But the cotidal lines are given approximately by the surface contour lines about the time of the mean level, and the co-range lines are given approximately by the surface contour lines about the time of high water. If high water occurs at maximum or at minimum current, it follows that the cotidal lines will have the direction of the current at the time of mean level, and that the co-range lines will have the direction of the current at high and low water. If also the maximum or minimum current is parallel to a coast-line, one of the two sets of cotidal and co-range lines will have the coastline for a member, while the other will have its mem bers striking the coastline at right angles. Now those lines which are per pendicular to the coastline will converge towards a cape and diverge in a bay. An example is afforded by the northern part of the Irish Sea. Here the minimum currents are practically zero and high water occurs at slack water, so that the cotidal lines are along the current lines and the co range lines are perpendicular to the current-lines.
Next examine the case in which the current at a place is non-rotatory, so that, with appropriate choice of axes and time-origin, we may take lc= U cosnt, v = o. The dynamical equations give— point and the co-range lines are similar and similarly situated ellipses with this point as centre. Such a point is called an amphidromic point, and fig. 2 provides two examples in the North Sea.
The frictional resistance of the sea floor will be directly opposite to the current and its M2 constituent will have a magnitude approximately proportional to the square of the speed of the M2 current. When the cur rent at any place reaches a maximum, so that the acceleration in this direction is zero, the frictional forces must be balanced by a surface-gra dient sloping downwards in the direction of the current. For a given cur rent the magnitude of the slope will be inversely proportional to the depth of the sea, so that the effects of friction will be greatest in shallow water. Suppose that the conditions are such that in the.absence of friction the maximum current and zero elevation would be simultaneous, as occurs in the broadest part of the Irish Sea. Then, owing to friction, there will be at the time of maximum current a downward gradient in the direction of the current. It follows that mean level will occur progressively later in the direction of the current, and therefore that there will be a progression of high water in this direction.
Next suppose that the depth increases away from the shore and that the currents are parallel to the shore. Also, suppose that the surface-gra dients parallel to the shore are zero at the same time T along a line per pendicular to the shore, as occurs in the North Sea between the Forth and the Humber. Then in the absence of friction the currents would reach
their maximum at the time T, but owing to friction they must have a backward acceleration at this time, and this acceleration will increase to wards the shore. It follows that the currents will reach their maximum somewhat before the time T, and that the effect will increase towards the shore. We thus see that the inshore currents will turn earlier than those in the offing.
Dynamical Explanation of Actual Tides.— (12) . Now consider a sea or ocean, and imagine definite limiting vertical sections transverse to the channels which connect it with other seas or oceans. Then the tides of this body of water are determined by the equations formulated above, when either the elevations or the normal currents are known functions of time over the limiting sections. In explaining the tides in any particular basin it is therefore necessary to assume, either from observation or from other theory, the tidal conditions over the limiting sections. The prob lem of determination is a purely mathematical one, but hitherto it has only been solved for those seas which are of elongated shape and fairly narrow. The simplifying principle concerning the tides of such basins is that the transverse components of current are usually negligible, so that the actual currents are non-rotatory and parallel to the general direction of the basin.
We will now illustrate this "narrow sea theory" by giving a general explanation of the M2 constituent in the English Channel. As the direct action of the generating forces may be neglected, the tides of the Channel are maintained by those of the Atlantic and North Sea. In the first place we shall leave out of account both the dynamical effects of the earth's rotation and the forces of friction. Imagine a fixed material barrier e placed directly across the Strait of Dover so as to cut off the currents in that strait. Under these conditions the tides of the Channel would con sist of a longitudinal standing oscillation (a see-saw motion) maintained by the tides of the Atlantic and illustrated in fig. 3, 8 hour and 11 hour. There would be a nodal line m from the neighbourhood of Bournemouth to that of Cherbourg along which there was never any tidal elevation. Along a line w from Cornwall to Brittany there would be no currents, and we shall take the vertical section through this line to form the western limit of the Channel. The positions of the lines m and w are determined by the distribution of depth in relation to the period. At 8 o'clock in lunar time the surface of the water would be everywhere level, with cur.