What is the green function for the Helmholtz equation?
2 Answers
The Green's function for the Helmholtz equation is a fundamental solution that helps in solving boundary value problems involving this partial differential equation. The Helmholtz equation is given by:
\[ \nabla^2 u(\mathbf{r}) + k^2 u(\mathbf{r}) = -\delta(\mathbf{r} - \mathbf{r}') \]
where \( \nabla^2 \) is the Laplacian operator, \( k \) is the wavenumber, \( u(\mathbf{r}) \) is the function to be solved for, and \( \delta(\mathbf{r} - \mathbf{r}') \) is the Dirac delta function centered at \( \mathbf{r}' \).
The Green's function \( G(\mathbf{r}, \mathbf{r}'; k) \) for the Helmholtz equation satisfies:
\[ \nabla^2 G(\mathbf{r}, \mathbf{r}'; k) + k^2 G(\mathbf{r}, \mathbf{r}'; k) = -\delta(\mathbf{r} - \mathbf{r}') \]
### Green's Function in Different Dimensions
1. **Three Dimensions**:
In 3D, the Green's function is given by:
\[ G(\mathbf{r}, \mathbf{r}'; k) = \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4 \pi |\mathbf{r} - \mathbf{r}'|} \]
This solution represents the response at point \( \mathbf{r} \) due to a point source located at \( \mathbf{r}' \).
2. **Two Dimensions**:
In 2D, the Green's function is:
\[ G(\mathbf{r}, \mathbf{r}'; k) = \frac{i}{4} H_0^{(1)}(k |\mathbf{r} - \mathbf{r}'|) \]
Here, \( H_0^{(1)} \) is the Hankel function of the first kind of order zero. This function characterizes the cylindrical symmetry of the problem in 2D.
3. **One Dimension**:
In 1D, the Green's function simplifies to:
\[ G(x, x'; k) = \frac{-i}{2k} e^{ik|x - x'|} \]
### Applications
The Green's function is used to solve the Helmholtz equation by expressing the solution \( u(\mathbf{r}) \) as:
\[ u(\mathbf{r}) = \int_{\text{domain}} G(\mathbf{r}, \mathbf{r}'; k) f(\mathbf{r}') \, d\mathbf{r}' \]
where \( f(\mathbf{r}') \) represents the source term or forcing function. This integral approach can handle various boundary conditions by incorporating appropriate boundary conditions into the Green's function or its formulation.
\[ \nabla^2 u(\mathbf{r}) + k^2 u(\mathbf{r}) = -\delta(\mathbf{r} - \mathbf{r}') \]
where \( \nabla^2 \) is the Laplacian operator, \( k \) is the wavenumber, \( u(\mathbf{r}) \) is the function to be solved for, and \( \delta(\mathbf{r} - \mathbf{r}') \) is the Dirac delta function centered at \( \mathbf{r}' \).
The Green's function \( G(\mathbf{r}, \mathbf{r}'; k) \) for the Helmholtz equation satisfies:
\[ \nabla^2 G(\mathbf{r}, \mathbf{r}'; k) + k^2 G(\mathbf{r}, \mathbf{r}'; k) = -\delta(\mathbf{r} - \mathbf{r}') \]
### Green's Function in Different Dimensions
1. **Three Dimensions**:
In 3D, the Green's function is given by:
\[ G(\mathbf{r}, \mathbf{r}'; k) = \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4 \pi |\mathbf{r} - \mathbf{r}'|} \]
This solution represents the response at point \( \mathbf{r} \) due to a point source located at \( \mathbf{r}' \).
2. **Two Dimensions**:
In 2D, the Green's function is:
\[ G(\mathbf{r}, \mathbf{r}'; k) = \frac{i}{4} H_0^{(1)}(k |\mathbf{r} - \mathbf{r}'|) \]
Here, \( H_0^{(1)} \) is the Hankel function of the first kind of order zero. This function characterizes the cylindrical symmetry of the problem in 2D.
3. **One Dimension**:
In 1D, the Green's function simplifies to:
\[ G(x, x'; k) = \frac{-i}{2k} e^{ik|x - x'|} \]
### Applications
The Green's function is used to solve the Helmholtz equation by expressing the solution \( u(\mathbf{r}) \) as:
\[ u(\mathbf{r}) = \int_{\text{domain}} G(\mathbf{r}, \mathbf{r}'; k) f(\mathbf{r}') \, d\mathbf{r}' \]
where \( f(\mathbf{r}') \) represents the source term or forcing function. This integral approach can handle various boundary conditions by incorporating appropriate boundary conditions into the Green's function or its formulation.