Unit 7: Boundary Value Problems with Cylindrical & Spherical Symmetry: Special Functions

1. General Metric Scale Factors

Let $(u_1, u_2, u_3)$ be an orthogonal curvilinear coordinate system related to Cartesian coordinates $(x, y, z)$. The differential displacement vector is:

where the scale factors (Lamé coefficients) are:

The gradient of a scalar field $\psi$ is:

The divergence of a vector field $\mathbf{A} = A_1 \hat{\mathbf{e}}_1 + A_2 \hat{\mathbf{e}}_2 + A_3 \hat{\mathbf{e}}_3$ is:

Combining $\nabla \cdot (\nabla \psi)$ gives the universal Laplacian in orthogonal curvilinear coordinates:

2. Cylindrical Coordinates $(r, \theta, z)$

The coordinates are $x = r\cos\theta, y = r\sin\theta, z = z$. The scale factors are:

Applying the Laplacian formula:

The Cylindrical Laplacian:

3. Spherical Coordinates $(r, \theta, \phi)$

The coordinates are $x = r\sin\theta\cos\phi, y = r\sin\theta\sin\phi, z = r\cos\theta$ (where $\theta$ is the polar colatitude and $\phi$ is the azimuthal longitude). The scale factors are:

The Spherical Laplacian:

1. Separation of Variables in Cylindrical Geometry

Consider the Helmholtz eigenvalue equation $\nabla^2 u + \lambda u = 0$ in polar coordinates $(r, \theta)$. Assume $u(r, \theta) = R(r) \Theta(\theta)$:

Dividing by $R \Theta$ and multiplying by $r^2$:

Multiplying out the radial part:

Let $x = \sqrt{\lambda} r$. By the chain rule, this reduces to Bessel's Differential Equation of order $m$:

2. Series Solution via the Frobenius Method

Since $x = 0$ is a regular singular point, assume a generalized power series $R(x) = \sum_{k=0}^\infty c_k x^{k+s}$. The indicial equation is $s(s-1) + s - m^2 = 0 \implies s^2 - m^2 = 0 \implies s = \pm m$. For $s = +m$:

Definition 7.1 (Bessel Function of the First Kind $J_m(x)$):

The series converges absolutely and uniformly for all $x \in \mathbb{C}$.

• $J_0(0) = 1$, and $J_m(0) = 0$ for all $m \ge 1$.
• As $x \to \infty$, $J_m(x)$ behaves asymptotically as an oscillating decaying cosine wave:

Definition 7.2 (Bessel Function of the Second Kind $Y_m(x)$ / Weber Function): The second linearly independent solution is defined for non-integer $\nu$ by:

and for integer $m$ by the limit $Y_m(x) = \lim_{\nu \to m} Y_\nu(x)$. Crucially, as $x \to 0^+$, $Y_m(x)$ diverges logarithmically:

Therefore, for any physical domain containing the center axis $r = 0$, $Y_m$ must be rejected, retaining only $J_m$.

1. Vibrations of an Elastic Circular Drumhead

Consider a circular membrane of radius $a$ clamped at its boundary $r = a$:

Separating variables $u(r, \theta, t) = R(r) \Theta(\theta) T(t)$ yields:

1. $\Theta_m(\theta) = A_m \cos(m\theta) + B_m \sin(m\theta)$ ($m = 0, 1, 2, \dots$)
2. $R_m(r) = J_m(k r)$, where the boundary condition requires:

where $\alpha_{m,n}$ is the $n$-th positive zero of $J_m(x)$.

1. The natural circular frequencies are:

Theorem 7.1 (Eigenmode Structure): Each normal mode $(m, n)$ of the vibrating circular membrane is given by:

The nodal lines where $u_{m,n} = 0$ at all times consist of:

• $m$ Nodal Diameters where $\cos(m\theta) = 0$.
• $n - 1$ Internal Nodal Circles where $r = \frac{\alpha_{m,k}}{\alpha_{m,n}} a$ for $k = 1, \dots, n - 1$ (plus the rim $r = a$).

1. Axisymmetric Laplace Equation in Spherical Coordinates

When a physical potential possesses azimuthal symmetry around the $z$-axis (no dependence on $\phi$), Laplace's equation in spherical coordinates reduces to:

Assume a product solution $u(r, \theta) = R(r) \Theta(\theta)$:

2. Legendre's Differential Equation

Consider the angular ODE:

Substitute the variable $\xi = \cos\theta$ (where $\xi \in [-1, 1]$ as $\theta \in [0, \pi]$). Then $\frac{d\xi}{d\theta} = -\sin\theta$ and $\sin^2\theta = 1 - \xi^2$. The chain rule gives:

Substituting into the angular equation:

This is Legendre's Differential Equation.

Theorem 7.2 (Quantization of $\lambda$ and Legendre Polynomials): Solutions to Legendre's equation that remain finite and non-singular at both poles $\theta = 0$ ($\xi = 1$) and $\theta = \pi$ ($\xi = -1$) exist if and only if the separation constant is quantized as:

Under this quantization, the bounded solutions are the Legendre Polynomials $P_n(\xi)$:

• $P_0(\xi) = 1$
• $P_1(\xi) = \xi = \cos\theta$
• $P_2(\xi) = \frac{1}{2}(3\xi^2 - 1) = \frac{1}{2}(3\cos^2\theta - 1)$
• $P_3(\xi) = \frac{1}{2}(5\xi^3 - 3\xi) = \frac{1}{2}(5\cos^3\theta - 3\cos\theta)$

Definition 7.3 (Rodrigues' Formula):

1. Orthogonality of Legendre Polynomials

Theorem 7.3 (Orthogonality Relation): On the interval $[-1, 1]$, the Legendre polynomials form a complete orthogonal set with respect to unit weight:

In terms of colatitude $\theta$:

2. Radial Solutions and General Axisymmetric Potential

For $\lambda = n(n + 1)$, the radial ODE is:

This is an Euler-Cauchy ODE with trial solutions $R(r) = r^\gamma$:

The roots are $\gamma = n$ and $\gamma = -(n + 1)$. Thus:

Theorem 7.4 (General Axisymmetric Harmonic Potential): Any axisymmetric solution to Laplace's equation $\nabla^2 u = 0$ in spherical coordinates is represented by:

• For interior problems (containing origin $r = 0$), all $B_n = 0$.
• For exterior problems (bounded as $r \to \infty$), all $A_n = 0$ (except possibly $A_0$ for potential at infinity).

3. Generating Function and Multipole Expansion

The generating function for Legendre polynomials is:

In electrostatics, the potential of a point charge $q$ at $\mathbf{r}' = (0, 0, d)$ observed at $\mathbf{r} = (r, \theta, \phi)$ with $r > d$ is:

• $n = 0$: Monopole potential $\propto 1/r$.
• $n = 1$: Dipole potential $\propto d\cos\theta / r^2$.
• $n = 2$: Quadrupole potential $\propto d^2 P_2(\cos\theta) / r^3$.