing

by Joseph L. Rose^*

Introduction
what is a guided wave dispersion and its corresponding dispersion curve? Guided waves are ultrasonic waves that propagate differently than the more commonly used longitudinal and shear waves. They are similar to rayleigh and lamb waves in that they propagate at the surface or in thin materials. Their velocity is not only dependent on the material (like longitudinal, shear and surface waves) but also the thickness of the material and frequency. Dispersion curves are used to describe and predict the relationship between frequency, phase velocity and group velocity, mode and thickness. Fundamental to the understanding of guided wave analysis in NDT is the generation or utilization of phase velocity, group velocity and attenuation dispersion curves. How do we use the curves and where do they come from?

Governing Equations
All engineering problems have associated with them a governing equation. In the case of wave propagation, the governing equation is Navier's equation. It is a second order linear partial differential equation for which we'd like to obtain a solution subject to certain boundary conditions. In trying to solve the second order partial differential equation, a variety of different techniques could be used. If we propose a particular solution, let's say for the longitudinal displacement component and lateral displacement component a harmonic solution (oscillating sinusoidal functions of time), we can substitute this into the governing equations to see what happens.

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Wave propagation possibilities must come from points on the dispersion curves and not points in between.

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We find that we do indeed satisfy the governing equation through this solution and the challenge really becomes one of satisfying the boundary conditions. As an example, in a stress free plate or tubular structure, we have to make sure that the stress components on the outside surface are zero. The boundary conditions utilize stress, strain and displacement and equations from the theory of elasticity. Once we have the relationships between stress, strain and displacement and a corresponding constitutive equation, in most cases considered as a generalized form of Hooke's law, we can now make an attempt to satisfy the boundary conditions. In our initially assumed harmonic solution there will be unknown coefficients. The coefficients are obtained by satisfying the boundary conditions.

 

Dispersion curves
With our assumed harmonic solution and an attempt to satisfy the boundary conditions, we end up with a system of homogeneous equations. In order to find a nontrivial solution, the system of homogeneous equations must have the determinant of the coefficient matrix set equal to zero. In extracting the roots, we end up with solutions for phase velocity versus frequency or versus frequency times thickness because of the scale factor of wavelength to thickness. These values are plotted on an engineering dispersion curve, so named because the velocity changes with frequency and the pulses tend to become stretched or dispersed as they propagate.

 

Phase Velocity Dispersion Curves
A typical phase velocity dispersion curve for lamb waves taken from Rose (1999) is illustrated in Figure 1a. A lot of information is presented in the dispersion curve: cp is phase velocity, f is frequency, d is thickness of the plate, c_R is the rayleigh surface wave velocity, c_T is the shear wave velocity and c_plate is the plate velocity. As the fd number increases, all the modes approach the rayleigh wave velocity. The waves formed by throwing a stone into water can help explain the difference between phase and group velocity. The velocity of a particular wave in the packet of waves that are propagating is the phase velocity and the group velocity is the packet velocity. Dispersion curves can be generated for all types of structures including plates, rods, tubes, multilayer structures, rails or any wave guide, whether isotropic or anisotropic.

Figure 1 - Dispersion curves for a traction free aluminum plate (the classic lamb wave problem): (a) phase velocity dispersion curves; (b) group velocity dispersion curves. 

 

In extracting the roots from the determinant, either analytically or through a numerical solution technique, we end up with possible wave propagation possibilities or wave resonances that might occur in the structure. Wave propagation possibilities must come from points on the dispersion curves and not points in between. The curves are for an infinite plane wave excitation which is modified somewhat for a finite sized transducer. We have to try to select a good phase velocity and frequency to carry out a test. The phase velocity can be evaluated by using, as an example, an angle beam entry into a structure. As we know, for oblique incidence into a structure, the governing equation is Snell's law. Snell's law can be used in this case to calculate the corresponding phase velocity and incident angle into the structure in producing the desired guided wave mode and frequency.

A comb type transducer could also be used to generate guided waves by using linear shaped crystals or elements that are spaced evenly apart (Rose et al., 1998). The modes or the curves are labeled for convenience as S0, S1, S2 and so on. S0, S1, S2 and S3 are the symmetric modes. A0, A1 and A2 are the antisymmetric modes. The reason for this naming is associated with the vibrational character of the structure. The symmetric modes have associated with them a compressional rarefaction expansion of the plate or tube in which the waves propagate. The antisymmetric nature is associated with flexural mode propagation as the wave travels through the structure. More details can be found in Rose (1999).

 

Group Velocity Dispersion Curves
Shown in Figure 1b, to accompany the phase velocity dispersion curve, is a group velocity dispersion curve. The phase velocity dispersion curve contains values associated with the extraction of roots from the determinant discussed earlier. However, when we take a group of waves traveling in a structure at approximately the same frequency, that particular wave packet travels with the group velocity. The group velocity can be much different than the phase velocity and changes drastically as we move along each mode as frequency is swept. It is this group velocity that we measure in a laboratory in order to carry out location analysis for a particular discontinuity. The group velocity is derived from the phase velocity curve, being related to the phase velocity values and slopes on the phase velocity dispersion curve. More information on this subject can be found in Rose (1999). When complex roots are extracted from the determinant referred to earlier, attenuation dispersion curves are often generated (Rose, 1999).

 

Guided Wave Testing
Any of the points on the dispersion curves could be used to carry out an ultrasonic guided wave test. Some points, however, are much better than others for improved reception, sensitivity and penetration power. One of the key variables associated with how good a particular point might be is related to the wave structure, that is the wave displacement characteristics across the thickness of the structure. We could think of the displacement characteristics as in plane or out of plane, sometimes referred to as the u and w displacements, with u directed along the plate in the wave propagation direction and w normal to the plate. The wave structure changes for every point on each particular mode on a dispersion curve as the frequency is swept.

In looking at lamb wave propagation, a sample result is illustrated in Figure 2, where the in plane and out of plane displacement component is plotted across the thickness of the structure versus frequency (or actually the frequency times thickness product ranging from 0.5 up to 3). For lower frequencies, it is easy to see that there is a dominant in plane displacement characteristic across the thickness, particularly noted on the outside surfaces. As the frequency increases, however, the variations become considerable to a point where for a frequency times thickness product of 2, the in plane displacement component is almost 0 on the outside surface of the structure. At this point, there is a dominant out of plane displacement characteristic and the waves would tend to propagate or leak into water, if present, and quickly attenuate the waves in the plate. The combination of in plane and out of plane displacement components leads to variations in energy and stress. The sensitivity to certain kinds of discontinuities depends on the wave structure, for example, whether the discontinuity is on the outside surface, in the center portion and so on.

Figure 2 - A wave structure for various points on the S0 mode of an aluminum plate, showing the in plane (u, solid line) and out of plane (w, dashed line) displacement profiles across the thickness of the plate: (a) fd = 0.5; (b) fd = 1; (c) fd = 1.5; (d) fd = 2; (e) fd = 2.5; (f) fd = 3. 

 

We have to select wave structures that have sufficient amounts of energy and displacement with respect to in or out of plane impingement onto a particular discontinuity. A great deal of work is being carried out to find out which of the variables are key with respect to improved sensitivity to certain kinds of discontinuities and also to overall improved penetration power in a structure. Penetration power is particularly important when we consider a multilayer structure or something that is water or fluid loaded.

A sample shear horizontal phase velocity dispersion curve is illustrated in Figure 3. We notice that the curves are simplified somewhat when compared to the lamb wave type dispersion curves; there are fewer modes involved and hence some simplicity with respect to the wave propagation in the structure because of less mode conversion potential. Also, it turns out that for the shear horizontal case, the wave structure is uniform over the entire mode for all frequencies.

Figure 3 - Shear horizontal mode phase velocity dispersion curves for an aluminum layer, c_T = 3.1 mm/µs (0.1 in./µs): solid curves denote symmetric modes and dashed curves denote antisymmetric modes. 

 

Acknowledgments
Thanks are given to Jack Spanner, Jr., of the Electric Power Research Institute, Charlotte, North Carolina, for discussions on this article.

 

References
Rose, J.L., Ultrasonic Waves in Solid Media, Cambridge, Cambridge University Press, 1999.

Rose J.L., S.P. Pelts and M.J. Quarry, "A Comb Transducer Model for Guided Wave NDE," Ultrasonics, Vol. 36, 1998, pp. 163-169.

  

 * Ultrasonic Laboratory, Department of Engineering Science and Mechanics, Pennsylvania State University, 212 Earth and Engineering Science Building, University Park, PA 16802; (814) 863-8026; fax (814) 863-8164; e-mail <jlresm@engr.psu.edu>.

 

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