## 1 Introduction  
The detection of surface corrosion defects and fatigue cracks is important in the petrochemical and infrastructure industries [1; 2]. Lack of timely and accurate assessment of these forms of distortion can eventually lead to either leakage or explosion, with potential financial and environmental consequences [3]. Ultrasonic testing is a common nondestructive testing technique carried out to reduce the risk of accidents. Inspections using ultrasound are often carried out in water baths and the pulsed acoustic beams launched by transducer are incident to the liquid-solid interfaces. The crack size is generally considered to be much larger than the geometry of irregular microstructures of the surface. The surfaces of workpieces are assumed to be perfectly smooth in previous theoretical analyses and in practical application [4; 5; 6]. However, when the size of the micro-crack is of the same order of magnitude as that of the microscopic surface fluctuation, wave scattering on the rough surface dramatically alters the shape of the received signal of the micro-crack [7; 8]. Therefore, understanding and accurately predicting the effect of surface roughness on the detection capacity of micro-cracks is the primary concern behind the work presented in this paper.  
Elastic wave scattering characteristics on rough surfaces can be scientifically identified and described [9; 10]. A variety of analytical models involving reflection and transmission waves such as the phase perturbation approximation [11; 12; 13] and Kirchhoff theory [14; 15; 16; 17; 18], have been developed to understand the scattered fields of different scatterers. [PERSON] explored the validity of the perturbation approximation for rough surface scattering using a Gaussian roughness spectrum [19]. [PERSON] discussed Kirchhoff approximation theory to estimate the complicated effects of wave scattering on rough defects [20; 21; 22]. [PERSON] and [PERSON] introduced theoretical formulae for diffuse fields of elastic waves on random rough solid-solid interfaces using the Kirchhoff approximation [23]. However, there is a consensus that none of the analytical models--which are based on mathematical assumptions--can describe rough surface scattering with a full solution, especially for some complex defect geometries. The finite element model (FEM) was used to accurately solve the problem of elastic wave scattering [24; 25; 26; 27]. This numerical method was used to measure the thickness distribution of corrosion surfaces [28; 29] and to analyze rough crack scattering [30; 31]. Many factors must be considered when ultrasonic testing for a crack scatterer is discussed, including incident frequency, transducer location, and the noise induced by backscatter of ultrasound from the microstructure. This investigation focuses primarily on the effects of surface roughness while other factors remain unchanged.  
In this work, the relationship between changes in ultrasound wave energy intensity for micro-crack detection and the roughness-modified transmission and reflection coefficients is briefly introduced. An ultrasonic simulation model used for detailed analysis is presented in Section 3. Two-dimensional sinusoidal curves are used to represent rough surface fluctuation. The signal-to-noise ratio loss of crack pulse-echo and signal distortion of micro-crack at different surface roughnesses \(\mathbf{R_{a}}\) and \(\mathbf{R_{sm}}\) are analyzed significantly. In Section 4, a number of experiments are designed to verify the effect of rough front surface and rough back surface. The results from experimental measurements are compared to theoretical analyses and simulations. An accurate description of the detection capacity of micro-cracks is obtained. Section 5 contains a brief conclusion.