I adjusted the solution you posted to be, in 1D Maple input syntax,
sol_Wouter := `ξr`(r,t) = sin(1/2*Pi*r)*cos(1/2*(kappa/Mu)^(1/2)*Pi*t);
pdetest(sol_Wouter, [eqn, ic, bc]);
[ /1 \ /Pi \ ]
[0, sin|- Pi r| - sin|---|, 0, 0, 0]
[ \2 / \2 r/ ]
So the first ic is not satisfied. Indeed, your first ic reads `ξr`(r, 0) = sin(Pi/(2*r)), not sin(Pi*r/2). Of course, if you change sin(Pi/(2*r)) by sin(Pi*r/2) in sol_Wouter, then the PDE (your eqn) does not cancel.
So, the solution you are expecting is not correct for the problem you posted.
Guessing what could be wrong in your post, from the output by pdetest above, if you change the right-hand side of your first ic from sin(Pi/(2*r)) to sin(Pi*r/2), then the (corrected above) solution you posted cancels all of eqn, ic and bc. But then pdsolve also returns the simpler solution:
# change your ic from sin(Pi/(2*r) to sin(Pi*r/2)
ic := `ξr`(r, 0) = sin(Pi*r/2), D(`ξr`)(r, 0) = 0:
pdsolve([eqn, ic, bc], Zeta(r, t));
`ξr`(r,t) = sin(1/2*Pi*r) * cos(1/2*kappa^(1/2)*Pi*t/Mu^(1/2))
In summary: to get from pdsolve the "simple" solution, you need to correct your input (the first ic). For the input you presented, there is no simple solution that I could see, and the infinite sum solution returned by pdsolve is correct.
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft