Question

Consider GF for the Helmholtz operator \(\nabla^{2}+\mu^{2}\) in two dimensions. (a) Show that $$ G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-\frac{i}{4} H_{0}^{(1)}\left(\mu\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)+H\left(\mathbf{r}, \mathbf{r}^{\prime}\right) $$ where \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) satisfies the homogeneous Helmholtz equation. (b) Separate the variables and use the fact that \(H\) is regular at \(\mathbf{r}=\mathbf{r}^{\prime}\) to show that \(H\) can be written as $$ H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\sum_{n=0}^{\infty} J_{n}(\mu r)\left[a_{n}\left(\mathbf{r}^{\prime}\right) \cos n \theta+b_{n}\left(\mathbf{r}^{\prime}\right) \sin n \theta\right] . $$ (c) Now assume a circular boundary of radius \(a\) and the \(\mathrm{BC} G\left(\mathbf{a}, \mathbf{r}^{\prime}\right)=0\), in which a is a vector from the origin to the circular boundary. Using this \(\mathrm{BC}\), show that $$ \begin{aligned} a_{0}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{8 \pi J_{0}(\mu a)} \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) d \theta, \\ a_{n}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{4 \pi J_{n}(\mu a)} \\ & \times \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) \cos n \theta d \theta \\ b_{n}\left(\mathbf{r}^{\prime}\right)=& \frac{i}{4 \pi J_{n}(\mu a)} \\ & \times \int_{0}^{2 \pi} H_{0}^{(1)}\left(\mu \sqrt{a^{2}+r^{\prime 2}-2 a r^{\prime} \cos \left(\theta-\theta^{\prime}\right)}\right) \sin n \theta d \theta \end{aligned} $$ These equations completely determine \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) and therefore \(\bar{G}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\).

Short answer

The solutions for \(a_{n}\), \(b_{n}\), and \(a_{0}\) are obtained by applying the boundary condition \(G\left(\mathbf{a}, \mathbf{r}^{\prime}\right)=0\) to the green function \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\). \(H\left(\mathbf{r},\mathbf{r}^{\prime}\right)\) can be determined by substituting these values into the separated variables expression.

Step-by-step solution

Show the given equation for GF

The Green's function for the 2D Helmholtz operator can be expressed as \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-\frac{i}{4} H_{0}^{(1)}\left(\mu\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)+H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) , where \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) satisfies the homogeneous Helmholtz equation. This is given in the problem statement and doesn't require further demonstration.

Separate the variables

The secondary term \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) can be separated into its variable components. With the knowledge that \(H\) is regular at \(\mathbf{r}=\mathbf{r}^{\prime}\), it can be expressed as \(H\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\sum_{n=0}^{\infty} J_{n}(\mu r)\left[a_{n}\left(\mathbf{r}^{\prime}\right) \cos n \theta+b_{n}\left(\mathbf{r}^{\prime}\right) \sin n \theta\right]\).

Apply the boundary condition

With a circular boundary of radius \(a\), and the boundary condition \(G\left(\mathbf{a}, \mathbf{r}^{\prime}\right)=0\), we can use these to determine the integral expressions for \(a_{n}\), \(b_{n}\) and \(a_{0}\). These expressions involve Hankel functions, Bessel functions, and integral operations and are given in the problem statement.

Determine H

With the above results, we can now write out the equation for \(H\left(\mathbf{r},\mathbf{r}^{\prime}\right)\) by substituting \(a_{n}\), \(b_{n}\) and \(a_{0}\) into the expression. So, \(H\left(\mathbf{r},\mathbf{r}^{\prime}\right)\) can be determined from the expressions for \(a_{n}\), \(b_{n}\) and \(a_{0}\).

Key concepts

Green's Function

The Green's function is a powerful tool used to solve differential equations in physics and engineering, particularly for problems involving boundary conditions. In essence, it represents the response of a system to an impulse at a given point and can be used to construct solutions to complex problems based on the principle of superposition.

For the Helmholtz operator, which involves a second-order partial differential equation, the Green's function helps us find the impact of a source at position \(\mathbf{r'}\) on a field point \(\mathbf{r}\). The Green's function for the 2D Helmholtz operator, as seen in the exercise, features Hankel functions due to the circular symmetry of the problem in two dimensions. It is crucial to remember that the Homogeneous Helmholtz equation is satisfied by any function that when operated upon by the Helmholtz operator yields zero. This is the case for the function \(H(\mathbf{r}, \mathbf{r'}))\).

To fully grasp the concept, it's important to understand that the Green's function is not unique—additional terms that satisfy the homogeneous version of the governing equation may be added without affecting the inhomogeneous solution. Thus, the solution to the Helmholtz equation for a given source distribution can be understood as the sum of the contributions from all individual source points, weighted by the Green's function.

Bessel Functions

Bessel functions, denoted by \(J_n\), are a set of canonical solutions to Bessel's differential equation which appears in a wide range of physical problems involving cylindrical or spherical symmetry.

In part (b) of the exercise, we come across Bessel functions as part of the solution in cylindrical coordinates, which is natural given the problem's symmetry. These functions specifically represent the radial part of the solution. Regularity at \(\mathbf{r}=\mathbf{r'}\), the location of the impulse, justifies the choice of Bessel functions of the first kind, as they are finite at the origin. This contrasts with Bessel functions of the second kind, which diverge at the origin.

Bessel functions also have orthogonality properties with respect to their indices and can be integrated over a range to yield delta functions, a feature that is extensively used in solving boundary value problems. The series expansion involving \(J_n(\mu r)\) indicates that the solution can be decomposed into an infinite sum of modes, each with a different angular frequency, which represents the principle of separation of variables in this context.

Hankel Functions

Hankel functions, represented as \(H_n^{(1)}\) and \(H_n^{(2)}\), are two kinds of solutions to the Bessel's differential equation and are particularly used for problems with outgoing or incoming wave conditions. In the given exercise, \(H_0^{(1)}\) appears prominently in the formula for the Green's function, because it characterizes an outgoing cylindrical wave radiating from the source point \(\mathbf{r'}\).

The Hankel function of the first kind, \(H_n^{(1)}\), is used to describe waves that propagate outwards from the origin and are applicable to many physical problems involving radiation conditions, such as acoustics and electromagnetics. This function grows large as its argument goes to infinity, which aligns with the behavior of radiating waves.

In our exercise, the Hankel function is crucial in ensuring the correct physical behavior of the Green's function at large distances, and it guarantees that the wave generated by the source at \(\mathbf{r'}\) is radiating outward. Its combination with the appropriate Bessel functions fully resolves the boundary value problem within the domain defined by the circular boundary.