Wave Envelope and Related Spectra

Abstract

Theoretical forms of spectra associated with the odd and even powers of the wave envelope are examined. It is shown that spectral densities for the even powers of the envelope admit exact forms, whereas those for the odd powers can only be expressed in series, involving first‐, second‐, and higher‐order terms. The implications of these results are discussed and contrasted with a number of similar results stated in previous studies. Certain discrepancies and points of concern encountered are clarified. The two leading terms to the envelope spectral density are then examined in detail and compared with simulated data typical of wide‐ and narrow‐banded waves. It is found that the sum of these two leading terms would represent the envelope spectral density with sufficient accuracy for most purposes. As an alternative to the relatively complex analytical forms implied by enveloperelated spectra, the possibility of constructing more practical approximations is considered.