DOI 10.1393/ncc/i2005-10134-1 IL NUOVO CIMENTO Vol. 28 C, N. 6 Novembre-Dicembre 2005 On the computation of the Benjamin-Feir Index( * ) M. Serio( 1 ), M. Onorato( 1 ), A. R. Osborne( 1 ) and P. A. E. M. Janssen( 2 ) ( 1 ) Dipartimento di Fisica Generale, Universit` a di Torino Via P. Giuria, 1 - 10125 Torino, Italy ( 2 ) European Centre for Medium-Range Weather Forecasts Reading RG2 9AX, Berks, England (ricevuto l’ 11 Luglio 2005; approvato il 19 Luglio 2005; pubblicato online il 19 Dicembre 2005) Summary. — Recently it has been shown theoretically, numerically and experimen- tally that the statistical properties (probability density function of wave amplitude and wave height) of long crested surface gravity waves depend not only on steepness but also on the Benjamin-Feir Index (BFI), which is the ratio between wave steep- ness and spectral bandwidth. The computation of this index requires the estimation of a number of parameters such as the spectral bandwidth and the peak frequency. For a given time series or a wave spectrum those parameters can be calculated using different methods, thus leading to different numerical values of the BFI. We analyze different approaches for computing the BFI and, based on numerical experiments with simulated spectra, we outline a unique robust methodology for its computation. PACS 92.10.Hm – Surface waves, tides, and sea level. PACS 47.35.+i – Hydrodynamic waves. 1. – Introduction It is well known that a Stokes wave in deep water is unstable to suitable small am- plitude long perturbations. If a 0 is the Stokes amplitude and k 0 is its wave number, then the wave is unstable whenever ak 0 is greater than K/(2 √ 2k 0 ), where K is the wave number of the perturbation. This instability, well known in other fields of physics as the modulational instability, is known in the field of surface gravity waves as the Benjamin-Feir instability. The effect of this instability is the following: as the Stokes wave becomes unstable, a single wave in the middle of the group begins to grow at the expense of the surrounding waves (actually borrowing mass from them), giving rise to a large amplitude wave. Recently it has been shown theoretically [1], numerically [2-4] and experimentally [5-7] that, if the wave steepness is sufficiently large and the spec- tral bandwidth is sufficiently small, this effect can take place also in a random spectra. ( * ) The authors of this paper have agreed to not receive the proofs for correction. c  Societ` a Italiana di Fisica 893