Module analytical 

Analytical solutions for wave equation validation

This module provides exact solutions to acoustic scattering problems used to validate numerical methods (BEM, FEM).

Available Solutions

• 1D: Plane wave, standing wave, damped wave
• 2D: Cylinder scattering (Bessel/Hankel series)
• 3D: Sphere scattering (Mie theory)

Modules

solutions_1d

1D Analytical Solutions

solutions_2d

2D Analytical Solutions

solutions_3d

3D Analytical Solutions

Structs

AnalyticalSolution

Analytical solution result

Point

Point in space (1D, 2D, or 3D)

Constants

SPEED_OF_SOUND

Default speed of sound (m/s)

Functions

classify_regime

Classify scattering regime based on ka

cylinder_directivity_2d

Pressure directivity pattern (far-field)

cylinder_scattering_2d

Cylinder scattering: rigid circular cylinder in a plane wave

cylinder_scattering_cross_section_2d

Total scattering cross-section (2D)

damped_wave_1d

1D wave with absorption: p(x) = exp(-(α + ik)x)

helmholtz_1d_mode

1D Helmholtz solution in a bounded domain [0, L]

legendre_p

Legendre polynomial Pₙ(x)

plane_wave_1d

1D plane wave: p(x) = exp(ikx)

plane_wave_2d

2D plane wave: p(x,y) = exp(ik(x cos θ + y sin θ))

plane_wave_3d

3D plane wave: p(x,y,z) = exp(ik·r)

point_source_3d

Point source (monopole): G(r) = exp(ikr)/(4πr)

sphere_rcs_3d

Radar Cross Section (RCS) for sphere

sphere_scattering_3d

Sphere scattering: rigid sphere in a plane wave (Mie theory)

sphere_scattering_efficiency_3d

Scattering efficiency Q_scat = σ / (πa²)

spherical_bessel_j

Spherical Bessel function jₙ(x)

spherical_bessel_y

Spherical Bessel function yₙ(x) (Neumann function)

standing_wave_1d

1D standing wave: p(x) = sin(kx)