Problem 13 - Advanced Engineering Mathematics

Question

In Problems 13 and 14, you are asled to find a product solution \(u(r, \theta, \phi)=R(r) \Theta(\theta) \Phi(\phi)\) of Helmholtz's partial differential equation \(\nabla^{2} u+k^{2} u=0\) where the Laplacian \(\nabla^{2} u\) is defined in (2). (a) Proceed as in Example 1 but using \(u(r, \theta, \phi)=\) \(R(r) \Theta(\theta) \Phi(\phi)\) and the separation constant \(n(n+1)\) to show that the radial dependence of the solution \(u\) is defined by the equation $$ r^{2} \frac{d^{2} R}{d r^{2}}+2 r \frac{d R}{d r}+\left[k^{2} r^{2}-n(n+1)\right] R=0 $$ (b) Now use the second separation constant \(m^{2}\) to show that the cemaining separated equations are $$ \begin{aligned} &\frac{d^{2} \Phi}{d \phi^{2}}+m^{2} \Phi=0 \\ &\frac{d^{2} \theta}{d \theta^{2}}+\frac{\cos \theta}{\sin \theta} \frac{d O}{d \theta}+\left[n(n+1)-\frac{m^{2}}{\sin ^{2} \theta}\right] \Theta=0 \end{aligned} $$ (c) Use the substitution \(x=\cos \theta\) to show that the second differential equation in part (b) becomes $$ \left(1-x^{2}\right) \frac{d^{2} \theta}{d x^{2}}-2 x \frac{d O}{d x}+\left[n(n+1)-\frac{m^{2}}{1-x^{2}}\right] \Theta=0 $$

Step-by-Step Solution

Answer

Radial: Bessel equation, Phi: Harmonic motion, Theta: Associated Legendre.

Step 1: Write the Laplacian in Spherical Coordinates

The Laplacian in spherical coordinates is \( abla^2 u = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial u}{\partial r}) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial u}{\partial \theta}) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 u}{\partial \phi^2} \). Substitute the separation of variables form \( u = R(r) \Theta(\theta) \Phi(\phi) \).

Step 2: Separate the Equation and Introduce Constants

Substitute \( u = R(r) \Theta(\theta) \Phi(\phi) \) into the Helmholtz equation \( abla^2 u + k^2 u = 0 \). Rearrange to obtain: \[ \frac{1}{R} \frac{1}{r^2} \frac{d}{dr}(r^2 \frac{dR}{dr}) + \frac{1}{\Theta} \frac{1}{r^2 \sin \theta} \frac{d}{d\theta}(\sin \theta \frac{d\Theta}{d\theta}) + \frac{1}{\Phi} \frac{1}{r^2 \sin^2 \theta} \frac{d^2 \Phi}{d\phi^2} + k^2 = 0 \] Assuming \( \lambda = n(n + 1) \), mind the dependency only on their respective variables.

Step 3: Analyze the Radial Dependency

Focus on the radial part: \[ r^2 \frac{d^2 R}{dr^2} + 2r \frac{dR}{dr} + \left(k^2 r^2 - n(n+1)\right) R = 0 \]. This is a form of the Bessel differential equation, confirming the radial dependence matches the stated form.

Step 4: Use the Second Separation Constant for Remaining Equations

Set \( \mu = m^2 \) as a separation constant for the \( \phi \) dependency. From \( \frac{1}{\Phi} \frac{d^2 \Phi}{d\phi^2} = -\mu \), analyze \( \frac{d^2 \Phi}{d\phi^2} + m^2 \Phi = 0 \).

Step 5: Derive the Theta Equation

For the theta part, separate \( \frac{1}{\Theta \sin \theta} \frac{d}{d\theta}(\sin \theta \frac{d\Theta}{d\theta}) = n(n+1) - \frac{m^2}{\sin^2 \theta} \). This becomes \( \frac{d^2 \Theta}{d\theta^2} + \frac{\cos \theta}{\sin \theta} \frac{d\Theta}{d\theta} + \left[n(n+1) - \frac{m^2}{\sin^2 \theta}\right] \Theta = 0 \).

Step 6: Substitute for Simpler Expression

Use the substitution \( x = \cos \theta \). Then \( \frac{d}{d\theta} = (1-x^2)^{-1/2} \frac{d}{dx} \). Substituting, the equation transforms to \( (1-x^2) \frac{d^2 \Theta}{dx^2} - 2x \frac{d\Theta}{dx} + \left[n(n+1) - \frac{m^2}{1-x^2}\right] \Theta = 0 \), which is a form of the associated Legendre differential equation.

Key Concepts

Laplacian in Spherical Coordinates

Understanding the Laplacian in spherical coordinates is essential for tackling problems involving symmetry in physics, such as the Helmholtz equation in spherical systems. In Cartesian coordinates, the Laplacian is the sum of second derivatives with respect to each spatial variable. However, spherical coordinates consider radial and angular dependencies and thus have a more complex form.
The Laplacian in spherical coordinates is expressed as:

• Radial part: \( \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u}{\partial r}\right) \)
• Poloidal part (\(\theta\)): \( \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta}\right) \)
• Azimuthal part (\(\phi\)): \( \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 u}{\partial \phi^2} \)

Together, these components form the Laplacian which accommodates for the curvilinear coordinate system. This adjustment allows for problem solutions in spherical domains, critical for understanding phenomena like waves or quantum particles in spherical potentials.

Bessel Differential Equation

The Bessel differential equation emerges in physics and engineering, especially when dealing with cylindrical or spherical symmetry. In the context of the Helmholtz equation with spherical coordinates, the radial part reduces to a Bessel form:
\( r^2 \frac{d^2 R}{dr^2} + 2r \frac{dR}{dr} + \left(k^2 r^2 - n(n+1)\right) R = 0 \).
This is a second-order linear differential equation. The radial dependency behaves like a Bessel function's solution, crucial for modeling wave-like phenomena or vibrations in circular membranes.

• Bessel functions represent solutions to this equation. They have oscillatory behavior, much like sine and cosine functions but adapted for circular geometries.
• Bessel's solutions are applicable in boundary-value problems with specific conditions like fixed boundary sizes or large distance limits.

By aligning the radial dependency of solutions from the Helmholtz equation, moments in physics with inherent symmetries or periodicity properties can be effectively described and understood.

Separation of Variables

Separation of Variables is a powerful method for solving partial differential equations (PDEs), like the Helmholtz equation. The technique assumes the solution can be expressed as a product of functions, each dependent solely on one variable.
For example, for a function \( u(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi) \), the separation allows the PDE to split into ordinary differential equations for each variable.

• This approach simplifies complex PDEs by reducing them to simpler, manageable ordinary differential equations.
• Separation constants, often denoted \( n(n+1) \) and \( m^2 \), emerge when breaking down the Helmholtz equation. They link the individual ordinary differential equations.

The beauty of this method lies in its ability to decouple intricate relationships between variables. Thus, it is widely used for various boundary value problems, especially in physics where symmetries simplify complex systems.

Associated Legendre Differential Equation

The Associated Legendre Differential Equation arises when solving the Helmholtz equation in spherical coordinates, specifically while examining angular dependencies. When transforming the \( \theta \)-related part through substitution, \( x = \cos \theta \), the equation becomes:
\( (1-x^2) \frac{d^2 \Theta}{dx^2} - 2x \frac{d\Theta}{dx} + \left[n(n+1) - \frac{m^2}{1-x^2}\right] \Theta = 0 \).

• This differential equation is pivotal for defining Legendre polynomials and associated functions, integral in mathematical physics.
• It is key in determining the angular part of the wave functions in quantum mechanics, influencing energy levels of quantum systems.
• Associated Legendre polynomials help in expanding functions over the sphere, aiding in representing potential functions or field distributions.

These mathematical functions are integral to solving problems in electromagnetism, quantum mechanics, and even gravitational systems where spherical symmetry is a key component.