86 REPORT— 1881. 



experiments. The paths' described in solitary waves by particles of the 

 fluid are parabolic arcs with axes vertical, constant horizontal chord 

 equal to the quotient of the volume of the wave by the primitive depth, 

 and height proportional to the initial height of the pai-ticle above the 

 bottom ; being, for a particle at the surface, equal to the height of the 

 wave. The latus rectum at the surface is four-thirds the primitive depth 

 of the fluid, and varies for other particles inversely as their height above 

 the bottom, and is independent of the height of the wave. Rayleigh^ 

 also gave, independently, a few years later, a theory of the solitary wave 

 agreeing with that of Boussinesq's. 



Rankine-^ has attempted to determine the velocity of propagation of 

 any possible kind of wave in a liquid of limited or unlimited depth 

 simply from the fact that the free surface is one of constant pressure. 

 But, in reality, he implicitly assumes that the waves are of a permanent 

 type. He proves that the velocity of propagation is equal to that 

 acquired by a body in falling through half the virtual depth. In this, the 

 velocity of the wave is defined as the mean of the velocities with which 

 the form advances relatively to particles in the crest and trough 

 respectively, and the virtual depth as follows : Suppose a stream flowing 

 Avith a velocity equal to the difference of the velocities of the particles at 

 the crest and trough ; then the depth of stream, in order that the amount 

 of horizontal disturbance should equal the whole amount of horizontal 

 disturbance in the actual fluid between two vertical planes, through the 

 crest and trough respectively, is the virtual depth. We have seen how 

 an analogous theorem can be deduced from the theory of dynamical 

 similarity. 



In all the foregoing, the waves have been regarded as following each 

 other in infinite series, or the whole extent of the fluid has been taken 

 into consideration. The theory would have to be modified, therefore, 

 when waves ai'e propagated into fluid at rest. This happens, for instance, 

 M'ith the trail of waves from the bow of a boat, or the group of waves 

 formed on the surface of water by pei-iodically disturbing it for a 

 short interval. It is then observed that the waves are not of the same 

 size, but that they advance in a group, of which the largest are in the 

 middle, and that the group itself progresses with only half the velocity 

 of the waves themselves. If a single wave be observed, it is seen to die 

 gradually out, while others are formed in its rear. The explanation of 

 this was given by Osborne Reynolds, at the meeting of this Association 

 at Plymouth, in 1877.'' The energy of a liquid trochoidal wave is half 

 kinetic and half potential, the latter of which is transmitted at the rate 

 of its amount in unit of length X the velocity of the wave. It is then easy 

 to see that, in a group of waves, which slightly decrease in size towards 

 the front, the form of wave is transmitted twice as fast as the energy, 

 and the velocity of the group is only one-half that of the waves composing 

 it. The propagation of groups of waves had been considered before 

 Reynolds, by Rayleigh,^ who treated them as compounded of two infinite 

 trains of waves, of equal amplitude and slightly diSerent wave-length ; 



' ' Addition au Memoire sur la Tlieorie,' &c. lAou. (2) xviii. p. 47. 

 = ' On Waves,' Phil. Ma^. (5) i. p. 262. 



' ' On Waves in Liquids,' Proc. Roy. Sue, xvi. p. 344 (1868). 

 * The paper is published in Nature, xvi. p. 343 : ' On the rate of progression ot 

 groups of waves, and the rate at which energy is transmitted by waves.' 

 ' Theory of So^ind, vol. i. p. 246. 



