|Description||The tides of coastal embayments derive their energy from the ocean tides rather
than from the direct action of lunar and solar gravitational forces. They are considered
to be part of co-oscilating systems in which the period is determined by the tide in the
outer sea, while the detailed character of the motion depends on the size and form of
the enclosed basin (Defant, 1925; Doodson and Warburg, 1941).
In narrow basins of simple form in which the influence of the earth's rotation is small,
the motions resemble standing waves. Ideally, such waves are characterized by the
simultaneous rise and fall of level on either side of a nodal line at which no change in
elevation takes place. The elevation at high water increases with distance from the
nodal line and slack water coincides with high and low water.
The properties of tides due to standing waves may be deduced by assuming the
motion to result from a primary progressive wave moving up the channel which undergoes
complete reflection at a barrier. Mathematically, this situation may be treated as the
interference of two identical progressive waves moving in opposite directions and so
related that both waves are in phase at the barrier. This treatment of standing waves
assumes the presence of total reflection, the absence of damping and the absence of
effects of the earth's rotation. Since these conditions are not realized in natural tidal
basins, the standing wave concept leads to oversimplification.
In coastal embayments the most striking departure from the expectations of the
standing wave concept is the discrepancy between times of high water and slack water,
which may be great near the mouths of the larger bays and sounds. High water does not
occur simultaneously within such enclosures but is earlier near the sea. Commonly, the
nodal line is represented merely by a region in which the tidal range is small. These are
effects which can be explained if damping of the primary and reflected waves by frictional
or other effects is taken into account.
According to these concepts, the problem of tidal behavior in embayments is to
determine numerically the properties of the primary and reflected waves so as to account
for the observed relations of amplitude and stream velocity of the actual tide and to
correlate these numerical properties with the geographical form of the embayment.
In the present paper an attempt is made to treat the tidal behavior in such a way that
the observed changes in elevation and motion of the water along the path of the wave
may be used to determine the distribution of phase of the primary and reflected waves
along the channel and to measure the damping.
The relations between the several aspects of a wave as it advances along a channel
of uniform depth and width have been developed theoretically so as to show the times
of high water and slack water, the range of the tide, and the phase relations of the
primary and reflected waves along the channel for any degree of damping. By expressing
the relationship of the several aspects of a reflected wave in a form in which the wave
period is taken as the unit of time and distance is given in terms of the related phase
changes, it is possible to eliminate the purely geographical dimensions and to obtain a
wholly general description of the tide which may be used to indicate how any given
channel distorts the behavior of the wave as it advances.
In the case of irregular channels, in order to justify the application of relations
deduced for uniform channels, in which the change in phase of the primary and reflected
waves and their damping is proportional to the distance traveled and in which the
velocity of the waves is constant, it is necessary to make the following assumptions:
1. That the effect of irregularities in cross section is to alter the velocity of the
primary and reflected waves; i.e., to distort the geographical distribution of phase
2.That damping is proportional to the phase change in the waves rather than to
the distance traveled.
3. That the damping coeffcient, as defined, is constant along the length of the