Changeset 2687 for documentation/experimentation/smf.tex

Timestamp:

Apr 10, 2006, 5:20:59 PM (17 years ago)

Author:

sexton

Message:

updating working documents

File:

• documentation/experimentation/smf.tex (modified) (3 diffs)

documentation/experimentation/smf.tex

@@ -16 +16 @@
 
 \usepackage{lscape} %landcape pages support
+%\input{definitions}
 
 \title{Application of SMF surface elevation function in inundation modelling}
@@ -55 +56 @@
 the system volume is conserved. As a result,
 
-$$\kappa' = {\rm erf} ( (x - x0)/ \sqrt \lambda_0 ) /
-{\rm erf} ( (x - \Delta x - x0)/ \sqrt \lambda_0 )]_{x_min}^{x_max}$$
+$$\kappa' = [{\rm erf} ( (x - x_0)/ \sqrt \lambda_0 ) /
+{\rm erf} ( (x - \Delta x - x_0)/ \sqrt \lambda_0 )]_{x_{\rm
+min}}^{x_{\rm max}} \ .$$
 
-\noindent with $\kappa' \ge 1$ for $\Delta x \ge 0$. Figure 2 in [1]
-could then be reproduced for appropriate values of $\kappa'$ to
+\noindent The relationship between $\kappa$ and $\Delta_x$ can be
+seen in Figure \ref{fig:vol_cons} where $\kappa$ approaches $\inf$
+quickly.Additionally, it is not possible for $\kappa' = 0.83$ as
+shown in Figure 2 of [1] as {\rm erf(x)} = 1 for ${\rm abs} x >
+5.93$. For the example described in Figures 2 and 3 of [1], whilst
+$\kappa'$ is technically less than 1 for $\Delta x < 5$ it is
+effectively equal to 1 for $0 \le \Delta x \approx 5$.
+
+
+Figure 2 in [1]
+could then be reproduced for appropriate values of $\kappa'$ and $\Delta_x$ to
 ensure conservation of mass within the system. Using the above
 formulation, the values of interest shown in Figure 2 of [1] would
-be ($\kappa', \Delta x) = (1,0) and (1.2, 0.0167)$. This function
-becomes unbounded for small $\Delta x $ thereby limiting both
-parameters.
+be ($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$ and shown in
+Figure \ref{fig:eta_vary}.
 
-%\caption Relationship between \kappa' and \delta x to ensure volume
-%conservation
+
+
+\begin{figure}[hbt]
+
+  %\centerline{ \includegraphics[width=75mm, height=75mm]{volume_conservation.ps}}
+
+  \caption{Relationship between $\kappa'$ and $\Delta x$ to ensure volume conservation.}
+  \label{fig:vol_cons}
+\end{figure}
+
+\begin{figure}[hbt]
+
+  %\centerline{ \includegraphics[width=75mm, height=75mm]{redo_figure.ps}}
+
+  \caption{Surface elevation functions for
+($\kappa', \Delta x) = (1,2), (1,4), (1.2, 13.48)$.}
+  \label{fig:eta_vary}
+\end{figure}
+
 
  {\bf TO DO:} Need a discussion in here on whether the
@@ -81 +108 @@
 recast as
 
-$$\kappa'  \approx {\rm erf} ( (x - x0)/ \sqrt\lambda_0 ) / {\rm erf} ( (x - 2 x0
+$$\kappa'  \approx {\rm erf} ( (x - x_0)/ \sqrt\lambda_0 ) / {\rm erf} ( (x - 2 x_0
 - x_g)/ \sqrt \lambda_0 )$$