Some aspects of the interpretation of spectra in meteorology

The meaningfulness of spectral presentations, and of spectral peaks in particular, is considered by the use of simple examples. First, we derive analytically the spectra of sinusoidal finite-length waves and subject the spectra to several area-conserving transformations. The peak of the logarithmic spectrum (power density per unit natural logarithm of frequency) is shown to be the most appropriate for defining the scales (or frequencies) of the waves. The advantage of the logarithmic spectrum becomes even more apparent when a wave consisting of the positive part of a sine wave is considered. In that case, the conventional frequency presentation is misleading because in addition to the erroneous location of the spectral peak, an increase of power density towards low frequencies occurs, giving the spectra the appearance of red noise. For the same wave, it is shown that the logarithmic spectrum has a single peak at the position corresponding with the actual wave frequency.