stationary-surface.tex (1828B)
1 \subsubsection{Stationary surface} 2 3 When a particle touches a surface given by 4 \begin{equation*} 5 \vec{S}=\vec{S}(a,b,t); 6 \qquad 7 a,b\in{}A=[0,1]; 8 \qquad 9 \vec{n}=\frac{\partial\vec S}{\partial a} \times \frac{\partial\vec S}{\partial b} 10 \end{equation*} 11 where \(a\) and \(b\) are parameters, boundary condition is written as 12 \begin{equation*} 13 \frac{d}{d t} \vec\nabla w \cdot\vec{n} = 0; 14 \qquad 15 \vec\zeta = \vec S 16 . 17 \end{equation*} 18 The third component of the normal vector is nought: \(\vec{n}=(n_1,n_2,0)\), 19 \(\left|\vec{n}\right|=1\). 20 The direction of incident wave is \(\vec{d}_i=(u,v,0)\) and the 21 direction of reflected wave is \(\vec{d}_r\) from 22 equation~\eqref{eq-dr}. We seek solutions of the form 23 \begin{equation} 24 \label{eq-w-all-0} 25 \begin{aligned} 26 w\left(\alpha,\beta,\delta,t\right) = 27 \, & 28 C_1\exp\left( \left(i\vec{d}_i+\vec{d}_{k}\right)\cdot\vec\zeta - i\omega 29 t\right) \\ + 30 \, & 31 C_2\exp\left( \left(i\vec{d}_r+\vec{d}_{k}\right)\cdot\vec\zeta - i\omega t\right). 32 \end{aligned} 33 \end{equation} 34 Here \(k\) is damping coefficient and \(\omega\) is wave velocity. 35 The coefficient is calculated from continuity equation. 36 We plug this expression into the boundary condition and get 37 \begin{equation*} 38 \left(\vec{d_i}\cdot\vec{n} \right) C_1 39 + 40 \left(\vec{d_r}\cdot\vec{n} \right) C_2 41 \exp\left( -i\vec{d}_s\cdot\vec\zeta_0 \right) = 0. 42 \end{equation*} 43 Hence 44 \begin{equation*} 45 C_1 = 46 \frac{1}{2} 47 \exp\left( -\frac{1}{2}i\vec{d}_s\cdot\zeta_0 \right) 48 \qquad 49 C_2 = 50 \frac{1}{2} 51 \exp\left( \frac{1}{2}i\vec{d}_s\cdot\zeta_0 \right) 52 \end{equation*} 53 Here we substituted \(\vec{d}_r\cdot\vec{n}\) with \(-\vec{d}_i\cdot\vec{n}\) 54 (see~sec.~\ref{sec:org653b1bb}). 55 Now plug the solution into continuity equation to get damping coefficients 56 and velocity: 57 \begin{equation*} 58 k = \pm\sqrt{u^2+v^2}; 59 \qquad 60 \omega^2 = g k. 61 \end{equation*} 62