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//! waves trait implementation
//!
//! defines the Waves trait and related functionality for wave propagation modeling
use crate::{angle::Angle, geonum_mod::Geonum};
use std::f64::consts::PI;
pub trait Waves: Sized {
/// propagates waves through spacetime using wave equation principles
/// conventional: numerical wave propagation with spatial/temporal discretization O(n²)
/// geonum: direct phase evolution via angle rotation O(1)
fn propagate(&self, time: Self, position: Self, velocity: Self) -> Self;
/// creates dispersive waves with wavenumber and frequency
/// conventional: wave packet construction with fourier transforms O(n log n)
/// geonum: direct dispersion relation encoding O(1)
fn disperse(position: Self, time: Self, wavenumber: Self, frequency: Self) -> Self;
/// computes frequency as a geometric number from temporal phase evolution
/// conventional: scalar frequency extraction via division
/// geonum: directional frequency preserving temporal geometry O(1)
fn frequency(&self, other: &Self, time_interval: Self) -> Self;
/// computes wavenumber as a geometric number from spatial phase evolution
/// conventional: scalar wavenumber extraction via division
/// geonum: directional wavenumber preserving spatial geometry O(1)
fn wavenumber(&self, other: &Self, spatial_interval: Self) -> Self;
}
impl Waves for Geonum {
fn propagate(&self, time: Self, position: Self, velocity: Self) -> Self {
// compute phase based on position and time using geometric operations
let velocity_time = velocity * time;
let phase = position - velocity_time;
// create new geometric number with same magnitude but adjusted angle
Geonum::new_with_angle(self.mag, self.angle + phase.angle)
}
fn disperse(position: Self, time: Self, wavenumber: Self, frequency: Self) -> Self {
// the dispersion relation φ = kx − ωt is the wave's ANGLE, the polar form
// E = [1, kx − ωt], so cos_sin reads the field straight off the angle
let k_x = wavenumber * position;
let omega_t = frequency * time;
let phase = k_x - omega_t;
// signed phase = the (kx − ωt) vector projected onto the real axis. rotating
// a unit wave by it carries φ in the angle, where cos_sin can recover it —
// storing φ in the magnitude (the earlier form) collapsed it to a sign
let phi = phase.mag * phase.angle.grade_angle().cos();
Geonum::new_with_angle(1.0, Angle::new(phi / PI, 1.0))
}
fn frequency(&self, other: &Self, time_interval: Self) -> Self {
// compute temporal phase difference
let phase_diff = *self - *other;
// frequency is phase difference per unit time
let magnitude = phase_diff.mag / time_interval.mag;
Geonum::new_with_angle(magnitude, Angle::new(1.0, 2.0))
}
fn wavenumber(&self, other: &Self, spatial_interval: Self) -> Self {
// compute spatial phase difference
let phase_diff = *self - *other;
// wavenumber is phase difference per unit distance
let magnitude = phase_diff.mag / spatial_interval.mag;
Geonum::new_with_angle(magnitude, Angle::new(1.0, 2.0))
}
}
#[cfg(test)]
mod tests {
use super::*;
use std::f64::consts::PI;
#[test]
fn it_propagates() {
// create a geometric number representing a wave
let wave = Geonum::new(1.0, 0.0, 1.0);
// define wave parameters as geonums
let velocity = Geonum::new(3.0e8, 0.0, 1.0); // speed of light
let time_1 = Geonum::new(0.0, 0.0, 1.0);
let time_2 = Geonum::new(1.0e-9, 0.0, 1.0); // 1 nanosecond later
let position = Geonum::new(0.0, 0.0, 1.0);
// propagate wave at two different time points
let wave_t1 = wave.propagate(time_1, position, velocity);
let wave_t2 = wave.propagate(time_2, position, velocity);
// prove magnitude is preserved during propagation
assert_eq!(wave_t1.mag, wave.mag, "propagation preserves amplitude");
assert_eq!(wave_t2.mag, wave.mag, "propagation preserves amplitude");
// compute actual phase changes from the propagate method
// phase = position - velocity * time as Geonum operations
let velocity_time_1 = velocity * time_1;
let phase_1 = position - velocity_time_1;
let velocity_time_2 = velocity * time_2;
let phase_2 = position - velocity_time_2;
// test that wave angles are correctly updated
assert_eq!(
wave_t1.angle,
wave.angle + phase_1.angle,
"phase at t1 evolves according to position - velocity * time"
);
assert_eq!(
wave_t2.angle,
wave.angle + phase_2.angle,
"phase at t2 evolves according to position - velocity * time"
);
// prove phase difference between two time points
let phase_diff = wave_t2.angle - wave_t1.angle;
let expected_phase_diff = phase_2.angle - phase_1.angle;
assert_eq!(
phase_diff, expected_phase_diff,
"phase difference matches expected value"
);
// prove propagation in space
let position_2 = Geonum::new(1.0, 0.0, 1.0); // 1 meter away
let wave_p2 = wave.propagate(time_1, position_2, velocity);
let phase_p2 = position_2 - velocity_time_1;
assert_eq!(
wave_p2.angle,
wave.angle + phase_p2.angle,
"phase at p2 evolves according to position - velocity * time"
);
}
#[test]
fn it_disperses() {
// a plane wave is E = [1, kx − ωt]: the dispersion relation lives in the
// ANGLE, so cos_sin reads the field. k = 2π gives wavelength 1; null
// dispersion ω = ck carries the wave at the speed of light
let c = 3.0e8;
let k = 2.0 * PI; // 2π rad/m, wavelength 1 m
let omega = c * k; // ω = ck for light
let wavenumber = Geonum::scalar(k);
let frequency = Geonum::scalar(omega);
// at the origin the phase is 0 — the wave sits at its crest, cos = 1
let crest = Geonum::disperse(
Geonum::scalar(0.0),
Geonum::scalar(0.0),
wavenumber,
frequency,
);
assert!(crest.near_mag(1.0), "dispersed waves have unit amplitude");
let (cos_crest, _) = crest.angle.cos_sin();
assert!(
(cos_crest - 1.0).abs() < 1e-12,
"phase 0 at the origin — the wave's crest, cos = 1"
);
// a quarter wavelength out kx = π/2: the phase lands at grade 1, a node
let node = Geonum::disperse(
Geonum::scalar(0.25),
Geonum::scalar(0.0),
wavenumber,
frequency,
);
assert_eq!(
node.angle.grade(),
1,
"kx = π/2 lands at grade 1 — the node"
);
let (cos_node, _) = node.angle.cos_sin();
assert!(
cos_node.abs() < 1e-12,
"a quarter wavelength is a node — cos = 0"
);
// half a wavelength out kx = π: the phase is grade 2, the trough, cos = −1
let trough = Geonum::disperse(
Geonum::scalar(0.5),
Geonum::scalar(0.0),
wavenumber,
frequency,
);
assert_eq!(
trough.angle.grade(),
2,
"kx = π lands at grade 2 — the trough"
);
let (cos_trough, _) = trough.angle.cos_sin();
assert!(
(cos_trough + 1.0).abs() < 1e-12,
"half a wavelength is the trough — cos = −1"
);
// one period later t = 2π/ω the wave returns to the same phase — periodic,
// the angle's blade arithmetic handling the wraparound with no manual modulo
let period = 2.0 * PI / omega;
let later = Geonum::disperse(
Geonum::scalar(0.0),
Geonum::scalar(period),
wavenumber,
frequency,
);
let (cos_later, _) = later.angle.cos_sin();
assert!(
(cos_later - cos_crest).abs() < 1e-9,
"the wave repeats after one period T = 2π/ω"
);
// the phase velocity ω/k recovers the speed of light — the null dispersion
let wave_speed = frequency.mag / wavenumber.mag;
assert!(
(wave_speed - c).abs() / c < 1e-10,
"ω/k = c — the dispersion relation yields lightspeed"
);
}
}