## Waves

The purpose of this category is to display the time evolution of a "wave-form", by which form, we mean a real (or complex) valued function u(x,T) of the "time" T, and a "space" variable x.

At any time T the wave has a given "shape" or "profile", namely the graph of the function x ---> u(x,T), and the program displays this graph with x as abscissa and u(x,T) as ordinate

[NOTE: The resolution of this graph is tResolution, and its domain runs from tMin to tMax. That is to say, the interval [tMin,tMax] is divided into tResolution points x_i, i = 1, ... , tResolution with spacing xStep = (tMax - tMin) / (tResolution - 1) and the graph is plotted at the points (x_i,y_i) where x_i = tMin + (i-1) * xStep and y_i = u(x_i,T), and then these plot points are joined into a polygonal graph. Note that the t of tMax tMin and tResolution have nothing to do with the time T! Thus to make the graph smoother, you must increase tResolution in the Set Resolution & Scale... dialog, and to change the domain of the graph, you must change tMin and tMax in the Set t,u,v Ranges... dialog.]

The time evolution of the wave-form is shown by the standard animation technique; namely, the above graph is first plotted for T=InitialTime, then for T=InitialTime + StepSize, then T=InitialTime + 2*StepSize, etc., until the user clicks the mouse. The variables InitialTime and StepSize are set in the dialog brought up by choosing ODE Settings... from the Settings menu. To make the animation slower (but smoother) decrease StepSize, and conversely increasing StepSize will make the animation of the wave-form evolution proceed more rapidly (but less smoothly).

As usual, the formulas defining a wave form u(x,T) can depend on the parameters aa,bb, ..., ii as well as on x and T, and for each of the canned wave-forms these formulas can be checked by choosing About This Object... from the Waves main menu.

In trying to understand various features of a wave-form, it helps to see not only the animated evolution as above, but also to display the graph of the function u(x,T) as a surface, over the (x,T)-plane or to show "time-slices" of this graph. Therefore the Waves menu has choices to permit the user to switch between these display methods. (These two alternate formats, being three dimensional objects, can be viewed in stereo.)

There is, of course, a User Wave Form... item in the Wave menu, which brings up a dialog permitting the user to enter a formula for u(x,t) as a function of x,t, aa, ..., ii.

Interesting wave-forms generally arise as solutions of so-called "wave equations". These are partial differential equations of evolution type for a function u(x,t). That is, the function u(x,t) is determined as the solution of a PDE satisfying some initial conditions. Perhaps the simplest example is the so-called "linear advection equation" u_t + v*u_x = 0 (where v is some constant "velocity"). This has the general solution u(x,t) = f(x - v*t). But there are also many interesting non-linear wave equations, in particular the so-called soliton equations, including the Korteweg-DeVries (KDV), Sine-Gordon, and Cubic Schroedinger equations. Many of our examples are pure soliton solutions of these latter examples.