following iteration scheme for the non-linear wave train has to be run:

is the initial phase spectrum from Airy theory

As a test case we chose a transient wave packet mea- sured at two positions — the first location close to the wave board (x = 8。82 m) where the wave train is linear and the second position where the waves are already steeper and cannot be calculated by linear transform anymore (x = 85。03 m)。 Fig。 1 shows the linear wave train and its envelope。

According to Airy wave theory a wave train at an arbi- trary position xl is transformed to another position xl+k

and Cij is the modified phase calculated from the theory

adequate to the investigated case。 Here the following equations  have  to  be solved  to  calculate  the  kij (see

e。g。 Kinsman (1965), Skjelbreia (1959)):

1。deep water d/L0 ≥ 0。5:

ω2 2

j  = gkij (1 + (kijai)  ) (8)

(Stokes III) — solved by Cardan formulae

Fig。 3: Iteration of wave numbers kij (ωj, a(ti)) as func- tion of the instantaneous wave envelope a at time step ti。 Propagation velocity cij = ωj/kij increases with ”wave amplitude” ai (see Eqs。 8-10)。

2。intermediate water depth 0。04 < d/L0 < 0。5:

Fig。 4: Non-linear transformation of  wave  train  in Fig。 1 to downstream positions (showing selected it- eration  steps):  Comparison  with  measured  data  at x = 85。03 m is satisfactory (see also Fig。 5)。

140 145 150 155 160 165 170

Fig。  5:  Wave  train from Fig。 1 is transformed to  posi-

(Stokes III) — solved by fix point  iteration

3。shallow water d/L0 ≤ 0。04:

j  = gkij tanh(kijd) (10) (linear wave theory)

Our test case is a transient wave packet measured at the Hamburg Ship Model Basin with a water depth of  d  =  5。6  m。  Thus  deep  water  limit  frequency is ω = 2。34  rad/s,  the  shallow  water  limit  frequency ω = 0。44 rad/s。

kij is subject to the temporary envelope ai = a(ti) = H(ζi)。 Thus the required Hilbert transform for the par- ticular xl is calculated at each time step ti since it rep- resents the instantaneous wave height at a particular point in time and space。 It also considers the fact that the wave height increases on the way through the tank and non-linearities gain more and more influence。 Fig。 3 gives an impression of the iteration of the kij 。

In accordance with Stokes III wave theory the corre- sponding wave components at xl are:

tion x = 85。03 m using the described non-linear calcu- lation procedure (iteration step 105) and compared to measurements。

xl is given。 Note that the phase velocity depends not only on frequency but also on wave elevation which is represented by the instantaneous envelope and its lin- ear amplitude distribution。 The correct shape is also composed of higher order components (bounded waves

— Eq。 12 and  13)。

The calculation of Cij , Eq。 7-13, is repeated twice to average kij from the first and second step。 The (6x)l are chosen such that they decrease with increasing non- linearity。  In  our  example  the  iteration  is  done with 2 × 105 steps in space and 1024 steps in time。 Fig。 4 presents some iteration steps。 The result of the calcula- tion procedure is shown in Fig。 5 and compared to the measured wave train。 Agreement with the measured time series is good。 Compared to Fig。 2 the higher fre- quency terms show the adequate propagation speed and a pronounced non-linear shape with steep crests and flat troughs。