Computer Algebra Recipes

250 CHAPTER 6. LINEAR PDE MODELS. PART 2
The
P
transverse displacement of the string at arbitrary time t is Ã(x; t) =
1
n=1 Ãn. For animation purposes, Vectoria takes L =10,h =1,andc =1.
> L:=10: h:=1: c:=1:
To obtain a good approximation to the initial string pro¯le, Vectoria keeps
terms in the series up to n = 25, only the partial output being shown here.
> psi:=sum(psi[n],n=1..25);
³
¼x
´ μ
¼t
8sin cos
10 10
à :=
¼2 ¡ 8
μ μ
3 ¼x 3 ¼t
sin cos
10 10
9 ¼2 + 8
³
¼x
´ μ
¼t
sin cos
2 2
25 ¼2 + ¢¢¢¢¢¢+ 8
μ μ
21 ¼x 21 ¼t
sin cos
10 10
441
¼2 ¡ 8
μ μ
23 ¼x 23 ¼t
sin cos
10 10
529
¼2 + 8
μ μ
5 ¼x 5 ¼t
sin cos
2 2
625
¼2 The displacement à is animated over the time interval t = 0 to 20, with 100
equally spaced time frames.
> animate(plot,[psi,x=0..L],t=0..20,frames=100);
On running the animation command, Vectoria is at ¯rst surprised by the behavior
of the string, but on thinking about it decides that it makes sense. After
executing the recipe, does the behavior make sense to you? What should Vectoria
write in her explanation of the wave motion?
PROBLEMS:
Problem 6-1: Another plucked string
A light homogeneous horizontal string of length L ¯xed at its ends initially
has a parabolic shape with height h in the middle. If it is released from rest,
solve the wave equation to determine its subsequent transverse displacement at
arbitrary time t. Animate the vibrations for parameters of your own choice and
choose enough frames to produce a smooth animation.
Problem 6-2: Struck piano string
If a horizontal piano string ¯xed at x =0andx = L is struck in such a way
that its initial displacement Ã(x; 0) is zero and its initial transverse velocity is
8
< 4 vx=L; 0