ON WAVES, 369 



One observation wliich I have made is curious. It is, that in the case of 

 oscillating waves of the second order, I have found that the motion of pro- 

 pagation of the whole group is different from the apparent motion of wave 

 transmission along the surface ; that in the group whose velocity of oscilla- 

 tion is as observed 3*57 feet per second, each wave having a seeming velocity 

 of 3*57, the whole group moves forward in the direction of transmission with 

 a much slower velocity. The consequence of this is a difficulty in observing 

 these waves (especially such as are raised by the wind at sea), namely, that 

 as the eye follows the crest of the wave, this crest appears to run out of 

 sight, and is lost in the small waves in which the group terminates. The 

 termination of these groups in a series of waves becoming gradually smaller 

 and smaller, yet all continuous with the large wave, is curious and leads to a 

 curious conclusion. It is plain that if these large waves are moving with the 

 same velocity as the small ones, this result would be inconsistent vvith the 

 other experiments. But if we conceive each to be transmitted with the 

 velocity due to its breadth, we shall have the velocity of oscillation varying 

 from point to point in the same group of waves, but it will be impossible 

 always to measure this velocity directly as it may be continually changing. 

 There is to be observed, therefore, this distinction in a group of waves of the 

 second order, between the velocity of individual wave transmission and the 

 velocity of aggregate wave propagation. 



I have not I'ound it possible to measure this velocity of aggregate propa- 

 gation of a group of waves, from want of a point to observe. If I fix my eye 

 upon a single wave, ^ follow it along the group, and it gradually diminishes 

 and then disappears ; I take another and follow it, and it also disappears. 

 My eye, in following a wave crest, follows the visible velocity of transmission 

 merely. After one or two such observations, I find that the whole group of 



motion of transference of the particles. In siiort, tiiey become moving waves of tlie third order, 

 the common waves of the sea. 



From M. Gerstner's investigations we obtain the following results, for oscillating waves which 

 correspond to our second order : — 



1 . Waves of the same amplitude are described in equal times independently of their height. 

 (This corresponds vvith the results of our experiments.) 



2. Waves are transmitted with velocities which vary as the square roots of their amplitudes. 



3. The waves on the surface are of the cycloidal form, always elongated, never compressed; 

 the common cycloid being the limit between the possible and impossible, the continuous and 

 the broken wave. j». 



4. The particle paths in the standing waves of running water are cycloids, which on the sur- 

 face are identical with the wave form, and below the surface have the same character with the 

 wave lines of the surface, the height of the waves only diminishing with the increase of depth. 



5. The particle paths of moving waves in standing water are circles corresponding to the 

 circles of height of the cycloidal paths ; the diameters of these circles of vertical oscillation di- 

 minish in depth as follows. Let 0, Uy^'i u, 3 ii, &c. be depths increasing in arithmetical pro- 



u 2 m _ 3 « 



gression, then i, i 6 ",6s a'bt ", which decrease in geometrical proportion, are the 

 ratio of the diminishing diameters of vertical o^c)llation. Thus, if 0, ^ a, f a, f a, &c. be depths, 

 a, 0-6065 a, 0-3679 a, 0-2231 a, 0-1353 a, are the ranges. 



6. The forms of these paths and the circles of oscillation are shown in Plate X. fig. 1, which 

 has been drawn with geometrical accuracy from the data of M. Gerstner's theory, and it is at 

 the same time the most accurate representative I am able to give of my observations on the 

 wave of the second order. 



7. The period of wave oscillation is t = a--\^/ ^ " . 



8. The velocity of wave propagation is v = ^/2ag, a being the radius of the wave cycloid 

 generating circle. ^ 



9. It follows that the length of a pendulum isochronous with the wave is less than the wave 

 length in the ratio of the diameter of a circle to its semi -circumference. Newton made these 

 equal. These last three rtsults are inconsistent with my observations on transmissior;. 



1844. 2 B 



