GraphingCalculator 4;
Window 52 29 818 1440;
PaneDivider 390;
FontSizes 14;
Slider 0 20;
SliderSteps 400;
SliderControlValue 320;
2D.Scale 0.1 0.25 5 1;
2D.BottomLeft -2.390625 -4.33125;
2D.GraphPaper 0;
Text "<size=17>Travelling waves, dispersive medium.

Version 0.65, 4-8-13 

Define amplitude, wavelength, frequency, wave direction, phase constant, wavenumber and angular frequency.

Since the wave speed V = Lf is not fixed – L and f may be independently specified – this may simulate a dispersive medium.
Note for equal amplitudes, the phase velocity of the superposed wave is v_p=w_ave/k_ave = (f_ave/L_ave)*(L_1L_2) and the group velocity v_g=(Delta w)/(Delta k)=-(Delta f/Delta L)*(L_1L_2).  

Both L and f must be different in order for the group velocity to be (i) nonzero and (ii) different from the phase velocity.  Illustrative values are L1=1, f1=1, and L2 = 0.5.  Then with f2 = 1.5, vp>vg; with f2 = 2, vp=vg, and with f2=2.5, vp<<vg.";
MathPaneSlider 2;
Expr A_1=slider([0,10,20]);
Color 7;
MathPaneSlider 2;
Expr A_2=slider([0,10,20]);
Color 4;
MathPaneSlider 2;
Expr L_1=slider([0,10,20]);
MathPaneSlider 2;
Expr L_2=slider([0,10,20]);
Color 6;
MathPaneSlider 2;
Expr f_1=slider([0,10,20]);
Color 5;
MathPaneSlider 2;
Expr f_2=slider([0,10,20]);
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;
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;
Color 2;
Expr V_p=(f_2+f_1)/(L_2+L_1)*L_1*L_2,V_g=-((f_2-f_1)/(L_2-L_1)*L_1*L_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=17>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=17>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=17>Point on wave<size=18>; <size=16>window on it<size=17>:";
Color 8;
MathPaneSlider 4;
Expr X=slider([0,5,40]);
Color 4;
MathPaneSlider 36;
Expr h=slider([-1,1]);
Color 5;
MathPaneSlider 33;
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.   Z is the x-coordinate of a point with phase Q at time n.";
Color 3;
MathPaneSlider 11;
Expr Q=slider([0,2*pi]);
Color 7;
Expr Z=a_1*(w_1/k_1)*n+Q/k_1;
Color 17;
Expr vector(Z,function(Y_1,Z,a_1));
Text "<size=17>Envelope of beat oscillations:";
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]);
Text "<size=17>Resultant wave y1+y2=E*P <I>i.e.</I> P modulated by envelope E.";
Color 6;
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 [=Lf] must be different in order for the phase velocity [w_ave/k_ave] to be different from the group velocity [Delta w/Delta k].   Otherwise P moves at the same speed as E and the wave retains its shape as it evolves.)";
Color 17;
Expr y^2=function(E,x)^2;
Color 17;
Expr function(E,x);
Color 17;
Expr function(P,x);
Color 17;
Expr function(E,x)*function(P,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/>";