Modified non-linear theory for modelling wave propagation

The modified non-linear approach combines empirical and analytical wave models to allow a fast and precise prediction of non-linear wave propagation。 It can be ap-

plied both to ”forward” (downstream) and ”backward” (upstream) prediction of wave trains。 The ”forward” prediction at arbitrary positions of the model tank (sim- ilar to the below presented numerical wave tanks) in- cludes the representation of wave trains in the moving reference frame of a cruising vessel。 The ”backward” calculation is used for the transformation of given tar- get wave trains to the location of the wave maker: This is a unique feature of the proposed procedure。

The method starts with a linear wave train ζ0(t), either measured close to the wave maker or known from cal- culating the control signal of the wave maker。 Thus, as a first step the wave train is checked with regard to lin-

Fig。 1: Transient wave packet measured close to the wave board at x = 8。82 m: Linear wave theory is still acceptable for its description。

Fig。 2: Transient wave packet at x = 85。03 m: Compar- ison of registration with calculated data (linear trans- formation from x = 8。82 m — see Fig。 1) proves that

< 0。05 over the entire wave length range。 As a

linear wave theory gives inaccurate results。

further step in pre-processing the wave train is written

as Fourier series and time mapped with respect to the Shannon theorem:

n/2ζ0(ti) = 。 Aj cos(ωjti + ϕ0j ), i = 0, 1, 。 。 。 n − 1    (1)

by linear phase shift (the Fourier spectrum remains the same):

。 F (ωj, xl)ei(ωjti−k(xl+k−xl))6ω。

Aj  = |Fj|6ω = 6ω| 。 ζ0(ti0)e−iωjti0 6t|, (2)

i0=0

j = 0, 1, 。 。 。 n0/2

is the Fourier  spectrum of ζ  (t) with     ω =    2π  , ω   =

n6tj6ω,  and  i0   =  0 。 。 。 n0   denotes  the  initial  time  map-

ping。 The corresponding initial phase spectrum is also calculated by Fourier transform of the initial linear wave train:

Propagation of higher waves  cannot be described    by

Airy theory since the propagation velocity increases with the instantaneous wave height。 Also wave asym- metry and mass transport are introduced as consider- able quantities。 Fig。 2 shows the wave train from Fig。 1 transformed to x = 85。03 m by means of linear wave theory。 Note that Airy theory is not adequate any- more。 Especially, the higher frequencies deviate ob- viously since they propagate faster than predicted by linear wave theory。 Also the shape does not corre- spond with the measured wave train (flat troughs, steep

ϕ0j = arctan( «

R(Fj )

), j = 0, 1, 。 。 。 n/2。 (3)

crests)。

Our non-linear semi-analytical approach is based    on

The Hilbert transform of a function f is defined as

。

Stokes III。 It can be replaced by other terms from dif- ferent theories as well。

H(f ) := IFFT(

(FFT(f ))2  + (FFT(f )eiπ/2)2) (4)

Adapting Eq。 6, the phase Cij is adjusted to the non- linear wave celerity cij 。   For each step l in space the

where  ”(I)FFT”  is  the  abbreviation  of  the  (inverse)

Fourier Transform (Eq。 2), calculated by the Fast Fourier Transform algorithm。  The inverse FFT gives: