Traveling Waves

1. Introduction and Instructions

In this exercise you will investigate the behavior of traveling waves. Section 2 is an introduction to the Mathematica  notebooks that we will be using throughout the semester and demonstrates how one defines functions, plots, prints, and makes 'movies'. In Section 3 you will explore the behavior of several functions that are defined in Section 2.1 and determine which ones are traveling waves. Finally, you will conjure up a traveling wave function of your own and subject it to the same tests. Include a purpose, proofs, figures, and a discussion in your report.

2. Mathematica Notebooks and Commands

Mathematica  documents are known as notebooks and are used to manage your work session.  Each notebook in turn consists of several "cells" containing different parts of the session. Below you should see several subsection titles for the different parts of this exercise. To open any section or cell place the pointer over the small box located at the bottom of the bracket immediately to the right of each section title. You should see the pointer change shape. When this occurs double-click the mouse by pressing down momentarily on the mouse button twice in succession and then releasing. Now open section 2.1 and follow the instructions there.

2.1 Mathematical Syntax and Test Functions

The lines below display definitions of the functions from your assignment that you are to plot in their Mathematica  format. Please note several things. The software interprets a blank space or the character "*" as multiplication (each character is used randomly in the definitions below). Notice the need for the trailing underscore, "x_", when defining the function and the use of ":=". To actually define these functions in Mathematica  place the pointer anywhere in the cell (note cell bracket on the right) and click by pressing down briefly on the mouse button. The editing cursor should now appear and will be located somewhere within the cell. Now  press "Enter" (not "Return") on the keyboard. Notice how the bracket of the cell changes shape. This means Mathematica   is executing the commands you have given it. When it is done the bracket will return to its original shape. Now open section 2.2 and follow the instructions there.

psi1[x_] := A*Cos[w*t]*Sin[k*x + B]
psi2[x_] := A*Cos[k*(x - c*t)]
psi3[x_] := A*Exp[-0.02*(x + c*t)^2]*Cos[k*(x + c*t)]

2.2 Plotting Commands

The lines below display some of the commands needed to plot the functions defines above. First one must assign values to the constants in the problem. Place the pointer in the cell below and click. After the editing cursor appears hit "Enter".

A = 1.0
k = 1.0
B = 3.14
c = 1.0
w = 1.0
t = 41.5

In the cell below is the command to plot the first function defined above (psi1[x]). Several options of the "Plot" command have been used to illustrate their function and to produce a clear, well-labelled graph. The Mathematica  book by Wolfram on the online Help in Mathematica contain detailed descriptions of the meaning of each option.To evaluate the commands in the cell place the pointer anywhere in the cell and click. After the editing cursor appears hit "Enter". Try different values for the numbers to see how they affect the plot.

Plot[psi1[x], {x,0,10}, 
       Frame->True,
       PlotRange->{-1,1},
       FrameLabel->{"x",
                    "psi3",
                    "Fig 4. t = 41.5",
                    " "},
       RotateLabel->True]

2.3 Printing

To print a portion of your work highlight the brackets on the right of the notebook. This is done by placing the pointer over the top bracket first. The pointer will change shape. Now press and hold the mouse button down and drag the pointer through the brackets you want to print. The brackets will be highlighted if this is done correctly. Release the mouse button and go to the "File" menu at the top of the screen. Click and drag down to "Print Selection". Highlight and release the button. You will be presented with a dialog box. Click the “Print” button. Your output will soon appear at the printer.

2.4 Making Movies

If you really want to demonstrate your mastery of the Mathematica  package, use the following commands to create a 'movie' of a function developing in time. After the 'MoviePlot' command is executed select all the different frames by clicking on the bracket that encloses the movie frames. Then, choose the command 'Animate Selected Graphics' from the 'Graph menu above.

f[x_,time_]:=Sin[x]+0.2*Sin[3.0(x-0.2*time)]
p1=Animate[Plot[f[x,time],{x,0,10}],
{time,0,40*π, 2*π/100},
AnimationRunning->False]

3. The Time Evolution of Functions

You can now complete the assignment using the commands demonstrated above. Which of the mathematical forms defined in Section 2.1 describes a lone, traveling wave and which does not? Use Mathematica's plotting capability to support your assertions graphically and then prove this analytically, i.e. construct a mathematical argument to support the result you demonstrated visually.Describe in clear language the time evolution of the three wavefunctions listed above using appropriate figures (that you generate) to support your assertions. 

Once you have done this for the three plots defined in Section 2.1, invent a fourth wave function of your very own that corresponds to a traveling wave and is different from the ones listed above. How does it satisfies the analytical and graphical tests you used above? Include a purpose, proofs,  figures, and a discussion in your report.