Computer Algebra Recipes

5.1. CHECKING SOLUTIONS 211
where μ ´ ! (r ¡ a)=c ¡ !tand c is the speed of sound in the °uid.
(a) Check that the solution satis¯es the boundary condition.
(b) The velocity V is related to the velocity potential © by V = ¡@©=@r.
Determine the radial dependence of the velocity potential.
(c) Since © depends only on the distance r from the center of the sphere, it
satis¯esthewaveequationintheform
@ 2 (r ©)
@r 2
1
=
c2 @2 (r ©)
@t2 :
Verify that © satis¯es the wave equation, thus ensuring that it is the
correct solution to the pulsating sphere problem.
(d) Taking the nominal values U =1,a =1,c =1,and! = 1, animate the
analytic formula for V in the region outside the spherical surface.
(e) How far from the surface does the velocity oscillation amplitude drop to
5% of the value at the surface?
5.1.2 Play It, Sam
You just pick a chord, go twang, and you've got music.
Syd Vicious, British rock musician (1957{1979)
In a famous scene from the classic movie Casablanca, Humphrey Bogart is
annoyed by the musical piece that the nightclub pianist is playing and doesn't
want to hear it. But Ingrid Bergman turns to the piano player, and says \Play
it, Sam." Humphrey Bogart then echoes her, by saying \If she can stand it, I
can. Play it."
To a certain undergraduate physics student watching this old movie, an
individual who tends to look for deeper understanding rather than simply enjoying
the movie and the music, the sounds that emanate are of course due to
the transverse vibrations of the piano strings as each is successively struck by
a piano hammer. This student is our old friend Vectoria, who is spending a
lonely evening by herself, since her ¯ance Mike is out of town at a mathematics
conference. Hoping that Mike will phone before it gets too late, Vectoria
decides in the meantime to look at the mathematical vibrations of an elastic
string ¯xed at its ends.
Not yet having taken the necessary mathematics prerequisites to study the
topic of vibrating strings in depth, Vectoria consults a physics text that gives
no derivation but simply the formula for the transverse displacement of a light,
horizontal, elastic string of length a ¯xed at both ends when struck and given a
certain initial velocity pro¯le. The information that Vectoria gleans from this
particular text is as follows.