Question

The equation $$ v_{x x}+v_{y y}+k^{2} v=0 $$ is a generalization of Laplace's equation, and is sometimes called the Helmholtz \((1821-1894)\) equation. (a) In polar coordinates the Helmholtz equation is $$v_{r r}+(1 / r) v_{r}+\left(1 / r^{2}\right) v_{\theta \theta}+k^{2} v=0$$ If \(v(r, \theta)=R(r) \Theta(\theta),\) show that \(R\) and \(\Theta\) satisfy the ordinary differential equations $$ r^{2} R^{\prime \prime}+r R^{\prime}+\left(k^{2} r^{2}-\lambda^{2}\right) R=0, \quad \Theta^{\prime \prime}+\lambda^{2} \Theta=0 $$ (b) Consider the Helmholtz equation in the disk \(r

Short answer

Question: Analyze the Helmholtz equation in polar coordinates and show that the functions R(r) and Θ(θ) satisfy certain ordinary differential equations. Then find a solution that satisfies specific conditions in a disk of radius less than c. Answer: The general solution of the Helmholtz equation in polar coordinates in a disk of radius less than c is given by: $$v(r, \theta) = R(r)\Theta(\theta) = \left[c_1 J_n(kr)\right]\left[C_1\cos(n\theta) + C_2\sin(n\theta)\right],$$ where R(r) satisfies the Bessel equation of order n and Θ(θ) is periodic with period 2π, given by Θ(θ) = Θ(θ + 2π). The coefficients c_1, C_1, and C_2 can be determined by imposing the boundary conditions for a specific problem.

Step-by-step solution

(Step 1: Separation of Variables)

We are given that the Helmholtz equation in polar coordinates is: $$v_{r r}+(1 / r) v_{r}+\left(1 / r^{2}\right) v_{\theta \theta}+k^{2} v=0$$ and we assume a solution in the form of $$v(r, \theta)=R(r) \Theta(\theta)$$ Now, we have to compute the partial derivatives of v and substitute them into the Helmholtz equation.

(Step 2: Compute Partial Derivatives)

Compute the second-order partial derivatives of v with respect to r and θ: $$v_{r r} = R''(r)\Theta(\theta), \quad v_{r} = R'(r)\Theta(\theta), \quad v_{\theta \theta} = R(r)\Theta''(\theta)$$ Now substitute these partial derivatives into the Helmholtz equation.

(Step 3: Substitute Partial Derivatives in the Equation)

Substituting the computed partial derivatives, we get: $$R''(r)\Theta(\theta)+\frac{1}{r}R'(r)\Theta(\theta)+\frac{1}{r^2}R(r)\Theta''(\theta)+k^2R(r)\Theta(\theta)=0$$ Now, divide by \(R(r)\Theta(\theta)\): $$\frac{R''(r)}{R(r)}+\frac{1}{r}\frac{R'(r)}{R(r)}+\frac{1}{r^2}\frac{\Theta''(\theta)}{\Theta(\theta)} + k^2 = 0$$

(Step 4: Separate the Equations)

Notice that the first two terms depend on r, the third term depends on θ, and the last term is a constant. To separate the equations, set: $$\frac{\Theta''(\theta)}{\Theta(\theta)} = -\lambda^2$$ and $$\frac{R''(r)}{R(r)}+\frac{1}{r}\frac{R'(r)}{R(r)} = -k^2 + \lambda^2$$ Solving these two equations, we get: $$\Theta''(\theta) = -\lambda^2\Theta(\theta)$$ $$r^2R''(r)+rR'(r)+(k^2r^2-\lambda^2)R(r) = 0$$

(Step 5: Solve the Boundary Value Problem)

Now, we need to solve the problem in the disk r < c, with periodic boundary conditions and with the boundary condition v(c, θ) = f(θ). First, let's handle the θ equation. Since Θ(θ) is periodic with period 2π, Θ(θ) = Θ(θ + 2π). Now, solve the θ equation by using the ansatz: Θ(θ) = cos(nθ) and Θ(θ) = sin(nθ), where n is an integer. The equation in R(r) is a Bessel equation of order n. To solve this problem, recall that the general solution of the Bessel equation of order n is a linear combination of the Bessel functions of the first kind, J_n, and the second kind, Y_n: $$R(r) = c_1 J_n(kr) + c_2 Y_n(kr)$$ For the solution to be bounded in the disk r < c, we need to choose \(c_2 = 0\). Moreover, since \(v(c, \theta) = R(c)\Theta(\theta) = f(\theta),\) we have $$R(c) = \frac{f(\theta)}{\Theta(\theta)}$$ Hence, the general solution of the Helmholtz equation in the disk r < c is $$v(r, \theta) = R(r)\Theta(\theta) = \left[c_1 J_n(kr)\right]\left[C_1\cos(n\theta) + C_2\sin(n\theta)\right]$$ where the coefficients c_1, C_1, and C_2 can be determined by imposing the boundary conditions for a specific problem.

Key concepts

Boundary Value Problems

When studying physical phenomena such as heat conduction, wave propagation, or quantum mechanics, boundary value problems (BVPs) often arise. BVPs involve differential equations with conditions specified at the boundaries of the domain.

A classic example is the Helmholtz equation, which appears in contexts like acoustics and electromagnetism. To solve a BVP like the Helmholtz equation in a disk, one must find a function that satisfies the differential equation throughout the domain and also meets specified constraints along the boundary. In our exercise, the solution must remain bounded inside the disk and match a given function on the boundary.

Importance of Boundary Conditions

BVPs are only well-posed when the boundary conditions are stipulated. These conditions dictate the unique solution of the equation, and without them, the problem may have multiple solutions or none at all. For instance, the periodic and boundedness conditions in our problem are critical to finding the correct form of the solution.

Bessel Functions

Bessel functions play a quintessential role in solving wave equations in cylindrical or spherical geometries. They are named after Friedrich Bessel and are solutions to Bessel's differential equation.

Bessel functions come in various types, but for physics-related problems, the Bessel functions of the first kind, denoted as \( J_n(x) \), and the second kind, denoted as \( Y_n(x) \), are particularly relevant. These functions describe radially symmetric oscillatory behavior.

Solving BVPs with Bessel Functions

In BVPs like the Helmholtz equation within a circular domain, Bessel functions are utilized because of their appropriateness for radial problems. The Bessel function of the first kind is chosen for bounded solutions since the Bessel function of the second kind becomes unbounded near the origin, which would violate the boundedness condition of our problem.

Separation of Variables

Separation of variables is a method used to reduce partial differential equations (PDEs), which involve multiple independent variables, into a set of simpler ordinary differential equations (ODEs) that can be solved more readily.

In context to the Helmholtz equation in polar coordinates, the solution is divided into a radial part \( R(r) \) and an angular part \( \Theta(\theta) \). This technique assumes the solution can be expressed as a product of functions of each variable separately.

Reduction to ODEs

Separating the variables allows each part to be solved independently. For the Helmholtz equation, the method leads to two separate ODEs: one for \( R(r) \) that includes a Bessel equation and another for \( \Theta(\theta) \) which is a simple harmonic oscillator equation. The ODEs are easier to handle and are the building blocks of the general solution to the PDE.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a fixed point—usually called the origin—and an angle from a fixed direction—commonly the positive x-axis.

In polar coordinates, problems with circular or spherical symmetry, like the Helmholtz equation within a disk, become easier to address. Instead of dealing with the Cartesian coordinates \( x \) and \( y \), one works with the radial distance \( r \) and the angle \( \theta \), which naturally describe the geometry of such problems.

Advantages in BVPs

The use of polar coordinates in BVPs such as the Helmholtz equation simplifies the process, especially when applying the separation of variables method, and allows for more direct incorporation of radially symmetric boundary conditions.