# Numerical Simulation of the Kelvin Wake Patterns
## 2 Problem Formulation
Utilizing the mean value theorem of integrals, the result can be described as  
\[\
abla^{2}\;\Phi^{\prime}=\frac{\partial^{2}\;\Phi^{\prime}}{\partial x^{ \prime 2}}+\frac{\partial^{2}\;\Phi^{\prime}}{\partial y^{\prime 2}}+\frac{ \partial^{2}\;\Phi^{\prime}}{\partial z^{\prime 2}}=0\quad\text{for}\;z^{ \prime}<\zeta\big{(}x^{\prime},y^{\prime}\big{)}. \tag{1}\]7 Schematic diagram of the fluid volume and its bounding surface. \[\
abla^{2}\;\Phi^{\prime}=\frac{\partial^{2}\;\Phi^{\prime}}{\partial x^{ \prime 2}}+\frac{\partial^{2}\;\Phi^{\prime}}{\partial y^{\prime 2}}+\frac{ \partial^{2}\;\Phi^{\prime}}{\partial z^{\prime 2}}=0\quad\text{for}\;z^{ \prime}<\zeta\big{(}x^{\prime},y^{\prime}\big{)}. \tag{1}\]8  
Thirdly, as the spherical surface \(S_{1}\) collapses to the source point (0, 0, \(-1\)), the result of this partial integration can be described as  
\[\
abla^{2}\;\Phi^{\prime}=\frac{\partial^{2}\;\Phi^{\prime}}{\partial x^{ \prime 2}}+\frac{\partial^{2}\;\Phi^{\prime}}{\partial y^{\prime 2}}+\frac{ \partial^{2}\;\Phi^{\prime}}{\partial z^{\prime 2}}=0\quad\text{for}\;z^{ \prime}<\zeta\big{(}x^{\prime},y^{\prime}\big{)}. \tag{1}\]9  
Finally, Equation (13) yields  
\[\Phi^{\prime}\sim x^{\prime}-\frac{\epsilon^{\prime}}{4\pi\sqrt{x^{\prime 2 }+y^{\prime 2}+(z^{\prime}+1)^{2}}}\quad\text{as}\;(x^{\prime},y^{\prime},z^{ \prime})\rightarrow(0,0,-1). \tag{2}\]0  
where point \(P\) now lies on the punctures surface \(S_{T}\). The integral in Equation (17) can be rendered less singular by subtraction of the following term, which is equal to zero by utilizing Gauss flux theorem,  
\[\Phi^{\prime}\sim x^{\prime}-\frac{\epsilon^{\prime}}{4\pi\sqrt{x^{\prime 2 }+y^{\prime 2}+(z^{\prime}+1)^{2}}}\quad\text{as}\;(x^{\prime},y^{\prime},z^{ \prime})\rightarrow(0,0,-1). \tag{2}\]1  
Thus, Equation (17) can be re\(-\)written, as follows:  
\[ \begin{array}{rl}2\pi\Bigl{(}\Phi^{\prime(Q)}-x^{\prime}\Bigr{)}=&-\frac{ \epsilon^{\prime}}{\left[{x^{\prime}}^{2}+{y^{\prime}}^{2}+\left(\zeta^{\prime }(Q)+1\right)^{2}\right]^{\frac{1}{2}}}\\ &+\iint_{S_{T}}\bigl{(}\bigl{(}\;\Phi^{\prime}-x^{\prime}\bigr{)}_{P}-\bigl{(} \;\Phi^{\prime}-x^{\prime}\bigr{)}_{Q}\bigr{]}\frac{\partial}{\partial n_{P}} \Bigl{(}\frac{1}{R^{\prime}_{PQ}}\Bigr{)}dS_{P}\\ &-\iint_{S_{T}}\Bigl{[}\frac{1}{R^{\prime}_{PQ}}\frac{\partial}{\partial n_{P }}\bigl{(}\;\Phi^{\prime}-x^{\prime}\bigr{)}_{P}\biggr{]}dS_{P},\end{array}  \tag{19}\]  
where the symbol \(n\) denotes the downward pointing unit normal vector to the free surface and can be written, as follows:  
\[n=\frac{\xi^{\prime}x\dot{t}+\xi^{\prime}y\dot{t}-\mathbf{k}}{\left(1+{\xi^{ \prime}}^{2}_{x}+{\xi^{\prime}}^{2}_{y}\right)^{\frac{1}{2}}}, \tag{20}\]  
Then the surface integrals become the double integrals by projection onto the \(x-y\) plane using the following formula:  
\[ds=\frac{dxdy}{|\mathbf{k}\cdot\mathbf{n}|}. \tag{21}\]  
On the free surface, the velocity potential can be regarded as a function simply of the two independent variables \(x^{\prime}\) and \(y^{\prime}\) and will be denoted by the symbol \(\phi^{\prime(x^{\prime},y^{\prime})}=\;\Phi^{\prime}(x^{\prime},y^{\prime}, \xi^{\prime}(x^{\prime},y^{\prime}))\); the final boundary integral equation can be described, as follows:  
\[ \begin{array}{rl}2\pi\Bigl{(}\phi^{\prime(Q)}-x^{\prime}\Bigr{)}=&-\frac{ \epsilon^{\prime}}{\left[{x^{\prime}}^{2}+{y^{\prime}}^{2}+\left(\zeta^{\prime }(Q)+1\right)^{2}\right]^{\frac{1}{2}}}\\ &+\int_{0}^{\infty}\int_{-\infty}^{\infty}[\phi^{\prime}(P)-\phi^{\prime}(Q) -\rho^{\prime}+x^{\prime}]K^{(1)}(\rho^{\prime},\sigma^{\prime},x^{\prime},y ^{\prime})d\sigma^{\prime}d\rho^{\prime}\\ &+\int_{0}^{\infty}\int_{-\infty}^{\infty}\xi^{\prime}_{\rho}(P)K^{(2)}(\rho^ {\prime},\sigma^{\prime},x^{\prime},y^{\prime})d\sigma^{\prime}d\rho^{\prime}, \end{array}  \tag{22}\]where the kernel functions \(K^{(1)}(\rho^{\prime},\sigma^{\prime},x^{\prime},y^{\prime})\) and \(K^{(2)}(\rho^{\prime},\sigma^{\prime},x^{\prime},y^{\prime})\) are described, as follows:  
\[