# Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack
## 2 Analytical Model
### Derivation of Displacement Potentials. The displacement field as the final theoretical model can be expressed as two potential functions, a scalar potential \(\varphi(x,y)\) and a vector potential \(\psi(x,y)\) by Helmholtz decomposition [14]:  
\[u=\
abla\varphi+\
abla\times\psi \tag{1}\]  
where \(u\) is the displacement field of scattered waves and \(\
abla\) is the Nabla or Del operator. Computing the displacement potentials is the main part of defining the theoretical scattering model. The displacement field is defined in the \(z\)-invariant condition. The waves in the half-plane are \(z\)-invariant waves where all functions are dependent on the \(z\)-direction. The displacement potentials are defined in the scattered field and satisfy a radiation condition of the elliptic boundary-value problem by [PERSON] [15] as the equation including the exponential and small '\(o\)' functions:  
\[ \begin{split}\varphi&=A_{\pm}e^{\pm ik_{R}x-(k_{R} ^{2}-k_{T})^{2}\frac{1}{y}}+A(\theta)(k_{L}r)^{-\frac{1}{2}}e^{ik_{L}r}+o\Big{(} (k_{L}r)^{-\frac{1}{2}}\Big{)}\\ \psi&=B_{\pm}e^{\pm ik_{R}x-(k_{R}^{2}-k_{T})^{1/2} y}+B(\theta)(k_{T}r)^{-\frac{1}{2}}e^{ik_{L}r}+o\Big{(}(k_{T}r)^{-1/2} \Big{)}\end{split}  \tag{2}\]  
where \(A\) and \(B\) are constant; \(k_{R}\), \(k_{L}\) and \(k_{T}\) are the Rayleigh wavenumbers, longitudinal waves and transverse waves, respectively. \(r\) and \(\theta\) are the components of the polar coordinate (\(r\), \(\theta\)). The \(\pm\) signs denote forward direction, \(+\) and backward direction, \(-\). The part including the polar coordinate in the Equation (2) is ignored since the incident wave angle is 90 degrees. The coefficient \(A\) and \(B\) satisfy the equation and are given as:  
\[A_{+}/B_{+}=-A_{-}/B_{-}=\left(2k_{R}{}^{2}-k_{T}{}^{2}\right)/2 ik_{R}(k_{R}{} ^{2}-k_{L}{}^{2})^{1/2} \tag{3}\]  
The constant B can be obtained by Equation (3) when the constant \(A\) can be calculated as an explicit formula given [PERSON]'s approach [16], which can be expressed as:  
\[A_{\pm}=\frac{\left(2k_{R}{}^{2}-k_{T}{}^{2}\right)^{2}k_{R}}{2k_{T}{}^{2}S} \int_{a}^{b}dy\Big{(}\pm d_{y}(y)P(y)+id_{x}(y)Q(y)\Big{)} \tag{4}\]  
Where  
\[P(y)=2k_{R}\Big{\{}e^{-(k_{R}{}^{2}-k_{L}{}^{2})^{1/2}y}\big{(}k_{R}{}^{2}-k_{ T}{}^{2}\big{)}^{1/2}-e^{-(k_{R}{}^{2}-k_{T}{}^{2})^{1/2}y}\big{(}k_{R}{}^{2}-k_{L}{ }^{2}\big{)}^{1/2}\Big{\}} \tag{5}\]  
Figure 1: Illustration of subsurface crack on the two-dimensional \(x-y\) half-plane. \(a\) is the distance between the ground and the top of the crack tip, \(b\) is the distance between the ground and the bottom of the crack tip and \(r\) and \(\theta\) are components of the polar coordinate. The image shows the directions of the incident waves and scattered waves influenced by different \(a\) and \(b\) distances. \[Q(y)=e^{-({k_{R}}^{2}-{k_{L}}^{2})^{1/2}}\frac{\Big{(}2{k_{R}}^{2}-2{k_{L}}^{2}+{k_ {T}}^{2}\Big{)}\Big{(}{k_{R}}^{2}-{k_{T}}^{2}\Big{)}^{1/2}}{\big{(}{k_{R}}^{2}-{k _{L}}^{2}\big{)}^{1/2}}-e^{-({k_{R}}^{2}-{k_{T}}^{2})^{1/2}y}\Big{(}2{k_{R}}^{2}- {k_{T}}^{2}\Big{)} \tag{6}\]  
\[S=\Big{(}4{k_{R}}^{2}-{k_{T}}^{4}\Big{)}\Big{(}{k_{R}}^{2}-{k_{L}}^{2}\Big{)}^{ 1/2}\Big{(}{k_{R}}^{2}-{k_{T}}^{2}\Big{)}^{1/2}-2{k_{R}}^{4}\Big{(}2{k_{R}}^{2} -{k_{L}}^{2}-{k_{T}}^{2}\Big{)} \tag{7}\]  
The components of integration in Equation (4) are given as:  
\[\int_{a}^{b}dy\Big{\{}\pm d_{y}(y)P(y)+id_{x}(y)Q(y)\Big{\}} \tag{8}\]  
where, \(d_{y}(y)\) and \(d_{x}(y)\) are the displacement functions in \(y\)-direction and \(x\)-direction, respectively. The displacement functions can be obtained by [PERSON]'s law, \(d=F/K\), where \(d\) is a displacement matrix, \(F\) is a force matrix and \(K\) is a stiffness matrix. The force matrix can be computed by the stress field given by [PERSON] [3] as:  
\[