252
I. Gankevich and A. Degtyarev
AR model was successfully applied to predict evolution of propagating wave profiles based on instantaneous wave recordings. In [13] AR model is used to predict
swell waves to control wave-energy converters (WEC) in real-time. In order to make
WEC more efficient its internal oscillator frequency should match the one of ocean
waves. The authors treat wave elevation as time series and compare performance
of AR model, neural networks and cyclical models in forecasting time series future
values. AR model gives the most accurate prediction of low-frequency swell waves
for up to two typical wave periods. It is an example of successful application of AR
process to ocean wave modelling.
The feature that distinguishes present work with respect to afore-mentioned ones
is the study of three-dimensional (2D in space and 1D in time) ARMA model, which
is mostly a different problem.
1. Yule–Walker system of equations, which are used to determine AR coefficients,
has complex block-block structure.
2. Optimal model order (in a sense that target spectrum agrees with initial) is determined manually.
3. Instead of PM spectrum, analytic formulae for standing and propagating waves
ACF are used as the model input.
4. Three-dimensional wavy surface should be compatible with real ocean surface
not only in terms of spectral characteristics, but also in the shape of wave profiles.
So, model verification includes distributions of various parameters of generated
waves (lengths, heights, periods etc.).
Multi-dimensionality of investigated model not only complexifies the task, but also
allows to carry out visual validation of generated wavy surface. It is the opportunity
to visualise output of the programme that allowed to ensure that generated surface is
compatible with real ocean surface, and is not abstract multi-dimensional stochastic
process that is real only statistically.
Pressure Field Determination Formulae
Small Amplitude Waves Theory
In [14–16] the authors propose a solution for inverse problem of hydrodynamics of
potential flow within the framework of small-amplitude wave theory (under assumption that wave length is much larger than height: í µí¼ ≫ h). In that case inverse problem
is linear and reduces to Laplace equation with mixed boundary conditions, and equation of motion is solely used to determine pressures for calculated velocity potential
derivatives. The assumption of small amplitudes means the slow decay of wind wave
coherence function, i.e. small change of local wave number in time and space compared to the wavy surface elevation (z coordinate). This assumption allows to calculate elevation z derivative as í µí¼ z = kí µí¼ , where k is wave number. In two-dimensional
case the solution is written explicitly as
I. Gankevich and A. Degtyarev
AR model was successfully applied to predict evolution of propagating wave profiles based on instantaneous wave recordings. In [13] AR model is used to predict
swell waves to control wave-energy converters (WEC) in real-time. In order to make
WEC more efficient its internal oscillator frequency should match the one of ocean
waves. The authors treat wave elevation as time series and compare performance
of AR model, neural networks and cyclical models in forecasting time series future
values. AR model gives the most accurate prediction of low-frequency swell waves
for up to two typical wave periods. It is an example of successful application of AR
process to ocean wave modelling.
The feature that distinguishes present work with respect to afore-mentioned ones
is the study of three-dimensional (2D in space and 1D in time) ARMA model, which
is mostly a different problem.
1. Yule–Walker system of equations, which are used to determine AR coefficients,
has complex block-block structure.
2. Optimal model order (in a sense that target spectrum agrees with initial) is determined manually.
3. Instead of PM spectrum, analytic formulae for standing and propagating waves
ACF are used as the model input.
4. Three-dimensional wavy surface should be compatible with real ocean surface
not only in terms of spectral characteristics, but also in the shape of wave profiles.
So, model verification includes distributions of various parameters of generated
waves (lengths, heights, periods etc.).
Multi-dimensionality of investigated model not only complexifies the task, but also
allows to carry out visual validation of generated wavy surface. It is the opportunity
to visualise output of the programme that allowed to ensure that generated surface is
compatible with real ocean surface, and is not abstract multi-dimensional stochastic
process that is real only statistically.
Pressure Field Determination Formulae
Small Amplitude Waves Theory
In [14–16] the authors propose a solution for inverse problem of hydrodynamics of
potential flow within the framework of small-amplitude wave theory (under assumption that wave length is much larger than height: í µí¼ ≫ h). In that case inverse problem
is linear and reduces to Laplace equation with mixed boundary conditions, and equation of motion is solely used to determine pressures for calculated velocity potential
derivatives. The assumption of small amplitudes means the slow decay of wind wave
coherence function, i.e. small change of local wave number in time and space compared to the wavy surface elevation (z coordinate). This assumption allows to calculate elevation z derivative as í µí¼ z = kí µí¼ , where k is wave number. In two-dimensional
case the solution is written explicitly as