Observational note explores waveform invariance and amplitude scaling in odd-harmonic waves, suggesting underlying mathematical principles.
Observational note. For the isolated peak wave formed by superposing constant-amplitude odd harmonics on the half-wavelength phase interval, giving two copies a common relative phase 2a applied identically to all harmonics yields, by the product-to-sum identity, psi = 2 cos(a) S_N, hence I = 4 cos^2(a) I_N. Two elementary consequences follow: (i) waveform invariance - the normalized waveform is independent of the relative phase and coincides with that of a single copy; (ii) a contrast law - the relative phase scales the squared amplitude by cos^2(a) without changing the waveform. No physical interpretation is given.
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Noriaki Kihara (2026) studied this question.
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