10 REPORT—1846, 
give an idea of the nature of the calculations and methods of explanation 
employed, and to mention some of the principal results. 
On account of the great length of the tide wave, the horizontal motion of 
the water will be sensibly the same from top to bottom. This circumstance 
most materially simplifies the calculation. The partial differential equation 
for the motion of long waves, when the motion is very small, is in the simplest 
case the same as that which occurs in the theory of the rectilinear propaga- 
tion of sound; and in Mr. Airy’s investigations the arbitrary functions which 
occur in its integral are determined by the conditions to be satisfied at the 
ends of the canal in which the waves are propagated, in a manner similar to 
that in which the arbitrary functions are determined in the case of a tube in 
which sound is propagated. When the motion is not very small, the partial 
differential equation of wave motion may be integrated by successive ap- 
proximations, the arbitrary functions being determined at each order of ap- 
proximation as before. 
To proceed to some of the results. The simplest conceivable case of a 
tidal river is that in which the river is regarded as a uniform, indefinite canal, 
without any current. The height of the water at the mouth of the canal will 
be expressed, as in the open sea, by a periodic function of the time, of the 
form asin (mt+a). The result of a first approximation of course is that 
the disturbance at the mouth of the canal will be propagated uniformly up 
it, with the velocity due to half the depth of the water. But on proceeding to 
a second approximation*, Mr. Airy finds that the form of the wave will alter 
as it proceeds up the river. Its front will become shorter and steeper, and 
its rear longer and more gently sloping. When the wave has advanced suf- 
ficiently far up the river, its surface will become horizontal at one point in 
the rear, and further on the wave will divide into two. At the mouth of the 
river the greatest velocities of the ebb and flow of the tide are equal, and 
occur at low and high water respectively ; the time during which the water 
is rising is also equal to the time during which it is falling. But at a station 
up the river the velocity of the ebb-stream is greater than that of the flow- 
stream, and the rise of the water occupies less time than its fall. If the sta- 
tion considered is sufficiently distant from the mouth of the river, and the 
tide sufficiently large, the water after it has fallen some way will begin to 
rise again: there will in fact be a double rise and fall of the water at each 
tide. This explains the double tides observed in some tidal rivers. The 
velocity with which the phase of high water travels up the river is found to 
be V gk (1+3b), k being the depth of the water when in equilibrium, and 
bk the greatest elevation of the water at the mouth of the river above its 
mean level. The same formula will apply to the case of low water if we 
change the sign of 6. ‘This result is very important, since it shows that the 
. interval between the time of the moon’s passage over the meridian of the 
river station and the time of high water will be affected by the height of the 
tide. Mr. Airy also investigates the effect of the current in a tidal river. He 
finds that the difference between the times of the water's rising and falling 
is increased by the current. 
When the canal is stopped by a barrier the circumstances are altered. 
When the motion is supposed small, and the disturbing force of the sun and 
moon is neglected, it is found in this case that the tide-wave is a stationary 
waveT, so that there is high or low water at the same instant at every point 
of the canal; but if the length of the canal exceeds a certain quantity, it is 
high water in certain parts of the canal at the instant when it is low water 
* Art. 198, &c, 7 Art. 307. 
