ON RECENT RESEARCHES IN HYDRODYNAMICS. 5 
_ February 1839*, Mr. Green has given the theory of the motion of long waves 
in a triangular canal with one side vertical. Mr. Green found the velocity of 
__ propagation to be the same as that in a rectangular canal of half the depth. 
_ Ina memoir read before the Royal Society of Edinburgh in April 1839f, 
Prof. Kelland has considered the case of a uniform canal whose section is of 
any form. He finds that the velocity of propagation is given by the very 
% 
simple formula a/ Has where A is the area of a section of the canal, and 
b the breadth of the fluid at the surface. This formula agrees with the ex- 
periments of Mr. Russell, and includes as a particular case the formula of 
Mr. Green for a triangular canal. 
Mr. Airy, the Astronomer Royal, in his excellent treatise on Tides and 
Waves, has considered the case of a variable caual with more generality than 
Mr. Green, inasmuch as he has supposed the section to be of any form}. If 
A, 6 denote the same things as in the last paragraph, only that now they are 
supposed to vary slowly in passing along the canal, the coefficient of horizontal 
8.1 ‘ eae, eee 
displacement o¢ A~ * 5*, and that of the vertical displacement OC A” * 67%, 
while the velocity of propagation at any point of the canal is that given by 
the formula of the preceding paragraph. Mr. Airy has proved the latter 
formula § in a more simple manner than Prof. Kelland, and has pointed out 
the restrictions under which it is true. Other results of Mr. Airy’s will be 
more conveniently considered in connection with the tides. 
Theory of Oscillatory Waves.—When the surface of water is covered with 
an irregular series of waves of different sizes, the longer waves will be con- 
tinually overtaking the shorter, and the motion will be very complicated, and 
_ will offer no regular laws. In order to obtain such laws we must take a 
simpler case: we may for instance propose to ourselves to investigate the 
motion of a series of waves which are propagated with a constant velocity, 
and without change of form, in a fluid of uniform depth, the motion being in 
two dimensions and periodical. A series of waves of this sort may be taken 
as the type of oscillatory waves in general, or at least of those for which the 
motion is in two dimensions: to whatever extent a series of waves propagated 
in fluid of a uniform depth deviates from this standard form, to the sume ex- 
tent they fail in the characters of uniform propagation and invariable form. 
The theory of these waves has long been known. In fact each element of 
the integrals by which MM. Poisson and Cauchy expressed the disturbance 
of the fluid denotes what is called by Mr. Airy a standing oscillation, and a 
_ progressive oscillation of the kind under consideration will result from the 
_ superposition of two of these standing oscillations properly combined. Or, 
if we merely replace products of sines and cosines under the integral signs 
by sums and differences, each element of the new integrals will denote a 
progressive oscillation of the standard kind. The theory of these waves how- 
ever well deserves a more detailed investigation. The most important formula 
connected with them is that which gives the relation between the velocity of 
__ propagation, the length of the waves, and the depth of the fluid. Ife be the 
__ velocity of propagation, A the length of the waves, measured from crest to 
on 
Risa 9 
rest, i the depth of the fluid, and m = =, then 
1 
1s 
es g ina p ET a 
a es — a Cpe tr ° . . . . . (B.) : 
& im ems | sm 
i * Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 
T Transactions of the Royal Society of Edinburgh, vol. xiv. pp. 524, 530. 
t Encyclopedia Metropolitana, article ‘ Tides and Waves.’ Art. 260 of the treatise. 
§ Art. 218, &c. 
