It is well known that the Fourier transform (FT) may be used to describe the motion of a localized linear packet of waves on the infinite interval. In this context one may demonstrate that a packet may be viewed as the linear superposition of infinitely many, infinitely long sinusoids («elementary wavelets»). It is possible, through the nontrivial introduction of generalized function theory, to show that each sinusoid corresponds to a Dirac delta-function Fourier wave number spectrum (the corresponding «elementary spectrum»). In this paper we investigate an analogous problem for the spectral or scattering transform (ST), with specific application to nonlinear wave behaviour governed by the KortewegdeVries equation (on the infinite interval) in the absence of solitons. We are forced by the nonlinear theory to consider only finite-amplitude spectra and this eliminates the possibility of a delta-function representation. But, by assuming a finite amplitude, rectangular elementary spectrum (with infinitesimal width and centred on a single wave number) we find, to second order, that the resultant elementary wavelet is the Stokes series solution to the KdV equation. This suggests the (as yet unproven) hypothesis that a nonlinear wave packet solution to the KdV equation might be representable as the nonlinear superposition of infinitely many, infinitely long Stokes waves. We also derive an approximate expression for the Stokes wave in shallow water which is accurate to a surprisingly high degree of nonlinearity.