# Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack
## 2 Analytical Model
### Numerical Integral Solutions of an Analytical Model  
The ultimate objective of this study is to calculate and verify displacement variations, which is defined by the ratio of displacements as scattered wave and incident wave occurrences. The displacements are calculated from the potentials derived by the singular integral Equation (4) Notably, numerical integration methods are key to solve the analytical model to provide reliable and accurate displacement variation. Five Gaussian quadrature numerical integration methods are employed: GLEQ, GCQ, GJQ, GHQ and GLAQ.  
The key to the Gaussian quadrature method is to define the node (\(x\)) and weight (\(w\)), which is the additional coefficient known as \"weight\" at the element and given by the weight function, to calculate integration in numerically as described in Equation (12). The rule of integration is stated as \[\int_{-1}^{1}f(x)dx\approx\sum_{i=1}^{n}f(x_{i})w_{i} \tag{12}\]  
where \(f(x)\) is a continuous function, \(f(x_{i})\) is a dependent variable at a discrete node \(x_{i}\) and \(w_{i}\) is the weight function at the discrete node \(x_{i}\).  
To define the node or the reference point in the \(x\) coordinate, an orthogonal polynomial is utilized. Depending on the applied orthogonal polynomial, the five numerical integration methods are classified. The Gauss-Legendre quadrature is the simplest integration problem by Legendre polynomials on [-1, 1], and the weight function is:  
\[w_{i}=\frac{2}{(1-x_{i}^{2})[P_{n}(x_{i})]^{2}} \tag{13}\]  
where \(P_{n}(x)\) is the Legendre polynomial, which is the solution of the Legendre differential equation \(\big{(}1-x^{2}\big{)}y^{\prime\prime}-2 xy^{\prime}+n(n+1)y=0\).  
The Gauss-Chebyshev quadrature uses Chebyshev polynomials with respect to the weight function, which is given as:  
\[w_{i}=\sqrt{(1-x^{2})} \tag{14}\]  
The Gauss-Jacobi quadrature is for an approximate integral with the Jacobi polynomials and weight function expressed as:  
\[w_{i}=(1-x)^{\alpha}(1+x)^{\beta} \tag{15}\]  
where \(f\) is a smooth function on [-1, 1] and \(\alpha\), \(\beta>-1\). The interval [-1, 1] can be replaced by any other interval by a linear transformation. The Gauss-Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind, \(\int_{-\infty}^{+\infty}e^{-x}f(x)dx\) and the weight function is given as:  
\[w_{i}=\frac{2^{n-1}n!\sqrt{\pi}}{n^{2}[H_{n-1}(x_{i})]^{2}} \tag{16}\]  
where \(H_{n}(x)\) is a Hermite polynomial, which is the solution to [PERSON]'s differential equation, \(y^{\prime\prime}-2 xy^{\prime}+2 ny=0\).  
The Gauss-Laguerre quadrature is an extension of the Gaussian quadrature method for approximating the value of integrals for the following kind \(\int_{0}^{+\infty}e^{-x}f(x)dx\) and the weight function is:  
\[w_{i}=\frac{x_{i}}{(n+1)^{2}[L_{n+1}(x_{i})]^{2}}, \tag{17}\]  
where \(L_{n}(x)\) is the Laguerre polynomial, which gives the solution to [PERSON]'s equation of a second-order linear differential equation, \(xy^{\prime\prime}+(1-x)y^{\prime}+ny=0\). Weight functions of each numerical integration method are described in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Integral Method** & **Weight Function** \\ \hline [PERSON]–[PERSON] quadrature & \(w_{i}=\frac{2}{(1-x_{i}^{2})[P_{n}(x_{i})]^{2}}\) \\ [PERSON]–[PERSON] quadrature & \(w_{i}=\sqrt{(1-x^{2})}\). \\ [PERSON]–[PERSON] quadrature & \(w_{i}=(1-x)^{\alpha}(1+x)^{\beta}\) \\ [PERSON]–[PERSON] quadrature & \(w_{i}=\frac{2^{n-1}n!\sqrt{\pi}}{n^{2}[H_{n-1}(x_{i})]^{2}}\) \\ [PERSON]–[PERSON] quadrature & \(w_{i}=\frac{x_{i}}{(n+1)^{2}[L_{n+1}(x_{i})]^{2}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Weight function of integral methods.