(* \input mathlily
\heading{Introducing D'Alembert's solution of the wave equation}
\synopsis{These commands are primarily intended to be typed in as a 
demonstration, rather than input as a file.}
 \section{Discussion}
 *)

 (* This demonstration deals with a function of the form $B(x-t)$.
 We verify that it solves the two-variable wave equation,
 $${\partial ^2u\over\partial x^2} ={\partial ^2u\over\partial 
t^2}\,.$$
 Then we plot $B(x-t)$ as a function of $x$ for a sequence of 
values of $t$ and animate the results, verifying that the solution 
 represents a traveling wave. *)


(* \section{Demonstration} *)

 (* \subsection{Define the basic function.} *)

 (*  We choose an example that has a
 localized peak and approaches 0 at the sides, since that will 
 best demonstrate the traveling-wave behavior.
 (Generally speaking, we use lower case for  {\sl 
Mathematica\/} variable names, to avoid possible conflicts with
 built-in {\sl Mathematica\/} function names.) *)

 b[z_] := 1/(1 + z^2)

 (* \subsection{Verify the differential equation.} *)

 (* Calculate a second derivative. *)

 D[b[x-t], {x,2}]

 (* Subtract the corresponding derivative with respect to the other 
variable. *)

 % - D[b[x-t], {t,2}]

(* \newpage *)
(*  \subsection{Plot and animate.} *)

   (* First try a single plot. *)

 Plot[b[x-1], {x,-10,10}]

 (* Animation is hardware-dependent. *)

 (* [[ On a Sun-3 running SunView: *<

 
<<Graphics`Animation`
Animate[Plot[b[x-t], {x,-10,10}, PlotRange->{0, 1.2}], 
 {t,-12,12,2}] 


>* The plots will briefly appear individually, then the animation 
  will start.  ]] *)

 (* [[ On a NeXT: *)

 Do[Plot[b[x-t], {x,-10,10}, PlotRange->{0, 1.2}], 
 {t,-12,12,2}]

 (* This will display all the plots sequentially.  Use the mouse to 
mark the large bracket (at right of the window) surrounding the 
entire list of plots.  Then click on the menu item ``graphs" and the 
submenu item ``animate selected graphics".  Use the buttons at the 
bottom of the animation display window to slow down the animation 
for clarity. ]] *)

 (* \endlily *)