# 3.7, Example 1, rectangular membrane, dimensions a=b=1,
# speed c=1/Pi,
# u_{tt} = c^2(u_{xx}+u_{yy}),
# u=0 on the edges of the membrane, 
# u(x,y,0)=f(x,y)=x(x-1)y(y-1),
# u_t(x,y,0)=g(x,y)=0.
A:=unapply(2*((-1)^m-1)/(Pi^3*m^3),m);
B:=unapply(4*A(n)*A(m),(m,n));
C:=unapply(sqrt(m^2+n^2),(m,n));
V:=unapply(B(m,n)*sin(m*Pi*x)*sin(n*Pi*y)
   *cos(C(m,n)*t),(m,n,x,y,t));
S:=(x,y,t,N)->
sum(sum(V(2*k+1,2*j+1,x,y,t),k=0..N),j=0..N);
plot3d(S(x,y,0,16),x=0..1,y=0..1);
plot3d(S(x,y,2,16),x=0..1,y=0..1);
plot3d(S(x,y,10,16),x=0..1,y=0..1);