## 1 Introduction  
The ultrasonic nonlinear technique is considered a promising nondestructive evaluation method to evaluate material degradation [1]. The nonlinear interaction between a material and a propagating ultrasonic wave distorts the waveform of the incident ultrasonic wave owing to the nonlinear elasticity of the material, and the second-order harmonic frequency component is generated. This nonlinear elasticity can be quantified by measuring the ultrasonic nonlinearity parameter, \(\beta\), determined from the ratio of the second-order harmonic displacement amplitude to the square of the fundamental frequency displacement amplitude [1, 2]. The ultrasonic nonlinearity parameter is closely related to the microstructural changes in a material, such as precipitates [3] and dislocations [4]. Thus, it can be used to evaluate the microstructural deformations caused by fatigue damage [5, 6, 7], creep [8], and thermal aging [9, 10, 11]. This parameter can be of the absolute or relative type. The absolute nonlinearity parameter is defined using the displacement amplitude, whereas the relative nonlinearity parameter is defined using the received signal amplitude, regardless of whether there is a displacement [12]. Some techniques are available to measure the ultrasonic nonlinearity parameter, such as the piezoelectric [13], capacitive [14], and laser interferometer methods [15]. Among these methods, the piezoelectric method using a contact-type transducer is more widely used than other nonlinear measurement techniques because this method can reduce the influence of the surface roughness by applying a couplant between the material and transducer. However, if the surface is rough, the air gaps between the material and transducer are not completely eliminated even if a couplant is applied. Thus, the normal incident ultrasonic wave is reflected and scattered by the air gaps, leading to the attenuation and resonance frequency shift of the received ultrasonic signal [16; 17; 18; 19]. In other words, the receiving sensitivity at a certain interesting frequency determines that the ultrasonic nonlinearity parameter is changed. The amplitude of the second-order harmonic frequency component is very weak compared to that of the fundamental frequency component; thus, the surface condition of the material can considerably affect the measurement of the ultrasonic nonlinearity parameter. Therefore, the influence of surface roughness must be considered to accurately measure the ultrasonic nonlinearity parameter. Accordingly, studies to analyze the influence of the surface roughness on ultrasonic measurements have been conducted. [PERSON] et al. [17] analyzed the received electrical signal as a function of the roughness of the planar interfaces and confirmed that the amplitude of the received electrical signal decreases on rough surfaces. [PERSON] et al. [18] introduced an analytical model to evaluate the behavior of the reflected and transmitted waves generated on rough surfaces using a first-order phase perturbation technique, and the attenuation of the ultrasonic wave by the rough surface calculated using the proposed analysis model agrees well with the experimental results. [PERSON] et al. [19] studied the relationship between the amplitude and the surface roughness using an air-coupled ultrasonic transducer. The results showed that the amplitude decreases markedly and that the frequency spectra change as the surface roughness increases. However, these studies analyzed only the linear ultrasonic characteristics. Only a few studies have analyzed the influence of surface roughness on the measurement of the ultrasonic nonlinearity parameter. [PERSON] et al. [20] investigated the influence of surface roughness on the ultrasonic nonlinearity parameter measurements for 410 Cb stainless steel with rough surfaces on both sides of the specimens using the piezoelectric method.