# Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack
## 1 Introduction  
The propagation of disturbances known as wave propagation in solids has been studied in many branches of physical science and engineering. Wave propagation carries energy (e.g., kinetic and potential energies) through a solid medium and energy transmission over a radiation pattern is through the motion of particles in the wave motion phenomenon. Although wave motion in elastic solids is well-studied, adequately defining a wave scattering model by objects such as cracks still can be quite a challenge. Over the past decades, crack evaluations using a wave scattering approach in nondestructive evaluations (NDEs) have gained much attention [1; 2]. Many researchers in the early 1980s offered analytical solutions to define and explain the scattered surface waves caused by cracks (e.g., subsurface cracks) [3; 4; 5]. A vertical subsurface crack is considered capable of generating an efficiently scattered field since the crack is one of the most influential damages of surface wave propagation. The analytical model of a vertical subsurface crack [3] is obtained by establishing an integral representation for the scattered field expressed in terms of the fundamental potentials first introduced by [PERSON] [6]. Furthermore, numerical methods have been devised to solve the singular integral equations of the analytical model. For example, the Gauss-Chebyshev quadrature rule as a numerical method was adopted by [PERSON] and [PERSON] [7]. Understanding the numerical solution of a singular integration is vital to providing an accurate and reliable solution to the analytical wave scattering model. Such studieshave focused on scattered wave motions presented in relation to the displacement variations caused by different frequencies and geometrical crack parameters. Based on this approach, a well-defined relationship can be developed to evaluate causes and estimate behavioral patterns in cracks. Although using the relationship between displacement variations and frequencies can lead to evaluate cracks by solving the inverse problem, research is needed to verify displacement variations in the existing analytical solutions. Meanwhile, high-performance computers and simulation software have facilitated efficient wave modeling studies [8; 9], which perform with experimental studies under the constraints of time, environment, sources, etc. Several methods have been conducted to simulate the mechanical wave propagation in solid mediums. Among them, finite element (FE) simulation is widely used for higher accuracy and adaptability. [PERSON] and [PERSON] [8] performed a comparative study of FE simulation of wave propagation and laser interferometry experiment with a cracked steel plate to show the reliability of FE simulation. FE results allow simulation of waveform change between the incident wave and scattered wave based on the resultant crack(s). One challenge of FE simulation of wave propagation is that it requires a large-sized simulation model to avoid wave reflection at the model's boundary. Furthermore, the model should have enough distance between the wave source and listening nodes to obtain pure waveforms (e.g., surface wave). Typically, a minimized model has been designed to streamline computational efficiency. [PERSON] and his colleagues [10] conducted practical FE simulations using a damper and energy-absorbing element known as an infinite element to minimize the model size. They proposed a combination of damper and infinite element with a required absorbing thickness and damping factor, so dampers could prevent wave reflection. Both analytical and simulation methods (e.g., FE) are powerful approaches for wave modeling, which aim to predict or provide a specific parametric relationship in a system (e.g., variations of a scattered wave amplitude by different frequencies and crack sizes). The simulation method is beneficial when a complex wave model is needed (e.g., complicated geometry and various incident waves).