Simulation of Standing and Propagating Sea
Waves with Three-Dimensional ARMA
Model
Ivan Gankevich and Alexander Degtyarev
Introduction
Studying behaviour of a ship at sea is often based on some model of external
excitations—any disturbance that displaces the vessel from equilibrium—major
component of which is wind waves. Currently, the most popular sea wave simulation models are based on the linear expansion of a stochastic moving surface as
a system of independent random variables. Such models were studied by St. Denis
and Pearson [1], Rosenblatt [2], Sveshnikov [3], and Longuet-Higgins [4]. The most
popular model is that of Longuet-Higgins (LH), which approximates propagating
sea waves as a superposition of elementary harmonic waves with random phases í µí¼ n
and random amplitudes c n :
í µí¼ (x, y, t) =
∑
n
c n cos(u n x + v n y − í µí¼ n t + í µí¼ n ),
(1)
where the wave number (u n , v n ) is continuously distributed on the (u, v) plane, i.e.
the unit area du × dv contains an infinite number of wave numbers. The frequency
í µí¼ n associated with wave numbers (u n , v n ) is given by a dispersion relation
í µí¼ n = í µí¼(u n , v n ).
The phase í µí¼ n are jointly independent random variables uniformly distributed in the
interval [0, 2í µí¼].
I. Gankevich ( ✉ ) ⋅ A. Degtyarev
Saint Petersburg State University, Universitetskii prospekt 35, Petergof,
Saint Petersburg 198504, Russia
e-mail: i.gankevich@spbu.ru
A. Degtyarev
e-mail: a.degtyarev@spbu.ru
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_18
249
Waves with Three-Dimensional ARMA
Model
Ivan Gankevich and Alexander Degtyarev
Introduction
Studying behaviour of a ship at sea is often based on some model of external
excitations—any disturbance that displaces the vessel from equilibrium—major
component of which is wind waves. Currently, the most popular sea wave simulation models are based on the linear expansion of a stochastic moving surface as
a system of independent random variables. Such models were studied by St. Denis
and Pearson [1], Rosenblatt [2], Sveshnikov [3], and Longuet-Higgins [4]. The most
popular model is that of Longuet-Higgins (LH), which approximates propagating
sea waves as a superposition of elementary harmonic waves with random phases í µí¼ n
and random amplitudes c n :
í µí¼ (x, y, t) =
∑
n
c n cos(u n x + v n y − í µí¼ n t + í µí¼ n ),
(1)
where the wave number (u n , v n ) is continuously distributed on the (u, v) plane, i.e.
the unit area du × dv contains an infinite number of wave numbers. The frequency
í µí¼ n associated with wave numbers (u n , v n ) is given by a dispersion relation
í µí¼ n = í µí¼(u n , v n ).
The phase í µí¼ n are jointly independent random variables uniformly distributed in the
interval [0, 2í µí¼].
I. Gankevich ( ✉ ) ⋅ A. Degtyarev
Saint Petersburg State University, Universitetskii prospekt 35, Petergof,
Saint Petersburg 198504, Russia
e-mail: i.gankevich@spbu.ru
A. Degtyarev
e-mail: a.degtyarev@spbu.ru
© Springer International Publishing AG, part of Springer Nature 2018
M. G. Velarde et al. (eds.), The Ocean in Motion, Springer Oceanography,
https://doi.org/10.1007/978-3-319-71934-4_18
249