### A12. Period with changing current velocity

As in A11 we consider an infInite domain with a uniform wave field. An initial spectrum is given (narrow both in σ and in θ). In test A12 the depth is constant (=5 m), and the current velocity is varied as function of time. It is seen from theory that wave number and direction of the wave field have to remain constant. A table is written giving depth, current velocity, wavelength, direction and period (both relative and absolute) as function of time.

The analytical solution for this case is: $T = 2 \pi / \left\{\left[g k tanh\left(k d\right)\right]1/2+ U.k\right\}$ (where T is absolute period).
We use the following Initial conditions: k0 = 0.1 m-1, d0 = 5 m and $\theta$0 = 90o. It follows that T0 = 9 s.

Since Swan uses a spectrum formulated in terms of $\sigma$; there is transport of energy in $\sigma$-direction; there is also transport in $\theta$. There may be a change in the shape of the spectrum due to numerical inaccuracies. Plot file A12.plt shows (absolute) period, wavelength and direction as function of current velocity (the component in y-direction since the direction of the wave field is 90o.

1 case is considered:

A12a
time-varying current velocity