GraphingCalculator 4;
Window 45 0 863 1431;
PaneDivider 401;
FontSizes 14;
Slider 0 4;
SliderControlValue 0;
2D.Scale 0.25 0.5 1 2;
2D.BottomLeft -5.46875 -10.9375;
2D.Axes 0;
2D.GraphPaper 0;
Text "<size=17>Travelling waves – reflection and transmission at a free boundary.

Version 0.2 [12-11-08]
To do: 
(i) Improve standing wave demo by incorporating reflection/transmission at left boundary as well.


Define amplitude of the incident wave, wave speed in each medium,  frequency, angular frequency, wavelengths (expressed in terms of wave speed and frequency), and wavenumbers.  

Medium boundary is at x=0.  V1 is the speed to the left of 0 and V2 the speed to the right.   The frequencies must agree at the boundary.
[This is enforced by equality of time derivatives (vertical velocity) at boundary.   Equality of waves and spatial derivatives (slopes) at boundary give reflection/transmission coefficients.]

Since the wave speed V=Lf is fixed this is a non-dispersive medium.  ";
Color 6;
MathPaneSlider 8;
Expr A=slider([0,10,40]);
Color 4;
MathPaneSlider 16;
Expr V_1=slider([0,10,40]);
MathPaneSlider 4;
Expr V_2=slider([0,10,40]);
Color 8;
MathPaneSlider 30;
Expr f=slider([0,5,100]);
Color 6;
Expr W=2*pi*f;
Text "<size=17>Incident, reflected, and transmitted wavenumbers.   (The denominator of k_t is a kludge to prevent k_t from becoming undefined when V2->0.) ";
Color 2;
Expr k_i=W/V_1,k_r=-W/V_1,k_t=W/(V_2+d);
Color 5;
Expr d=sn(1,-8);
Text "<size=17>Reflected and transmitted amplitudes.";
Color 7;
Expr A_r=(k_i-k_t)/(k_i+k_t)*A,A_t=2*k_i/(k_i+k_t)*A;
Text "<size=17>Incident, reflected, and transmitted waves.";
Color 3;
Expr function(Y_i,x)=A*sin([k_i*x-(W*n)]);
Expr function(Y_r,x)=A_r*sin([k_r*x-(W*n)]);
Color 7;
Expr function(Y_t,x)=A_t*sin([k_t*x-(W*n)]);
Color 17;
Expr y=function(Y_i,x),x<0;
Color 17;
Expr y=function(Y_r,x),x<0;
Color 3;
Expr y=function(Y_i,x)+function(Y_r,x),x<0;
Color 4;
Expr y=function(Y_t,x),x>0;
Text "<size=17>Point at boundary:";
Color 17;
Expr vector(0,function(Y_t,0));
Text "<size=17>Medium 2:";
Color 7;
Expr x>0;
Text "<size=17>Note the limit V2->0 yields a fixed boundary condition, while V2/V1 >>1  approaches a free boundary. 

Boundary on the left.  When V2=0, you always get a standing wave, but only certain lengths will ""fit"" within the two boundaries and hence satisfy the boundary conditions at the other end.   (For V2 small but not zero, can also illustrate approximately fixed boundary at the right as one has when a vibrator is driving a string e.g. in the lab.)";
Color 2;
Expr L=slider([-20,0]);
Color 7;
Expr x<L;
Color 17;
Expr y=0;
Text "<size=17>

<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/>";