Helmholtz Equation Solver Calculator

Example data table

Formula used

This solver targets the Helmholtz equation:

∇²u + k²u = f

For 2D, a five-point stencil is used at interior nodes:

(uE+uW-2u)/hx² + (uN+uS-2u)/hy² + k²u = f

Rearranged update (used inside SOR):

u ← (1-ω)u + ω * [ (uE+uW)/hx² + (uN+uS)/hy² - f ] / (2/hx² + 2/hy² + k² )

For 1D, the second derivative finite difference produces a tridiagonal system solved directly.

How to use this calculator

1. Select 1D or 2D, then set the domain size and grid points.
2. Enter the wavenumber k and choose a source type f.
3. For 2D, pick boundary modes and amplitudes for each edge.
4. Adjust iteration controls: tolerance, maximum iterations, and relaxation ω.
5. Press Solve to compute results shown above the form.
6. Use the download buttons to export CSV or a PDF summary.

Notes for advanced use

• Higher k increases oscillation; refine the grid accordingly.
• If SOR stalls, try ω between 1.3 and 1.9.
• Nonzero boundary modes can excite resonant patterns in the domain.
• For strict accuracy, validate against analytical solutions for simple cases.

Professional guide

1) What the solver computes

The Helmholtz equation, ∇²u + k²u = f, models steady spatial fields such as acoustic pressure, electromagnetic components, and vibrating membranes. This calculator approximates u on a 1D line or a 2D rectangle using finite differences, producing nodal values, simple statistics, and exportable tables.

2) Wavenumber and physical scaling

The wavenumber k sets the intrinsic length scale. The associated wavelength is λ = 2π/k (for k>0). If you are matching a frequency-domain problem, keep units consistent: when x is meters, k is rad/m, and f has the same units as ∇²u.

3) Grid resolution guidelines

Accuracy depends strongly on resolving oscillations. A practical rule is 12–20 points per wavelength: h ≤ λ/12 (preferably h ≤ λ/20) where h is the grid spacing. For example, if k=12, then λ≈0.524 and a spacing h≈0.03 gives about 17 points per wavelength. Under-resolution can cause phase errors and artificial damping.

4) Boundary conditions drive modes

Dirichlet boundaries are specified on all edges. Constant boundaries impose offsets, while sine/cosine boundaries inject smooth spatial modes. In rectangular domains, boundary modes can align with natural standing patterns; sweeping k and observing peak interior amplitudes is a useful way to explore near-resonant behavior.

5) Choosing a source term

The source f controls forcing. A Gaussian source approximates a localized emitter; keep σ several grid cells wide (e.g., σ ≥ 3h) to avoid aliasing. The sinusoidal source is convenient for controlled benchmarks because it produces smooth, predictable patterns on uniform grids.

6) Convergence behavior and settings

The 2D option uses Successive Over-Relaxation. The reported residual estimate decreases when the iterate becomes consistent with the discrete stencil. Typical choices are ω=1.3–1.9, tolerance 10⁻⁶ to 10⁻⁸, and iteration limits that scale with grid size. If convergence stalls, lower ω or start with a coarser grid, then refine.

7) Interpreting the output

Use min/max to detect overshoot and boundary inconsistency. The center value is a quick indicator for symmetry checks. For 2D, compare edge slices to confirm boundary enforcement. When exporting, the CSV is best for post-processing, while the PDF is suited for reporting inputs, settings, and summary statistics.

8) Validation and good practice

Validate with simple cases: zero source with homogeneous boundaries should converge to near-zero, and sinusoidal forcing should remain smooth. Perform grid refinement by doubling Nx and Ny; stable results should change modestly. For engineering decisions, always document k, grid spacing, tolerance, and boundary choices alongside the exported results.

FAQs

1) What does a larger k do to the solution?

Larger k shortens the wavelength (λ=2π/k) and increases oscillation. You usually need a finer grid to avoid phase errors, and the iterative 2D solve can take more iterations.

2) How many grid points should I use?

Target 12–20 points per wavelength. Estimate λ, then choose h ≤ λ/12. In 2D, increase Nx and Ny together to keep spacing balanced.

3) Why can SOR fail to converge?

Convergence can degrade with high k, coarse grids, or an aggressive relaxation ω. Reduce ω, relax tolerance temporarily, or solve on a coarser grid and refine.

4) What is the difference between the 1D and 2D options?

1D builds a tridiagonal system and solves it directly, so it is fast and robust. 2D uses iterative SOR on a five-point stencil, which is flexible but may require careful settings.

5) How should I pick σ for a Gaussian source?

Choose σ wide enough to be resolved: a practical minimum is σ ≥ 3h. Very narrow sources can look like numerical spikes and may slow convergence.

6) Can I model mixed boundary types?

This tool applies Dirichlet boundaries on all edges, with constant, sine, or cosine profiles. To approximate other types, you can emulate them by choosing boundary profiles that match measured edge values.

7) What should I report with exported results?

Include domain size, grid (N or Nx×Ny), k, source parameters, boundary modes/values, tolerance, max iterations, and ω. These control accuracy and reproducibility.