## 2 Background
### Detection of Back-Surface Crack with Rough Surface  
A general schematic diagram for the ultrasonic detection of micro-crack with rough surface is shown in Figure 3a. A pressure wave generated from the immersed ultrasonic-focused transducer is obliquely incident on the rough front surface of the workpiece. Generally, there are only transmitted shear waves inside the solid that improve detection accuracy and sensitivity. The oblique incident angle is therefore set as 17\({}^{\circ}\), which is located between the first critical angle and second critical angle. Since the wavelength of shear wave is about half the wavelength of the longitudinal wave at the same low frequency, the influence of surface roughness on detection sensitivity of the shear wave is more serious than that of the longitudinal wave. Currently, in terms of the industrial ultrasonic testing system for steel pipes, it is difficult to detect micro-cracks of depth less than 5% of the wall thickness for pipes with large surface roughnesses. On the other hand, there is a possibility for inner and outer walls of steel pipe, such as hot-rolled steel pipes changing to rough surfaces that are rougher than bearing steel pipes. In these cases, incoherent waves scattering in all directions are generated when a longitudinal wave interacts with the rough surface. This leads to significant attenuation of energy intensity of the transmitted shear wave and severe degradation of amplitude of the micro-crack signal.  
To further analyze the transmitting process of acoustic wave for micro-crack detection when the front surface and back surface are both rough, Figure 3b performs energy intensity changes of acoustic waves at different stages. The transmission coefficients \(T_{1}\) and \(T_{2}\) are modified by surface roughness [12]. The reflection coefficient \(R_{1}\) reflects the interaction of transmitted shear wave with the  
Figure 3: (a) Schematic diagram of ultrasonic detection of micro-crack with rough surface; (b) changes in energy intensity of acoustic wave for micro-crack detection. \(I_{0}\)–\(I_{5}\) represent the changes in energy intensity from incident wave to crack echo. \(T_{1}\) represents the transmission coefficient of incident acoustic wave, \(R_{1}\) represents the reflection coefficient of transmitted shear wave, and \(T_{2}\) represents the retransmission coefficient of the reflected wave of back-surface crack.  
micro-crack and rough back surface. Considering absorption attenuation and scattering attenuation caused by the backscatter of microstructures inside the solid, the relationship between \(I_{0}\) and \(I_{5}\) can be simply given by:  
\[I_{5}=I_{0}\cdot T_{1}\cdot R_{1}\cdot T_{2}\cdot\exp(-4\delta x) \tag{3}\]  
where \(\delta\) is the attenuation coefficient and \(x\) represents the propagation distance of the acoustic wave in the solid. For steel plate with uniform thickness, the absorption attenuation and scattering attenuation remain unchanged. It can be seen that the energy intensity of the crack echo is mainly related to the transmission and reflection coefficients. When the frequency is unchanged, the reflection coefficient depends on the size of micro-crack and back-surface roughness. The transmission coefficients are exponentially affected by the surface roughness \(\mathbf{R_{a}}\) according to the phase-screen approximation. To further confirm the results of theoretical analysis, simulation and experimental studies are described in the following sections.