Problem 13 - Advanced Engineering Mathematics

Question

You are ashed 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 remaining separated equations are $$ \begin{aligned} &\frac{d^{2} \Phi}{d \phi^{2}}+m^{2} \Phi=0 \\ &\left.\frac{d^{2} \theta}{d \theta^{2}}+\frac{\cos \theta}{\sin \theta} \frac{d \theta}{d \theta}+\left[n(n+1)-\frac{m^{2}}{\sin ^{2} \theta}\right]\right]=0 . \end{aligned} $$ (c) Use the substitution \(x=\cos \theta\) to show that the second differeatial equation in part (b) becomes \(\left(1-x^{2}\right) \frac{d^{2} \Theta}{d x^{2}}-2 x \frac{d \theta}{d x}+\left[n(n+1)-\frac{m^{2}}{1-x^{2}}\right] Q=0\)

Step-by-Step Solution

Answer

Spherical harmonics determine the angular parts of the solution, while the radial equation is a version of Bessel's equation.

Step 1: Separation of Variables in Helmholtz Equation

Start by assuming a solution in the form of a product of functions: \( u(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi) \). Substituting this into the Helmholtz equation \( abla^{2} u + k^{2} u = 0 \), the Laplacian operator is applied to the product solution in spherical coordinates.

Step 2: Equation for Radial Part - R(r)

While separating variables, assign the separation constant \( n(n+1) \). Focus on the radial part:\[ r^{2} \frac{d^{2} R}{d r^{2}} + 2r \frac{d R}{d r} + (k^{2} r^{2} - n(n+1)) R = 0. \]This results from equating terms depending solely on \( r \), with other dependencies assumed independent.

Step 3: Equation for Azimuthal Part - \( \Phi(\phi) \)

Choose a second separation constant \( m^{2} \) to separate \( \phi \) from \( \theta \). For the azimuthal part:\[ \frac{d^{2} \Phi}{d \phi^{2}} + m^{2} \Phi = 0. \]This is derived by isolating terms depending only on \( \phi \). Solutions are sinusoidal, indicating periodicity.

Step 4: Equation for Angular Part - \( \Theta(\theta) \)

Isolate the parts involving \( \theta \) with the separation constant \( m^{2} \):\[ \frac{d^{2} \Theta}{d \theta^{2}} + \frac{\cos \theta}{\sin \theta} \frac{d \Theta}{d \theta} + \left[n(n+1) - \frac{m^{2}}{\sin ^{2} \theta}\right] \Theta = 0. \]This equation involves both \( \theta \) and the chosen separation constants.

Step 5: Substitute \( x = \cos \theta \) and Transform Equation

Make the substitution \( x = \cos \theta \), which transforms \( \Theta(\theta) \) to a function of \( x \). Using the chain rule, the derivatives change. Thus:\[ \left(1-x^{2}\right) \frac{d^{2} \Theta}{d x^{2}} - 2x \frac{d \Theta}{d x} + \left[n(n+1) - \frac{m^{2}}{1-x^{2}}\right] \Theta = 0. \]This is the associated Legendre differential equation in terms of \( x = \cos \theta \).

Key Concepts

Separation of Variables

The method of separation of variables is a powerful algebraic technique. It is often used to solve partial differential equations (PDEs) such as the Helmholtz equation, \( abla^{2} u + k^{2} u = 0 \). In this approach, we assume that the solution can be represented as a product of separate functions, each depending on a single variable. For the Helmholtz equation in spherical coordinates, we assume a solution of the form:
\[ u(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi) \]
This means we express the complex behavior of \( u \) as the combination of a radial part \( R(r) \), a polar angular part \( \Theta(\theta) \), and an azimuthal part \( \Phi(\phi) \).
By substituting this product into the Helmholtz equation, we can separate the variables so each equation involves only one of \( r \), \( \theta \), or \( \phi \). This separation allows us to turn a complex PDE into simpler ordinary differential equations (ODEs), each addressed one at a time.

• Helmholtz equation becomes a set of ODEs.
• Each resulting ODE can be solved individually.
• Simplification relies on suitable boundary conditions and separation constants.

Spherical Coordinates

Spherical coordinates provide a natural framework for problems involving spherical symmetry, such as the Helmholtz equation. They are expressed by three variables: radius \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \). In these coordinates, every point is represented by how far it is from the origin \( r \), its angle from the positive \( z \)-axis \( \theta \), and its angle in the \( xy \)-plane from the positive \( x \)-axis \( \phi \).
Using spherical coordinates, the Laplacian operator, \( abla^2 \), can be expanded as follows:
\[abla^2 u = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial u}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial u}{\partial \theta}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 u}{\partial \phi^2}\]
This expression clearly shows how the gradient and curvature of a function \( u \) are treated in spherical coordinates, allowing us to apply methods like separation of variables to solve complex physical problems.

• Useful for problems with inherent spherical symmetry.
• Simplifies the treatment of radial and angular components.
• Transforms complex geometry into manageable equations.

Legendre Differential Equation

The Legendre differential equation emerges when solving for the angular part, \( \Theta(\theta) \), of the solution to the Helmholtz equation in spherical coordinates. After separation and substitution \( x = \cos \theta \), the angular equation takes the form of a Legendre equation:
\[ (1-x^2) \frac{d^2 \Theta}{d x^2} - 2x \frac{d \Theta}{d x} + \left[n(n+1) - \frac{m^2}{1-x^2}\right] \Theta = 0 \]
This is a standard form of the associated Legendre differential equation, commonly encountered in physics. Its solutions, known as Legendre polynomials \( P_n(x) \) and associated Legendre functions, play significant roles in solving problems with spherical symmetry.
Key characteristics of Legendre polynomials include:

• Orthogonality over the interval \( -1 \leq x \leq 1 \).
• Generation of solutions for terms involving spherical harmonics.
• Use in expanding functions on spherical surfaces, which is vital in fields like quantum mechanics and electromagnetism.

Understanding the properties and applications of these functions is crucial for solving systems described by the Helmholtz equation in spherical geometries.