math-wave

Analytical solutions for wave and Helmholtz equations.

Installation

Add to your Cargo.toml:

[dependencies]
math-wave = { path = "../math-wave" }

Features

• 1D solutions: Plane waves, standing waves, damped waves
• 2D solutions: Cylinder scattering (Bessel/Hankel series)
• 3D solutions: Sphere scattering (Mie theory)
• Special functions: Spherical Bessel/Hankel, Legendre polynomials
• Green's functions: Helmholtz kernel and derivatives

Usage

1D Plane Wave

use math_wave::analytical::plane_wave_1d;
use std::f64::consts::PI;

let wave = plane_wave_1d(1.0, 0.0, 2.0 * PI, 100);
assert_eq!(wave.pressure.len(), 100);

3D Sphere Scattering

use math_wave::analytical::sphere_scattering_3d;
use std::f64::consts::PI;

let scatter = sphere_scattering_3d(1.0, 1.0, 20, vec![2.0], vec![0.0, PI / 2.0]);
assert!(scatter.pressure[0].norm() > 0.0);

Green's Functions

The Helmholtz Green's function for 3D:

G(x, y) = exp(ik|x-y|) / (4π|x-y|)

For 2D (cylindrical):

G(x, y) = (i/4) H₀⁽¹⁾(k|x-y|)

Modules

• analytical - Exact solutions for validation (1D, 2D, 3D)
• greens - Green's function implementations
• special - Spherical Bessel, Hankel, and Legendre functions

Key Types

Point

Represents a point in 1D, 2D, or 3D space with coordinate conversions:

use math_wave::Point;

let p = Point::new_3d(1.0, 2.0, 3.0);
let r = p.radius();
let theta = p.theta_3d();

AnalyticalSolution

Contains solution data with error computation:

use math_wave::AnalyticalSolution;

let l2_err = solution.l2_error(&reference);
let linf_err = solution.linf_error(&reference);
let rel_err = solution.relative_l2_error(&reference);

Use Cases

• Validating BEM (Boundary Element Method) solvers
• Validating FEM (Finite Element Method) solvers
• Reference solutions for convergence studies
• Test data generation for numerical methods