Seiche Calculator

The seiching of two coupled basins is examined using shallow water wave theory. The 2-D geometry is given in the figure below.

The free surface displacement in each basin is given by

where *k *is
the wavenumber in each basin and the basin lengths are b and c. The seiching
angular frequency (2 pi /T, the seiching period) is sigma. The wave amplitudes
a are arbitrary; the ratio between the two is important.

From the linear shallow water equations, the horizontal velocity in each basin is:

, where g is the acceleration of gravity and

At *x*=0, the free surface elevations
must be equal as must the volume flux of water: *u _{1}h_{1
}= u_{2}h_{2}. *These
two conditions lead to

The first of these equations gives the ratio between the wave amplitudes in each region and the second equation is a dispersion relationship for the seiching frequency. (This last equation has to be solved iteratively to find sigma; it is not an easy feat as the solution brackets zeros in the tangent function. This is ok for arguments of n pi, where n is an integer; but hard when the solution occurs near n pi/2.) Both of these equations are used in the applet. Note that sigma= k C by definition.

You may notice that the water surface slope at the junction between the basins (*x=0*) is not the same in both basins, as the velocities are not
matched at the junction--just the volume flux.

**Note** that the special
case of the same basin length and depths, the applet fails (use the previous
seiche applet). However you can get close withb=c+e, where e is a very
small amount. Also, presently not all modes are correctly represented--you will find some of the odd modes missing.

A few special cases are shown in Table 5.1 of
Dean and Dalrymple, **Water Wave Mechanics for Engineering and Scientists,**
World Scientific Press.

Comments: Robert Dalrymple