Computer Algebra Recipes

214 CHAPTER 5. LINEAR PDE MODELS. PART 1
solution isn't too bad, but not great either. Increasing the n value, Vectoria
observes an increasingly better ¯t. The reader should be the judge of how many
terms su±ce to give a good ¯t for the parameters chosen.
Although reasonably satis¯ed that the quoted Fourier series expansion is
correct, Vectoria still isn't entirely happy until the actual motion predicted
by the formula is observed. After all, it is not obvious from the series solution
exactly what the behavior of the string is after being struck. So the displacement
Ã(x; t) is animated for the time interval t = 0 to 50. First, the time variable t
must be unassigned. Otherwise, Maple will remember the value t =0usedto
check the initial velocity pro¯le.
> unassign('t'):
Finally, the animation command is given with 100 frames being used.
> animate(plot,[psi,x=0..a],t=0..50,frames=100,thickness=2);
On running the animation, Vectoria observes that the wave form begins to grow
in the region where the string was struck. This makes intuitive sense. Because
thewaveformiscreatedontheleftsideofthestring,itthenmoveswithwave
velocity c to the right and re°ects o® the boundary at x = a. On re°ection
the wave form is inverted, a characteristic feature for a ¯xed-ends boundary
condition. The wave then propagates to the left boundary at x =0before
inverting again and repeating the oscillatory behavior.
Finally, with a feeling of accomplishment, Vectoria is able to appreciate the
deeper content underlying the simple remarks made by Ingrid Bergman and
Humphrey Bogart. She will be even more content if Mike phones soon.
PROBLEMS:
Problem 5-3: Plucked string
An elastic string ¯xed between x =0andL, and initially at rest, is \plucked,"
its initial shape being given by the following symmetric triangular pro¯le,
Ã(x; 0) =
½ 2 hx=L; 0 · x · L=2;
2 h (L ¡ x)=L; L=2 · x · L:
Verify that the motion for t>0 may be described by the Fourier series solution
8 h
Ã(x; t) =
¼2 1X sin(n¼=2)
n
n=1
2 sin(n¼x=L)cos(n¼ct=L)
and animate the solution for parameter values of your own choosing.
Problem 5-4: A striking piano hammer
A piano string ¯xed between x =0 anda is struck by a piano hammer in a
region of width d centered at x=x0. Its initial velocity distribution is
½
Ã(x; _ v cos(¼ (x ¡ x0)=d); jx ¡ x0j d=2: