2.10 Guided waves
2.10.2 SH Waves
Shear Horizontal waves exist as a separate guided wave mode in the free plate situation, but here the particle displacement is polarised parallel to the surface of the waveguide [8, 64]. The guided wave is then generated by a superposition of up and down travelling bulk shear horizontal waves, with the conditions as in the Lamb case that the surfaces of the medium are traction free. This can be illustrated by considering the geometry in Figure 2.5 where the wave propagates in thex1direction, with the particle displacements polarised in thex3 direction. The superposition of the modes occurs in the remainingx2 direction through the thickness of the sample, here the thickness of the sample in the x2 direction is twice the distance from the surface to the mid point of the sample, d=2h [8].
Where k is the wavenumber of the modes and is defined as
Figure 2.5: Schematic diagram showing SH wave mode propagation in a plate, with the propagation of the wave in the x1 direction and the shear displacement of the particles along thex3 axis.
The particle displacement field for an isotropic medium was given in equation 2.13, and this equation can be used for shear horizontal modes considering vectors with only an x3 component such that the displacement in the other dimensions is equal to zero. This allows us to define the displacement vector for the wave as [8]:
u3(x1, x2, t) =f(x2)ei(kx1−ωt) (2.26) where ω is the circular frequency of the wave ω = 2πf. It should be noted here that the displacement is independent ofx3 such that the wavefront of the SH wave is infinite in the x3 dimension, travels in the x1 direction and has a fixed distribution in the x2 direction. Given the above conditions, the solution can be substituted into the equation in 2.13 and a general solution specified, such that the the general form of the displacement field is given by:
u3(x1, x2, t) = [Asin(qx2) +Bcos(qx2)]ei(kx1−ωt) (2.27) Where q is defined as q= s ω2 c2T −k 2 (2.28)
and A and B are arbitrary constants.
The total displacement field equation can then be separated into two different components relating to the symmetric and antisymmetric components with respect tox2 :
us3(x1, x2, t) =Bcos(qx2)ei(kx1−ωt) (2.29)
Where the superscripts s and a denote a symmetric and antisymmetric mode respectively. Now boundary conditions can be imposed on the modes that the sur- faces of the layer atx2 =±h are traction-free. This allows the dispersion equations for a traction free surface for the symmetric and antisymmetric modes to be defined as
sin(qh) = 0 (2.31)
and
cos(qh) = 0 (2.32)
Explicit solutions to these equations can be obtained using knowledge of the sine and cosine behaviour in thatsin(x) = 0 when x is zero or an integer value ofπ
andcos(x) = 0 when x is a factor ofπ/2. In other words
sin(x) = 0 whenx=nπ(n= 0,1,2, ...) (2.33)
and
cos(x) = 0 whenx=nπ/2(n= 1,3,5, ...) (2.34)
such that
qh=nπ/2 (2.35)
Where the symmetrical SH modes are defined by n=(0,2,4,...) and the antisymmet- rical modes are defined by n=(1,3,5,...). By using the definition d=2h seen earlier and taking the real parts of the equation in equations 2.29 and 2.30, the displacement fields for the symmetric and antisymmetric modes can be respectively rewritten as:
us3(x1, x2, t) =Bcos(nπx2/d) cos(kx1−ωt) (2.36)
us3(x1, x2, t) =Asin(nπx2/d) cos(kx1−ωt) (2.37) This is an important result, as the sine and cosine terms do not depend on the frequency and wavenumber of the mode. This means that the displacement field across the thickness of the sample does not vary with the frequency of the mode generated but only with the order of the mode generated. This provides a marked difference to the Lamb wave case where the displacement profile changes with the frequency and means that different SH modes should be more sensitive to different defects at all frequencies rather the sensitivity varying with frequency. An important aspect to note from the displacement profiles of the first few modes
in Figure 2.6 is that although the displacement is always maximum at the outer surfaces of the sample, it does vary throughout the sample, falling to zero at certain points for the higher order modes. This can pose a problem for inspections as it will be less sensitive to defects in these regions. As the order of the mode increases so does the complexity of these profiles, with more points of minimal displacement, which limits their use in NDT inspections.
(a) SH0 displacement profile (b) SH1 displacement profile
(c) SH2 displacement profile
Figure 2.6: Displacement profiles through the thickness of the plate for SH wave forces.
Explicit solutions to the phase velocity vs frequency thickness (fd) curves can be calculated from the solutions to the previous equations. This allows the generation of the dispersion curves for the shear horizontal modes that dictate the speed of wave generated for a particular frequency [8]. Using the definition of q, the solutions to the dispersion equations and the definition for k, the dispersion equation can be written as:
ω2 c2 T −ω 2 c2 p = nπ 2h 2 (2.38)
Where the order of the SH mode in question is denoted by the integer number n. The phase speed can then be calculated using the equations set out for phase velocity wherecp=ω/k cp = ω k =±2cT f d q 4(f d)2−n2c2 T (2.39)
This develops the phase velocity curves for the infinite family of SH modes that can be generated in a plate. All of the modes in a plate show some dispersive nature except for the lowest order symmetric SH mode which is dispersionless and propagates with both a phase velocity and group velocity equal to the shear wave speed of the material. The group velocity of these modes is calculated from the definition of the group velocity seen earlier thatcg =dω/dk such that:
cg = dω dk =cT s 1− (n/2)2 (f d/cT)2 (2.40)
The phase velocity curves are shown in Figure 2.7 and group velocity curves are shown in Figure 2.8, with the SH0 mode in blue, SH1 mode in red, SH2 mode in black, SH3 mode in green, SH4 mode in magenta, SH5 mode in cyan and SH6 mode in yellow. An important feature that can be seen is that in addition to the dispersive behaviour these modes show, all but the fundamental SH mode have a cut off where the phase velocity becomes infinite and the group velocity drops to zero at a certain value of frequency thickness.
Figure 2.7: SH guided wave phase velocity dispersion curve for a steel plate These are known as cut off frequencies, here the wavenumber of the propagat- ing wave along thex1direction becomes imaginary and the wave becomes evanescent and nonpropagating [64]. The cut off frequency for the nth mode is:
Figure 2.8: SH guided wave group velocity dispersion curve for a steel plate
(f d)n=
ncT
2 (2.41)
When a higher order mode with a cut off frequency reaches a section of plate where the remaining thickness is less than the cut off thickness, then the mode is forced to mode convert and reflect at the interface. The extent to which these processes occur will be discussed later but it extremely important in guided wave inspections, being dependent not only on the thickness of the plate but the profile of the wall thickness change.
The area just before the cut off thickness in a sample is the most dispersive area of the curve, with the group speed decreasing for decreasing thickness which aids in identification of defects approaching the cut off thickness of the sample. As the frequency thickness product is increased, all modes tend towards the shear wave speed in the material, as the waves begin to propagate as if they were travelling in the bulk of a material.