Question

g(r,r0) = Sum[fmn(z, r0) Sin(n*Pi*x/a) Sin(m*Pi*y/a)
Show that fmn solves the following BVP:
fmn'' - k^2 fmn = 4/a^2 Sin(n*pi*x0/a)Sin(m*pi*y0/a)delta(z-z0)

fmn = 0 at z = 0 and z = L

Answer 1

\[g(r, r0) = \sum_{m,n = 1}^{infinity} f _{m,n}Sin(\frac{ n \pi x }{ a })Sin(\frac{ n \pi y }{ a })\]

Answer 2

\[f \prime \prime - k _{m,n}^{2}f = \frac{ 4 }{ a ^{2} } Sin(\frac{ n \pi x0 }{ a })Sin(\frac{ m \pi y0 }{ a })\delta(z - z0)\]

Answer 3

So I wanna show that fmn solves the second equation. I presume this starts with separation by variables. Where I get.... \[0 = \frac{ Z \prime \prime }{Z } + \frac{ Y \prime \prime }{ Y } + \frac{X \prime \prime }{X }\]

Answer 4

So I see how the constants are obtained. But I'm not quite sure where to go from there

Answer 5

And this is for a grounded cube of sides [0,a][0,a][0,L]