Dispersion of Rotonal LEDO Waves

1. What Dispersion Means in the LEDO-Field

In classical media, dispersion appears when phase or group velocity depends on frequency.
In the LEDO-Field it works differently:

• LEDO-waves originate from intrinsic rotational dynamics of Rotons.
• The modulation propagates through an energy-density continuum that reacts instantaneously.
• What carries inertia is the Roton’s axis, not the wave.

Thus:

Dispersion in the LEDO-Field does not arise from propagation velocity differences, but from frequency-selective resonance coupling.

2. No Classical Velocity Dispersion

In standard wave mechanics one has a dispersion relation
$$ v(\omega) = \frac{\partial \omega}{\partial k}. $$

But LEDO-waves do not have a meaningful $k(\omega)$ because:

• all frequency components propagate with the same baseline response,
• Rotons respond with different strength depending on frequency ratio, phase, and axis alignment.

Therefore the correct physical object is the coupling strength:

$$ C_{AB} = f!\left( \frac{\omega_A}{\omega_B},, \Delta\phi,, \text{axis alignment} \right) $$

This replaces the classical dispersion relation.

3. Dispersion as Resonance Selectivity

Rotonal dispersion is defined by how efficiently two Rotons exchange or absorb LEDO-modulation.
Strong coupling occurs when:

• rotational frequencies match in simple integer ratios ($1{:}1$, $2{:}1$, $3{:}2$, …)
• axes are parallel or antiparallel,
• phases align near multiples of $\pi$.

Thus:

The LEDO-field is a frequency-selective resonance medium, not a velocity-dispersive one.

4. Three Resonance Dispersion Regimes

4.1 Co-Axial Resonance (distance-independent)

• strongest coupling,
• highest frequency selectivity,
• essentially no dispersion,
• no spatial falloff.

Equivalent to a dispersionless coherent channel.

4.2 Parallel-Axial Resonance ($\sim 1/d$)

• weaker than co-axial,
• sensitive to axis misalignment and phase drift,
• higher frequencies couple less efficiently.

This produces mild dispersive behavior through geometric filtering.

4.3 Spherical / Omnidirectional Resonance ($\sim 1/d^2$ to $\sim 1/d^3$)

• broad but weak coupling,
• dominated by geometric spreading,
• strong frequency-dependent attenuation:

$$ C(\omega) \propto \omega^{-2} \quad \text{to} \quad \omega^{-3}. $$

This corresponds to strong dispersion: high $\omega$ components couple less effectively.

5. Emergent Dispersion Effects

Even though LEDO-waves themselves are non-dispersive, the responses of Rotons show clear dispersive effects:

5.1 Angular Precession Lag

High-$\omega$ Rotons resist axis perturbation more strongly → reduced precession for the same incoming signal.

5.2 Phase-Interference Chromaticity

A multi-frequency LEDO-emission becomes sorted by coupling strength, not by velocity.
This acts like a frequency-dependent filter.

5.3 Spatial Mode Scaling

Large Rotons couple more efficiently to low frequencies; small Rotons to high frequencies.
A geometric form of dispersion.

6. Compact Summary

Dispersion in Rotonal LEDO-Waves arises from the frequency-dependent resonance coupling between Rotons.
The field is dispersionless, but the interactions are strongly frequency-selective.

7. Further Work

Planned extensions:

• derivation of explicit coupling kernels $C(\omega)$,
• visualization of dispersion regimes,
• integration into general LEDO-field equations,
• comparison with electromagnetic dispersion in media and vacuum.