Hilbert transform and its application to modulation of capillary gravity waves

In this paper,the following important problem is answered:Under what condition can the Hilbert transform be put in the simple form.Then,using the Hilbert technique,it is demonstrated that,with some relative weak restrictions,the particle motion at the sea surface and energy transmission may be obtained from a wave record.The analysis shows that for the two-dimensional and narrow-band mo tion (or a physical phenomena,such as capillary waves breaking,which satisfy the narrow-band approx imation) the local kinetic energy fluctuates in time as the envelope squared multiplied by a constant factor and can be obtained directly from a wave record.It is pointed that the local fluctuations of the potential energy as well as the local fluctuations of the total energy can be separated into a slowly varying part and a more rapid oscillating part.Both parts can be evaluated by means of the Hilbert transform.Finally,a physical interpretation of the envelope of the two-dimensional capillary waves as well as the method for wave group analysis is presented.