Computer Algebra Recipes

6.1. WAVE EQUATION MODELS 251
6.1.2 Homer's Jiggle Test
The fundamental cause of trouble in the world today is that the stupid
are cocksure while the intelligent are full of doubt.
Bertrand Russell, British mathematician and philosopher (1872{1970)
Having successfully used Maple to solve the plucked string problem, Vectoria
now tackles the ¯rst problem on Professor Legree's wave equation assignment.
She is asked to determine the transverse displacement Ã(x; y; t) of a light, horizontal,
rectangular elastic membrane having sides of length a and 2 a that
is held ¯xed along all four edges. Speci¯cally, the boundary conditions are
Ã(0;y;t)=Ã(a; y; t) =0for0· y · 2 a and Ã(x; 0;t)=Ã(x; 2 a; t) =0for
0 · x · a. The membrane is given an initial displacement
Ã(x; y; 0) ´ f = 4 hx2 (a ¡ x) y3 (2 a ¡ y)
a7 and is released from rest. Professor Legree also requests that the membrane
displacement be animated for a =1,h = 1, and wave velocity c =1.
Vectoria realizes that this membrane problem is mathematically equivalent
to two ¯xed-ends string problems in the x and y directions. So her \shortcut"
approach to separating variables in the last recipe can be easily extended.
After loading the plots package, she enters the wave equation WE in two
dimensions.
> restart: with(plots):
> WE:=diff(psi(x,y,t),x,x)+diff(psi(x,y,t),y,y)
=(1/c^2)*diff(psi(x,y,t),t,t);
μ μ @
2
2
@ @
WE := Ã(x; y; t) + Ã(x; y; t) =
@x2 @y2 2
Ã(x; y; t)
@t2 c2 The pdsolve command is applied to WE with HINT=sin(p*x)*sin(q*y)*T(t),
where p and q are constants. The assumed form satis¯es the boundary conditions
along the ¯xed edges x =0andy =0.
> sol:=pdsolve(WE,HINT=sin(p*x)*sin(q*y)*T(t),INTEGRATE,build);
sol := Ã(x; y; t) =sin(px)sin(qy) C1 sin(c p p2 + q2 t)
+sin(px)sin(qy) C2 cos(c p p2 + q2 t)
The initial transverse velocity is zero, so once again only the term involving
cos(c p p2 + q2 t)iskept.
> sol2:=select(has,rhs(sol),cos);
sol2 := sin(px)sin(qy) C2 cos(c p p 2 + q 2 t)
At x = a, sin(pa)=0,sop = m¼=a,wherem is a positive integer. Similarly,
at y =2a, sin(q 2 a) =0,soq = n¼=(2 a), n being a positive integer. Vectoria
substitutes the p and q relations into sol2 , and temporarily removes all
coe±cients by setting C1 = C2 =1.