VISUALIZING NORMAL MODES

The text introduces us to normal modes in Chapter 11, section 3.  Below are a few animations designed to help visualize the discussion in the text.  Consider first the very simple function

y(x, t) = sin (x) cos (v t)

Every point on this curve will oscillate with the same period (2π/v), so that the shape of the curve remains constant while its amplitude changes. In other words, the shape will stay the same, but the curve will expand and shrink as t evolves.  Following the example in the text, we will set v = 2.  To begin the animation, you can either move the slide bar or click on the small gray square just to the right of the slide.  (When you hover over this square with the cursor you should see “Show Animation Controls”).  If you click on this, you will see a set of controls that will allow you to control the speed of the animation.

Clear[y]
y[x_,t_]:= Sin[x] Cos[2t]
Manipulate[Plot[y[x,t],{x,0,π}, PlotRange->{-1,1}], {t,0, π}]

Now, for the more complicated example of the sum of two normal modes.  Consider a function given by :

y(x,t) = 7 sin(x) cos(v t)+2 sin(8x) cos (8 v t)

Setting v = 2 again, we can observe the behavior of this function (this is best observed at low speeds) :

Clear[y]
y[x_,t_]:=7Sin[x] Cos[2 t]+2Sin[8x]Cos[16t]
Manipulate[Plot[y[x,t],{x,-5π,5π},PlotRange->{-9,9}], {t, 0, π}]

(* Animation of problem 11.52 in text *)
Clear[y,a,v,x,t,n]
v=2;
a[n_]:=a[n]=2/(π n^2)(Sin[n π/3]+Sin[2 π n/3])

y[x_,t_]:=Sum[a[n] Sin[n x]Cos[n v t],{n,11}]


Manipulate[Plot[y[x,t],{x,0,π},PlotRange->{-2,2}],{t,0,1.5,0.1}]