Non-Periodic Boundary conditions for wave equation

I already know that when we have a problem with spherical symmetry and periodic boundary conditions in both angles the Helmholtz equation (like the equations that we find in physics when we have wave or diffusion equation)

$$ left[ frac{1}{r^2} frac{partial}{partial r}left(r^2 frac{partial}{partial r}right)+frac{1}{r^2 sin(theta)} frac{partial }{partial theta} left( sin(theta) frac{partial }{partial theta} right)+frac{1}{r^2 sin^2(theta)} frac{partial^2}{partial varphi^2}right] Psi (r,theta,varphi)=k^2 Psi (r,theta,varphi)$$
with
$$ varphiin(0,2pi) quad theta in(0,pi) quad rin(0,a) $$
have the solution in terms of Spherical harmonics

$$ Psi (r,theta,varphi) = sum_{n=0}^infty sum_{l=0}^infty sum_{m=-l}^l R_{n,l}(r)cdot Y_{l,m}(theta,varphi) $$

Where

$$ Y_{l,m}(theta,varphi)=(-1)^m cdot left[ frac{2l+1}{4pi} frac{(l-m)!}{(l+m)!} right]^{1/2}cdot P^m_l[cos(theta)] cdot e^{imvarphi} $$

and where $P^m_l[cos(theta)]$ are the associated Legendre functions

I would like to know how can I find with this result (using the symmetry of the problem and the Spherical harmonics/associated Legendre functions properties) the solution for a not full sphere surface. Specifically, I would like for spheres with $varphiin(0,frac{2pi}{p})$ and $thetain(0,frac{pi}{2})$ with $pin mathbb{Z}$, for example:

1. The north pole of the sphere ($p=1$)
2. 3D first quadrant ($p=4$)
3. A maximum value of $varphi$ in first quadrant (for example, $p=6$)