= Bandwidth limitation in fixed-power filament optimization
At fixed leading <power of a periodic planar filament>, the family $a\sin(mkx+\Omega t)$ has <propulsive force of a periodic planar filament> increasing with spatial harmonic $m$. The formal quadratic objective is unbounded without a spatial cutoff; large harmonics eventually invalidate the small-slope model. With a finite admissible <Fourier series> band, force divided by power is a weighted average of the signed ratios $-(\xi_\perp-\xi_\parallel)k_j/(\xi_\perp\Omega_j)$. An allowed extremal ratio is attained by a <travelling wave> mode, or by a superposition with one common phase speed.
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