Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 4: 4.3 only

 4.3  Non-Symmetric Drumhead
====
 
  u_tt = c^2 ( u_rr + (1/r) u_r + (1/r^2) u_ww )  where w=theta
  u(a,w,t)=0 [clamped edge]
  u(r,w,0) = f(r,w) [shape]
  u_t(r,w,0)=g(r,w) [velocity]

  PRODUCT SOLUTION 
    u=R(r)W(w)T(t)  r=radial variable, w=theta, t=time

    THEOREM
      W'' + mu^2 W = 0, W(0)=W(2Pi), W'(0)=W'(2Pi)
      r^2 R'' + r R' + (lambda^2 r^2 - mu^2)R = 0, R(a)=0
      T'' + c^2 lambda^2 T = 0, T nonzero

    THEOREM
      R(r) = J_m( alpha_mn r/a ),
      alpha_mn = nth positive zero of J_m(x)
      m=0,1,2,3, ...
      n=1,2,3, ...

    THEOREM
      T(t) = linear combination of the Euler solution atoms
             cos(alpha_mn ct/a), sin(alpha_mn ct/a)

    THEOREM
      W(w) = linear combination of the Euler solution atoms
             cos(m w), sin(m w)
             m=0,1,2,3, ...
             w=theta variable

 EXAMPLE 1. General solution when the velocity is zero.
 EXAMPLE 2. Special problem f(r,w)=r(1-r^2)sin(w), g(r,w)=0, with 
            a=c=1.

   This problem was discussed on the Friday before Spring Break. See the 
   blackboard photos for details.

 EXAMPLE 3. Special problem a=c=1 with f(r,w)=r(1-r^2)sin(w), 
            g(r,w)=r^2(1-r^2)sin(2w).

   Half the answer comes from Example 2. This problem is better done by 
   computer algebra systems, due to its complexity. The results of 4.3 
   are used to develop the computer code.