### Problem 1. We solve damped wave equation along a cylinder ( D ) of radius ( a=1, H=4 ) [ left{begin{array}{l} partial_{t} u+2 xi u_{t}=Delta u \ u(t, text { bnd }(D))=0 \ u(0, r, theta, z)=r z \ partial_{t} u(0, r, theta, z)=0 end{array}right. text {. } ] a) As we know, we should first solve the eigenvalue problem [

Problem 1. We solve damped wave equation along a cylinder ( D ) of radius ( a=1, H=4 ) [ left{begin{array}{l} partial_{t} u+2 xi u_{t}=Delta u \ u(t, text { bnd }(D))=0 \ u(0, r, theta, z)=r z \ partial_{t} u(0, r, theta, z)=0 end{array}right. text {. } ] a) As we know, we should first solve the eigenvalue problem [