Wall

Wall Regular

The Wall is a function that is

\[ w(-\infty) = 0,\qquad w(0)=\epsilon, \qquad w(h)=1,\qquad w(\infty)=\infty \]

and grows quadratically for \(x>0\). It is a linear combination of lines and hyperbolas:

\[ w(x) \DEF \begin{cases} \dfrac{\epsilon}{1-z} & x < 0 \\[1em] \epsilon + a z \left( z-\dfrac{4}{3} \right) + \dfrac{bz}{\frac{1}{4}+z} + \dfrac{cz}{\frac{1}{2}+z} + \dfrac{dz}{\frac{3}{4}+z} + \dfrac{ez}{1+z} & \textrm{otherwise}, \end{cases} \]

where \(z\DEF x/h\) and the coefficients \(a\), \(b\), \(c\), \(d\) and \(e\) are defined as

\[ \begin{align*} a &\DEF \dfrac{315}{104} - \dfrac{1551}{208}\epsilon, \\[0.5em] b &\DEF \dfrac{385}{3328} - \dfrac{10055}{19968}\epsilon, \\[0.5em] c &\DEF \dfrac{243}{13}\epsilon - \dfrac{105}{26}, \\[0.5em] d &\DEF \dfrac{42525}{3328} - \dfrac{462105}{6656}\epsilon, \\[0.5em] e &\DEF \dfrac{1882}{39}\epsilon - \dfrac{70}{13} \end{align*} \]

The result is shown in Figure XO:fig:wall. The first and second derivative are respectively:

\[ \begin{align*} hw'(x)&= \begin{cases} \dfrac{e}{(1+z)^2}+\dfrac{12d}{(3+4z)^2}+\dfrac{2c}{(1+2z)^2}+\dfrac{4b}{(1+4z)^2}+a & x \geq 0 \\[1em] \dfrac{\epsilon}{(1-z)} & x < 0 \end{cases} \\[0.5em] h^2w''(x) &= \begin{cases} 2a-\dfrac{2e}{(1+z)^3}-\dfrac{96d}{(3+4z)^3}-\dfrac{8c}{(1+2z)^3}-\dfrac{32b}{(1+4z)^3} & x \geq 0 \\[1em] \dfrac{2\epsilon}{(1-z)^3}& x < 0 \end{cases} \end{align*} \]

The coefficient are evaluated so that \(w(h)=1\) and \(w(0)=\epsilon\).

Wall Smooth

The Wall is a function that is

\[ w(-\infty) = 0,\qquad w(0)=\epsilon, \qquad w(h)=1,\qquad w(\infty)=\infty \]

and grows quadratically for \(x>0\).

\[ w(x) = \dfrac{A}{2B^2}\left(\left(\frac{1}{2}+B^2x^2\right)(\erf(Bx)+1)+\dfrac{Bx}{\sqrt{\pi}}\exp(-B^2x^2)\right) \]

Wall Piecewise

The Wall is a function that is

\[ w(x) = 0,\quad x\in(-\infty,0), \qquad w(h)=1,\qquad w(\infty)=\infty \]

and grows quadratically for \(x>0\).

\[ w(x)= \dfrac{1}{3} \begin{cases} 0 & x<0 \\[1em] z^4 ( 5 - 2z) & x