On the truncation of spectra

Abstract

A rational method of truncating a Pierson and Moskowitz wave spectrum ${\mathit{\varphi }}_{\mathit{\xi }\mathit{\xi }}\mathrm{(}\mathit{\omega }\mathrm{)}$ of wave elevation is explained. The fourth order moment of the spectrum,

${\mathrm{m}}_4={\int }_0^{\infty }{\mathit{\omega }}^4{\mathit{\varphi }}_{\mathit{\xi }\mathit{\xi }}\mathrm{(}\mathit{\omega }\mathrm{)}\mathrm{d}\mathit{\omega }$

is rendered finite and its value is given, once the cut-off frequency has been selected. The implications of the theory as regards bandwidth parameter are examined and the particular features of the ITTC and ISSC formulations are discussed.