An Observation on the Isolated Peak Wave of a Constant-Amplitude Odd-Harmonic Sum on a Half-Wavelength Phase Interval and Its Localization

This short observational note records, as an elementary property of Fourier sums, that superposing constant-amplitude odd harmonics on the half-wavelength phase interval [-pi/2, pi/2] produces an isolated peak wave with a dominant main peak at the center and zeros at both ends. Starting from the constant-amplitude odd-harmonic sum S_N(phi)=sum_{m=0}^{(N-1)/2} cos((2m+1)phi) and deriving its closed form sin((N+1)phi)/(2 sin phi) independently (without presupposing the Dirichlet kernel), the note shows step by step that, under the magnified variable u=(N+1)phi, the peak-normalized shape approaches the universal function sin(u)/u, so that the localization width of the squared amplitude shrinks as 1/(N+1). A formula for the highest odd-harmonic order required to reach a prescribed localization half-width is also given, with a numerical-substitution method and an approximate closed form. The paper gives no physical interpretation; an appendix lists pure scale-ratio arithmetic only as a numerical example.