The factor 1 / 2 π π comes from Fourier transform conventions. The amplitude A (k) contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of u (x, t) evaluated at t = 0 by inverting the Fourier transform relation above: A ( k ) = 1 2 π π ∫ ∫ − − ∞ ∞ ∞ ∞ u ( x , 0 ) e − − i k x d x .