Question

For the Helmholtz operator \(\nabla^{2}-k^{2}\) in the half-space \(z \geq 0\), find the three-dimensional Dirichlet GF.

Short answer

The Green's function for this 3D Dirichlet problem is the inverse Fourier transform solution from step 5, which we can write as an integral: \(u(z)= \iint_{-\infty}^{\infty} -u(z)e^{-\sqrt{k^2-k_x^2-k_y^2}z}e^{i(k_xx+k_yy)} dk_x dk_y).\

Step-by-step solution

- Set up the 3D Helmholtz equation in Cartesian coordinates

The general 3D Helmholtz equation is \(\nabla^{2}u-k^{2}u=0\) . In Cartesian coordinates this becomes \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} - k^{2}u =0\)

- Apply Fourier Transform

In order to solve the problem in a half-space, we apply the Fourier transform to the Helmholtz equation which makes the equation easy to solve. We don't need to worry about \(x\) and \(y\) dimension mostly, so we transform those dimensions which give us: \( \int \int ( \frac{\partial^2 \tilde{u}}{\partial z^2} +k_x^2 \tilde{u} + k_y^2 \tilde{u} -k^2 \tilde{u}) e^{i(k_xx+k_yy)} dx dy =0\)

- Simplify and Solve

The above equation simplifies to an ordinary differential equation that can be solved via standard methods, \(\frac{\partial^2 \tilde{u}}{\partial z^2} + (k^2 -k_x^2 - k_y^2) \tilde{u}=0\). The solution to this equation is: \(\tilde{u}(k_x, k_y, z)= C(k_x, k_y)e^{-\sqrt{k^2-k_x^2-k_y^2}z}\) with \(C(k_x, k_y)\) being the Fourier transform amplitude spectrum.

- Implement Dirichlet boundary condition

Now, we implement the Dirichlet boundary condition, \(u(x, y, 0)=0\), which gives us \(\tilde{u}(k_x, k_y, 0)=0\). It means that \(C(k_x, k_y)=-u(z)\) since \(e^{0}=1\).

- Inverse Fourier Transform

Finally, we use the inverse Fourier Transform to get \(u(x, y, z)\) in spatial domain. The amplitude has negative sign and we multiply it with the exponential function. Hence, it returns us the Green's function in three dimensional Dirichlet space.

Key concepts

Dirichlet Boundary Condition

The Dirichlet boundary condition plays a crucial role, especially in solving partial differential equations (PDEs) like the Helmholtz equation. It essentially specifies the values that a solution must take along the boundary of the domain. In this exercise, the domain is the half-space where \( z \geq 0 \), and the Dirichlet boundary condition is given by \( u(x, y, 0) = 0 \). This means that the function \( u \), representing our solution at the boundary, must be zero when \( z = 0 \).

These conditions allow us to handle the problem in a controlled manner by setting the behavior of the function clearly at the boundary. This specific value simplifies the problem as it limits the possible solutions, making them easier to find and analyze. Utilizing the Dirichlet boundary condition is essential for obtaining a meaningful Green's function in the context of the Helmholtz equation.

Fourier Transform

The Fourier transform is a mathematical tool used to convert functions from their original domain (often time space or space domain) into a frequency domain. This transform simplifies many equations, especially those with periodic or oscillatory nature, by breaking down complex problems into simpler parts.

For solving the Helmholtz equation in this exercise, applying the Fourier transform to the spatial coordinates \( x \) and \( y \) allows us to essentially "filter out" these dimensions and focus on the \( z \)-dependent part of the problem. This gives us an equation with transformed variables, making it easier to solve differential equations using standard methods.

• Simplifies the PDE by turning spatial derivatives into algebraic forms
• Makes boundary value problems more manageable
• Provides a pathway for employing inverse transforms to find solutions in the original spatial domain

In the context of this problem, the Fourier transform's role is pivotal in simplifying the Helmholtz equation to solve for \( \tilde{u}(k_x, k_y, z) \), which is then transformed back to find the original solution \( u(x, y, z) \).

Green's Function

Green's functions are an essential concept in mathematical physics and engineering, useful for solving inhomogeneous differential equations. In this situation, it's about assisting with the Helmholtz equation under specific boundary conditions.

A Green's function for a differential operator \( L \) solves the equation \( L G(x, x') = \delta(x - x') \), which allows us to express the solution of a more complicated equation via convolution with this function. In this exercise dealing with the Helmholtz equation, the solution through Green's function aids us in understanding how waves propagate under specified conditions. Once the Green's function is determined with the Dirichlet condition on the boundary, it represents how the influence at one point affects the solution in the entire domain. This serves as a fundamental building block in constructing solutions to boundary value problems.

• Relates solutions of differential equations to delta functions
• Allows simplification over complex regions with boundary conditions
• Provides insight into the fundamental solutions governing physical phenomena

Therefore, finding the Green's function for our original problem is about determining a systematic approach to solve the wave-like behavior under spatial constraints.

Cartesian Coordinates

Cartesian coordinates provide a straightforward and familiar method for representing geometries and equations in three dimensions. By using the \( (x, y, z) \) system, one can easily navigate through physical space and represent mathematical problems, such as the Helmholtz equation, in a structured way.

In the context of this problem, the Helmholtz equation is expressed in Cartesian coordinates to break down the partial derivatives operator \( abla^2 \). This operator involves the three spatial derivatives corresponding to \( x \), \( y \), and \( z \). The transformation from a general form to Cartesian coordinates transforms our equation into:
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} - k^{2}u = 0 \]

• Allows precise and clear problem formulation
• Facilitates application of mathematical operations and transformations
• Forms the basis of connecting physical phenomena to mathematical models

Considering Cartesian coordinates enhances the accuracy and simplicity of solving PDEs in specified geometric contexts, illustrating how these coordinate systems underpin many scientific computations.