GraphingCalculator 4;
Window 46 11 862 1404;
PaneDivider 447;
FontSizes 14;
Slider 0 20;
SliderSteps 400;
SliderControlValue 0;
2D.Scale 0.1 0.25 5 1;
2D.BottomLeft -2.25 -4.2625;
2D.GraphPaper 0;
Text "<size=17>Travelling waves, non-dispersive medium.

Version 0.50, 4-14-15

Define amplitude, wave speed,  frequency, wave direction, phase constant, wavelength (expressed in terms of wave speed and frequency), wavenumber, and angular frequency.  

Since the wave speed V=Lf is fixed this is a non-dispersive medium.   (One may nonetheless set the speed of each wave independently to show some dispersive effects.)";
MathPaneSlider 2;
Expr A_1=slider([0,10,20]);
Color 7;
MathPaneSlider 2;
Expr A_2=slider([0,10,20]);
Color 4;
MathPaneSlider 8;
Expr V_1=slider([0,5,40]);
MathPaneSlider 8;
Expr V_2=slider([0,5,40]);
Color 6;
MathPaneSlider 10;
Expr f_1=slider([0,5,50]);
Color 5;
MathPaneSlider 10;
Expr f_2=slider([0,5,50]);
Color 5;
MathPaneSlider 1;
Expr a_1=slider([-1,1,1]);
Color 6;
MathPaneSlider 1;
Expr a_2=slider([-1,1,1]);
Color 4;
Expr P_1=slider([0,1,40]);
Color 8;
Expr P_2=slider([0,1,40]);
Color 3;
Expr p_1=2*pi*P_1,p_2=2*pi*P_2;
Expr L_1=V_1/f_1;
Color 2;
Expr L_2=V_2/f_2;
Color 2;
Expr k_1=2*pi/L_1,w_1=2*pi*f_1;
Color 7;
Expr k_2=2*pi/L_2,w_2=2*pi*f_2;
Text "<size=16>Wave moving to the right (a=+1) or left (a=-1):";
Color 3;
Expr function(Y_1,x,a)=A_1*sin([k_1*x-(a*w_1*n)+p_1]);
Color 4;
Expr y=function(Y_1,x,a_1);
Color 17;
Expr y=function(Y_1,x,-a_1);
Text "<size=18>Sum of wave moving to left and wave moving to right is standing wave:";
Color 17;
Expr function(Y_1,x,1)+function(Y_1,x,-1);
Text "<size=18>Superpose a second wave:";
Color 2;
Expr function(Y_2,x,b)=A_2*sin([k_2*x-(b*w_2*n)+p_2]);
Color 5;
Expr function(Y_2,x,a_2);
Color 17;
Expr function(Y_1,x,a_1)+function(Y_2,x,a_2);
Text "<size=18>Point on wave; <size=16>window on it<size=18>:";
Color 5;
MathPaneSlider 2;
Expr X=slider([0,5,40]);
Color 8;
MathPaneSlider 38;
Expr h=slider([-1,1]);
Color 6;
MathPaneSlider 11;
Expr l=slider([0,0.5]);
Color 17;
Expr vector(X,function(Y_1,X,a_1));
Color 17;
Expr vector(X,function(Y_2,X,a_2));
Color 17;
Expr vector(X,function(Y_1,X,a_1)+function(Y_2,X,a_2));
Color 17;
Expr abs(y)>A_1+A_2+h;
Color 17;
Expr abs(x-X)>l,abs(y)<A_1+A_2+h;
Text "<size=17>Point of constant phase Q on wave 1.   Z is the x-coordinate of a point with phase Q at time n (Q=kZ-awn+p).  Phase velocity of that point, with scaling s to improve visibility.";
Color 7;
MathPaneSlider 52;
Expr Q=slider([0,2*pi]);
Color 3;
MathPaneSlider 93;
Expr s=slider([0,1]);
Color 8;
Expr Z=[a_1*w_1*n+Q-p_1]/k_1;
Color 17;
Expr vector(Z,function(Y_1,Z,a_1));
Color 17;
Expr vector(Z,function(Y_1,Z,a_1)),vector(Z+s*V_1,function(Y_1,Z,a_1));
Text "<size=16>Transverse velocity of wave 1 (on wave, on axis, and at X), with scaling s to improve visibility:";
Color 8;
Expr function(V,x,a)=-(s*a_1*w_1*A_1*cos([k_1*x-(a_1*w_1*n)+p_1]));
Color 2;
Expr N=16,d=L_1/N;
Color 17;
Expr vector(b*d,function(Y_1,b*d,a)),vector(b*d,function(V,b*d,a)+function(Y_1,b*d,a)),in(b,set(0,ldots,2*N));
Color 17;
Expr vector(b*d,0),vector(b*d,function(V,b*d,a)),in(b,set(0,ldots,2*N));
Color 17;
Expr y=0,0<x<2*L_1;
Color 17;
Expr vector(X,function(Y_1,X,a)),vector(X,function(Y_1,X,a)+function(V,X,a));
Text "<size=17>Envelope of beat oscillations:";
Color 3;
Expr function(E,x)=2*A_1*cos([(k_2-k_1)/2*x-((w_2-w_1)/2*n)+(p_2-p_1)/2]);
Color 4;
Expr function(P,x)=sin([(k_2+k_1)/2*x-((w_2+w_1)/2*n)+(p_2+p_1)/2]);
Text "<size=17>(Note the V's must be different in order for the phase velocity to be different from the group velocity.)";
Color 17;
Expr y^2=function(E,x)^2;
Color 17;
Expr y=function(E,x);
Color 17;
Expr y=-function(E,x);
Color 17;
Expr function(P,x);
Color 17;
Expr function(P,x)*function(E,x);
Text "<size=15>
Option click on slider play button to get continuous motion in one direction.


<size=14>Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";