Science Direct Science  Direct Science   Direct Science    Direct

Science     Direct

Wheels were attached to the EMAT for laboratory tests to facilitate the linear movement in the axial direction, while continuously measuring the amplitude and phase. The gap from the EMAT case surface to the pipe surface is around 0.2 mm. The EMAT is put near one end of a sample pipe and is manually pulled toward another end in a smooth motion, taking approximately 10 s. The pipe can be in a vertical or horizontal position. During this period of time, the system acquires and stores approximately 500 sets of amplitude and phase of a selected SH mode. The amplitude represents the transmission efficiency and the phase lag represents the transit time at an individual cross section. An axial resolution of 1 mm is possible by averaging over two or three measurements.

3. Dispersion relation of SH modes
Measurements are interpreted with the dispersion relation of the guided SH waves in the pipe. The elastodynamic equation in the cylindrical coordinates is solved for only the axial component of displacement, satisfying the boundary condition at free surfaces (see Equation (1) in Ref. [10]). Numerical calculation revealed that a large diameter/thickness ratio reduces it to the dispersion relation in a flat plate [11,12], k² = (ωC_S)² 2 (nπ/t)², which is the basis of the following discussion. Here,ω is the angular frequency, k the wavenumber, t the thickness, and CS the shear wave speed. The integer n characterizes the dispersion nature and is used to define the mode as SH_n.

Fig. 3. Axial scanning data of the SH₁-mode amplitude and phase shift for dish-shaped defects (t = 6.3 mm). The defect profiles are determined with a dial gauge.

Fig. 2 shows dispersion curves of SH modes in plate, where symmetric group of even n is drawn with solid lines and antisymmetric group of odd n with broken lines. Except for the SH₀ mode, all are dispersive. Because the PPM-EMAT restricts the wavelength to & , the wavenumber is fixed to k = 2π/& for all modes. We can then select the probing SH mode by the frequency to drive the EMAT. The SH0 and SH1 modes were used in the present experiments. The SH0 mode is excited at 0.425 MHz and the SH₁ mode at 0.52 MHz when t = 5.4 mm. These operating points on a flawless portion are marked on Fig. 2. Because of the superheterodyne spectrometer and the fixed wavelength from the PPM-EMAT geometry, we obtain the acoustic data only of the intended mode and frequency by rejecting others that can be involved in the raw signals as noise or due to mode conversion.

Fig. 4. A simple schematic for the mode conversion at discontinuous thickness changes. The SH₁ mode is incident from the right-hand side and is detected on the left-hand side. The transmitted amplitude is larger for t_min > t_C than t_min < t_C.

In the SH₀ mode the shear deformation occurs uniformly through the thickness, while in the SH₁ mode, it is antisymmetric
around the midplane with maxima on both surfaces. Because of this difference in eigen functions, the two modes provide unique information on the defects, including the high sensitivity of SH₁ mode to surface anomalies.