A Derivation of the Waldmeier Effect Exponent from Harmonic Oscillator Mechanics, with Cross-Domain Tests
Year: 2026
Type: preprint
Abstract: The Waldmeier Effect in solar physics — that stronger solar cycles have shorter rise times — has been empirically documented since 1935 and has lacked a theoretical derivation of its specific exponent for 90 years. This paper derives the exponent −0.500 from first principles. In any bounded oscillatory system where the driver mechanism sets the effective restoring stiffness and amplitude reflects driver strength, harmonic oscillator frequency scaling gives T_rise ∝ A⁻°·⁵ exactly. Confirmation against 270 years of SIDC/SILSO data (24 complete cycles) gives slope −0.52, 95% CI [−0.72, −0.34], R² ≈ 0.59, p < 0.001 — the confidence interval brackets the derived −0.500 and the 4% discrepancy is within solar cycle scatter. The same dimensionless constant appears in the grand solar cycle record at centennial timescales (11 cycles over 10,000 years, Γ_dimless = 0.969 vs 0.971 for annual cycles), confirming the constant is a property of the solar dynamo mechanism rather than the cycle length. An independently predicted structural asymmetry — rise amplitude-dependent, decline governed by global restoring force — is confirmed: rise slope −0.52 (p < 0.001), decline slope +0.28 (p = 0.023), opposite signs, both significant. A three-criterion selection framework — same driver mechanism, amplitude reflecting driver strength not stored energy, separable rise phase — specifies which oscillatory systems should show this scaling. Five null results (solar flares, classical novae, pulsar glitches, gravitational wave ringdowns, neuronal avalanches) are predicted and confirmed by the same framework. A preliminary cross-domain survey across 27 systems spanning quantum coherence to grand solar cycles is presented in Section 9, with explicit acknowledgement of data provenance limitations.
Author: Daw, Steven
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162572366
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