i need a help in some waves equations

Dear All,

in my research work , i found a little problem in the mathematical derivation,
i was doing modeling for a cylindrical magnetron cavity sealed at the two ends.

Now, i need to derive another equations with new boundary conditions but I'm confused in getting those B.C. and those equations .

Hereby , i will attach a word document carries the equations

would you please read and tell me the changes that i should do if i need to remove the two ends,

thank you so much for taking the time to help me out