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: u1h1 = u2h2. 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