ON RECENT RESEARCHES IN HYDRODYNAMICS. ll 
in the remainder, and vice versd. The height of high water is different in 
different parts of the canal: it increases from the mouth of the canal to its 
_ extremity, provided the canal’s length does not exceed a certain quantity. If 
four times the length of the canal be any odd multiple of the length of a 
free wave whose period is equal to that of the tide, the denominator of the 
expression for the tidal elevation vanishes. Of course friction would pre- 
_ yent the elevation from increasing beyond acertain amount, but still the tidal 
oscillation would in such cases be very large. 
When the channel up which the tide is propagated decreases in breadth 
or depth, or in both, the height of the tide increases in ascending the channel. 
This accounts for the great height of the tides observed at the head of the 
Bristol Channel, and in such places. In some of these cases however the 
great height may be partly due to the cause mentioned at the end of the last 
paragraph. 
When the tide-wave is propagated up a broad channel, which becomes 
shallow towards the sides, the motion of the water in the centre will be of 
the same nature as the motion in a free canal, so that the water will be flow- 
ing up the channel with its greatest velocity at the time of high water. 
Towards the coasts however there will be a considerable flow of water to 
and from the shore; and as far as regards this motion, the shore will have 
nearly the same effect as a barrier in a canal, and the oscillation will be of 
the nature of a stationary wave, so that the water will be at rest when it is 
_ at its greatest height. If, now, we consider a point at some distance from 
_ the shore, but still not near the middle of the channel, the velocity of the 
water up and down the channel will be connected with its height in the same 
way as in the case of a progressive wave, while the velocity to and from the 
shore will be connected with the height of the water in the same way as in a 
stationary wave. Combining these considerations, Mr. Airy is enabled to 
explain the apparent rotation of the water in such localities, which arises 
_ from an actual rotation in the direction of its motion*. 
When the motion of the water is in two dimensions the mathematical cal- 
culation of the tidal oscillations is tolerably simple, at least when the depth 
of the water is uniform. But in the case of nature the motion is in three 
_ dimensions, for the water is distributed over the surface of the earth in broad 
_ sheets, the boundaries of which are altogether irregular. On this account a 
_ .eomplete theory of the tides appears hopeless, even in the case in which the 
_ depth is supposed uniform. Laplace’s theory, in which the whole earth is 
_ supposed to be covered with water, the depth of which follows a very pecu- 
liar law, gives us no idea of the effect of the limitation of the ocean by conti- 
_ nents. Mr. Airy consequently investigates the motion of the water on the 
_ supposition of its being confined to narrow canals of uniform depth, which 
in the calculation are supposed circular. The case in which the canal forms 
_ a great circle is especially considered. This method enables us in some de- 
gree to estimate the effect of the boundaries of the sea; and it has the great 
__ advantage of leading to calculations which can be worked out. There can 
be no doubt, too, that the conclusions arrived at will apply, as to their general 
_ nature, to the actual case of the earth. 
_ With a view to this application of the theory, Mr. Airy calculates the 
_ Motion of the water in a canal when it is under the action of a disturbing 
_ force, which is a periodic function of the time. The disturbing force at a 
point whose abscissa, measured along tle canal from a fixed point, is 2, is 
_ supposed to be expressed by a function of the form A sin (nt—ma+a). 
_ This supposition is sufficiently general for the case of the tides, provided the 
* Art. 360, &c. 
. 
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