A circular membrane clamped at radius $a$ vibrating at frequency $\omega$ has displacement:

$u(r,\theta,t) = J_m(k_r r)\cos(m\theta)\cos(\omega t)$

where $k_r = \omega/c$ and the boundary condition $u(a,\theta,t) = 0$ requires $J_m(k_r a) = 0$. The allowed frequencies are:

$\omega_{mn} = \frac{c\alpha_{mn}}{a}$

where $\alpha_{mn}$ is the $n$-th zero of $J_m$. The fundamental mode $(m=0, n=1)$ has frequency $\omega_{01} = 2.405c/a$. Higher modes create nodal circles and radial lines.

A Bessel beam maintains its profile while propagating:

$E(r,z) = J_0(k_\perp r)e^{ik_z z}$

where $k_\perp^2 + k_z^2 = k^2$. Unlike Gaussian beams, Bessel beams are diffraction-free (in the paraxial approximation) but require infinite energy. In practice, they're approximated over finite regions, useful in optical manipulation and microscopy.

A grounded conducting sphere of radius $a$ in external field $E_0\hat{\mathbf{z}}$ has potential:

$\Phi(r,\theta) = \begin{cases}-E_0 r\cos\theta + \frac{E_0 a^3}{r^2}\cos\theta & r > a \\ 0 & r \leq a\end{cases}$

Using $\cos\theta = P_1(\cos\theta)$, this is:

$\Phi(r,\theta) = -E_0\left(r - \frac{a^3}{r^2}\right)P_1(\cos\theta)$

The induced surface charge density is:

$\sigma(\theta) = -\epsilon_0\frac{\partial\Phi}{\partial r}\bigg|_{r=a} = 3\epsilon_0 E_0\cos\theta$

Earth's gravitational potential (oblate spheroid) can be expanded:

$\Phi(r,\theta) = -\frac{GM}{r}\left[1 - J_2\left(\frac{R}{r}\right)^2 P_2(\cos\theta) + \cdots\right]$

where $J_2 \approx 1.08 \times 10^{-3}$ is the quadrupole moment coefficient and $P_2(x) = (3x^2-1)/2$. This causes satellite orbit precession, crucial for GPS accuracy.

The angular part of the Schrödinger equation in 3D separates:

$\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right]Y(\theta,\phi) = -\ell(\ell+1)Y(\theta,\phi)$

Solutions are spherical harmonics $Y_{\ell m}(\theta,\phi)$. For hydrogen:

• $n=1, \ell=0$: $\psi_{100} \propto e^{-r/a_0}Y_{00}$ (1s orbital, spherical)
• $n=2, \ell=1$: $\psi_{21m} \propto re^{-r/(2a_0)}Y_{1m}$ (2p orbitals, lobed)
• $n=2, \ell=0$: $\psi_{200} \propto (2-r/a_0)e^{-r/(2a_0)}Y_{00}$ (2s orbital, radial node)

The angular momentum quantum number $\ell$ determines the orbital's shape through the spherical harmonic.

The radial Schrödinger equation for energy $E$ and angular momentum $\ell$:

$\frac{d^2u}{dr^2} + \left[k^2 - \frac{\ell(\ell+1)}{r^2} - \frac{2m}{\hbar^2}V(r)\right]u = 0$

where $u(r) = rR(r)$ and $k^2 = 2mE/\hbar^2$. For $V(r) = 0$, solutions are spherical Bessel functions $j_\ell(kr)$ and $n_\ell(kr)$:

$j_\ell(x) = \sqrt{\frac{\pi}{2x}}J_{\ell+1/2}(x), \quad n_\ell(x) = \sqrt{\frac{\pi}{2x}}Y_{\ell+1/2}(x)$

The scattering cross section involves phase shifts $\delta_\ell$ determined by matching boundary conditions.