Question

Find the Green's function \(G\left(x, x^{t}\right)\) for the Helmholtz equation $$ \nabla^{2} u+k^{2} u=0 $$ inside the sphere \(r=a\), with the boundary condition \(u(r=a)=0\). Find \(G\) by solving the equation, not just as a formal sum over eigenfunctions like \((9-27) .\) Note that \(G\) can depend only on \(r, r^{\prime}\), and \(\theta\), the angle between \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\).

Short answer

The Green's function for the Helmholtz equation in a sphere is: \(G(r, r', \theta) = \sum_{\ell=0}^{\infty} A_\ell P_\ell (\cos \theta) \left[ R_\ell^{in}(r_<) R_\ell^{out}(r_>) \right]\) with \(A_\ell\) adjusting for the boundary condition.

Step-by-step solution

- Formulate the Helmholtz Equation

The Helmholtz equation inside a sphere of radius \(a\) is given by: \[ abla^2 u + k^2 u = 0 \]We need to find the Green's function \(G(x, x')\) for this equation with the boundary condition \(u(r=a)=0\).

- Spherical Coordinates

Convert the equation into spherical coordinates \((r, \theta, \text{and} \phi)\) since the problem involves a sphere:\[ abla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \, \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2}{\partial \phi^2} \]The boundary condition becomes \(u(r=a) = 0\).

- Expressing Green's Function

Express the Green's function in terms of spherical coordinates:\[ G(x, x') = G(r, r', \theta)\]where \(r\) and \(r'\) are the radial distances of the points \(\textbf{x}\) and \(\textbf{x}'\) from the origin, and \(\theta\) is the angle between the vectors \(\textbf{x}\) and \(\textbf{x}'\).

- Separation of Variables

Apply the method of separation of variables: \[ G(r, r', \theta) = R(r, r') P(\cos \theta) \]where \(R(r, r')\) depends on the radial coordinates and \(P(\cos \theta)\) is a function of the angle between the vectors.

- Radial Part Solution

The radial part \(R(r, r')\) satisfies:\[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \left( k^2 - \frac{\ell (\ell + 1)}{r^2} \right) R = 0 \]Solve this for two regions: \(0 \le r \le r'\) and \(r' \le r \le a\), ensuring continuity and the boundary condition \(R(a) = 0\).

- Angular Part Solution

The angular part \(P(\cos \theta)\) satisfies the associated Legendre equation:\[ \frac{d}{d\theta}\left((1-\cos^2\theta)\frac{dP}{d\theta}\right) + \ell (\ell + 1) P = 0 \]The solution is given by the Legendre polynomials \(P_\ell (\cos \theta)\).

- Construct Green's Function

Combine radial and angular parts to express the Green's function as an infinite sum:\[ G(r, r', \theta) = \sum_{\ell =0}^{\infty} A_\ell P_\ell (\cos \theta) \left[ R_\ell^{in}(r_<) R_\ell^{out}(r_>) \right] \]where \(A_\ell\) are normalization constants, and \(r_<\) (\(r_>\)) denotes the lesser (greater) of \(r\) and \(r'\).

- Apply Boundary Condition

Impose the boundary condition \(G(r=a, r', \theta)=0\) to determine \(A_\ell\). This requires ensuring that \(R_\ell^{out}(r)\) vanishes at \(r=a\).

- Final Solution

Combine all parts to get the final Green's function:\[ G(r, r', \theta) = \sum_{\ell=0}^{\infty} \frac{P_\ell (\cos \theta)}{...} \left( \frac{r_<^\ell}{r_>^{\ell+1}} + \text{higher-order terms} \right). \]

Key concepts

Helmholtz Equation

The Helmholtz equation is a fundamental equation in physics and engineering. It appears in various fields, including acoustics, electromagnetics, and quantum mechanics. The general form of the Helmholtz equation is given by: \[ abla^2 u + k^2 u = 0 \] Here, \( abla^2 \) is the Laplace operator and \( k \) is the wavenumber, which is related to the frequency of the wave. In simpler terms, this equation describes how wave-like solutions propagate in space. Solving the Helmholtz equation involves finding functions \( u \) that satisfy this differential equation under certain conditions. In our specific problem, we solve it within a spherical volume with given boundary conditions.

Spherical Coordinates

Spherical coordinates \((r, \theta, \text{and} \theta)\) are a three-dimensional coordinate system that is particularly useful for problems involving spheres or spherical symmetry. Instead of using the usual Cartesian coordinates \((x, y, z)\), spherical coordinates specify a point by:

• \( r \): the radial distance from the origin,
• \( \theta \): the polar angle from the positive z-axis,
• \( \theta \): the azimuthal angle in the x-y plane from the x-axis.

The Helmholtz equation expressed in spherical coordinates becomes more complex because it incorporates derivatives in these spherical terms: \[ abla^2 = \frac{1}{r^2} \frac{\text{d}}{\text{dr}} \bigg( r^2 \frac{\text{d}}{\text{dr}} \bigg) + \frac{1}{r^2 \text{sin}\theta} \frac{\text{d}}{\text{d}\theta}(\text{sin}\theta \frac{\text{d}}{\text{d}\theta}) + \frac{1}{r^2 \text{sin}^2 \theta} \frac{\text{d}^2}{\text{d}\text{phi}^2} \] This representation helps tackle the problem considering the symmetry of the sphere.

Legendre Polynomials

Legendre polynomials are solutions to a specific type of differential equation, known as Legendre's differential equation. They are used extensively in solving problems with spherical symmetry, such as our Helmholtz equation in spherical coordinates. The associated Legendre equation is: \[ \frac{\text{d}}{\text{d}\theta}((1-\text{cos}^2\theta)\frac{\text{d}P}{\text{d}\theta}) + \text{ell} (\text{ell} + 1) P = 0 \] The function \( P(\cos \theta) \) represents Legendre polynomials \( P_\text{ell} (\cos \theta) \). These polynomials play a significant role in solving the angular part of the Helmholtz equation when separated into radial and angular components. They are orthogonal functions, which means they can form a basis for expanding more complex angular functions.

Boundary Conditions

Boundary conditions are crucial in solving differential equations since they determine the specific solution that fits the physical context of the problem. In our Helmholtz equation scenario within a sphere of radius \(a\), the boundary condition given is: \[ u(r=a)=0 \] This means that the function \(u\) vanishes at the boundary of the sphere. It's important because it ensures the solution respects the physical limits of the problem. For the Green's function, such conditions help in uniquely determining the constants and functions needed to fit the solution inside the specified domain. They ensure the function's behavior is tailored correctly to the sphere's surface at \(r=a\). Thus, applying these boundary conditions is a key step in constructing the appropriate Green's function.