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4. Interpretation of measurements
4.1. Dish-shaped defects
Complexity of the phenomena can be seen in Fig. 3, which plots a result of
scanning test with the SH1 mode along with the thickness
profile of the dish-shaped defects. Hereafter, phase data are plotted
relative to the measurements at flawless parts. Both the amplitude and
phase of the SH1 mode are shifted from the normal levels and, in this
sense, successfully indicate the presence of defects somewhere along the
circumference.We observe a decrease in amplitude at the defects, but the
magnitude is not directly related to the defect depth. We also observe a
decrease in phase for the shallow defect, but an increase for the deeper
defects. In practice, there is a complicated inter-mode energy transfer at
the positions of thickness change, including nonpropagating (imaginary)
modes whose amplitudes decay exponentially from the discontinuities
[11,12]. We intend to qualitatively explain the observation of Fig. 3 with
three mechanisms: that is, the mode conversion, the group velocity
dispersion and wave diffraction around the corroded area. An exact
explanation will only be available by solving this three-dimensional,
time-dependent elastic problem.
Fig. 5. Dependence of phase and
amplitude on tmin of the rectangular defect (t = 5.4 mm). The broken lines
indicate the cut-off thickness of the SH1 mode.
The defect behaves as an obstacle
for the propagating SH wave. When it impinges on the wall thinning, the
transmission,
the reflection, and the mode conversion take place (see a simplified
illustration in Fig. 4) and only a part of incident energy is eventually
allotted to the original propagation mode, which is detected by the EMAT
with less amplitude than the flawless parts. For fixed f, the wavelength
is adjusted in the thinner part so as to match the dispersion relation
there. This occurs in a straightforward manner and the velocity remains
unchanged with the SH0 mode, since t modifies the normalized frequency and
wavenumber at the same rate. But, it is not the case for the SH1 mode. The
group velocity diminishes with the thinning and this mode cannot propagate
for thickness less than the cut-off thickness defined by tC =
nπCS/ω, where ω satisfies the above dispersion relation. This
cut-off thickness tC takes the value of 3.1 mm for the normal thickness of
t = 5.4 and 3.3 mm for t = 6.3 mm. When the remaining thickness is larger
than tC, the defect supports the propagation both in the SH0 and SH1
modes, which are mode-converted back to the SH1 mode on the far side,
returning to the EMAT. Involvement of the SH0 mode of larger group
velocity has resulted in the earlier arrival and the phase drop of the SH1
mode relative to the flawless circumference. Since the thickness varies in
the axial direction and the group velocity decreases with the thickness,
the defect acts as a lens and focusing occurs [1]. This effect makes a
small amplitude rise at the center of the shallowest defect in Fig. 3.
When the remaining thickness is smaller than tC, the SH1 mode cannot
transmit the energy across the defect and the most incident energy is
reflected back to the near side. The SH1 wave takes the diffracted paths
around the defect, causing the delayed arrival and the phase jump. The
mode-converted SH0 wave seems to contribute little. |