## 2 Materials and Methods
### Flexible Ultrasonic Testing  
Ultrasonic probe usually requires a supplement of a coupling material for transmitting ultrasonic wave from the transducer to the test specimen. It complicates the inspection system and is waste of coupling material. In addition, the test specimen surface should be flat enough to maintain a positive lift-off (non-contact) for protecting the collision of the transducer with the test specimen. It is hard for the inspection of near-surface defects in the water-cooled tube because the changes of the tube wall could make an unpredictable lift-off that could lead to the collision and break the transducer. Therefore, we propose using a flexible transducer that the lift-off could be varied and not require using the coupling material [23]. At the head of a normal transducer, we attached a flexible membrane that was water-filled. The membrane has a sphere shape after filing the water and maintains contact with the tube even though the lift-off can vary. Also, the ultrasound wave can still propagate from the transducer to the water membrane and come to the test specimen.  
A sample flexible transducer is shown in Figure 6a. The transducer has a spring that keeps the contact between the membrane with the test specimen during the scan. The received time-domain signal of the transducer, which is A-scan signal (\(u(t)\)), is shown in Figure 6b. For a better signal-to-noise ratio, the spectrogram of the A-scan signal was processed (\(S(\tau_{f})\)) and extracted only the signal (\(S_{A}(\tau)\)) at the center frequency of the transducer (\(f_{c}\)), as shown in Figure 6c,d. The spectrogram (\(S(\tau_{f})\)) and extracted signals (\(S_{A}(\tau)\)) are calculated as expressed in Equations (10) and (11), respectively; where, \(h\) is a  
Figure 4: Signal processing block diagram of the magnetic flux leakage testing system. The block diagram is for a single Hall sensor.  
Figure 5: Configurations of the magnetic flux leakage testing system.  
sliding Gaussian window. The extracted signal (\(S_{A}(\tau)\)) was then stacked to form the B-scan signal while scanning the transducer along with the test specimen.  
\[S(\tau,f)=\left|\int\limits_{-\infty}^{\infty}u(t)h(t-\tau)e^{-j2\pi ft}dt\right| ^{2} \tag{10}\]  
\[S_{A}(\tau)=\left.S(\tau,f)\right|_{f=f_{c}} \tag{11}\]  
It is observed from the ultrasonic transducer signal that there are multiple peaks. The first peaks are the reflected wave from the specimen surface. It has a delay time of about 4 us (\(t_{1}\), \(u_{1}\)), which is the propagation time from within the probe membrane. This delay time could be varied due to the flexibility of the membrane (lift-off). Thus, it is necessary to eliminate this delay time by shifting the signal with an amount of time \(-t_{1}\). In addition, there are four peaks (\(t_{2}\), \(u_{2}\)), (\(t_{3}\), \(u_{3}\)), (\(t_{4}\), \(u_{4}\)), (\(t_{5}\), \(u_{5}\)) next to the specimen surface peak (\(t_{1}\), \(u_{1}\)), which correspond to the repetitions from the bottom surface of the specimen. The time intervals of these four peaks are the same and can be used to calculate the specimen thickness, as shown in Equation (12); where \(v\) is the speed of the ultrasound in the specimen.  
\[d=(t_{2}-t_{1})\times\frac{v}{2}=(t_{3}-t_{2})\times\frac{v}{2}=\cdots=(t_{5}- t_{4})\times\frac{v}{2} \tag{12}\]  
Figure 7 is a schematic of the flexible ultrasonic probe (FUP) for quantitatively measuring the specimen thickness. The FUP is an array of multiple transducers (i.e., 6) arranged for covering the tube wall, welding lines, and specimen membrane area. The FUP could be alternated the magnetizer and magnetic sensor array modules in Figure 5.  
Figure 6: (**a**) A single flexible ultrasonic transducer measuring a thickness of a specimen, (**b**) its time-domain signal, (**c**) spectrogram signal, (**d**) the cross-section view signal of the spectrogram, and (**e**) the stacked cross-section view signal (B-scan).