33 REPORT — 1869. 



the profile of a simple wave is trochoidal, and that the particles of water 

 move in circles in a vertical plane, at right angles to the ridges and valleys 

 of the waves. The consequences of this motion are briefly as follows, on the 

 assumption that the depth of water is unlimited. 



The diameter of the circle in which a surface-particle moves is the height 

 of the wave from hollow to crest. Particles which in still water would be at 

 a lower level, describe smaller circles in the same period. A horizontal plane 

 (in the still water) is thus converted into a wave-surface of the same period, 

 but of reduced amplitude of oscillation*. The velocity of the particles (and 

 on this depends the impact of a wave) is simply the circumference of one of 

 these circles divided by the periodic time. 



If we consider a column of particles which is vertical in still water, that 

 column oscillates in wave-water like corn-stalks in a gust of wind, and it 

 also oscillates vertically. But it always slopes towards the crest of the wave, 

 and the obliquity thus induced goes to enhance that due to the wave-slope ; 

 so that if we regard the profile of a wave, a small portion of water, rectan- 

 gular when still, undergoes a double deformation, the horizontal surfaces 

 following the wave-slope, and the vertical surfaces being deflected towards 

 the crest, both causes tending to increase the angular deformation instead 

 of to preserve rectaugularity. 



The crest of the wave being sharper than the hollow, and the quantity of 

 water invariable, the horizontal plane which lies halfway between valley and 

 crest is higher than the mean, or still- water, level ; and its elevation has 

 been shown to be equal to the height due to the velocity of revolution of Ihe 

 particles. 



Considered as trochoids, the wave-profiles are traced by a point within a 

 cii'cle rolling under a horizontal line. The line midway between valley and 

 crest is the line of centres. 



The particles of water above the line of centres are moving forwards, as 

 regards the direction of advance of the wave; those below that line back- 

 wards. The particles in the front face of the wave are rising, and those in 

 the rear-face falling. 



The wave whose period is -th of a second has a length of \ = ^ — r, 



whence we find the number of waves to a second to be ?i = a / — ^-_. The 



V 27r\ 



velocity of wave-propagation, that is to say, of the apparent advance of the 

 wave in a deep sea, is n \ = a / ^ =t~—. In other words, the speed 



of the wave-crest varies as the periodic time, and the length of the wave 

 varies as their product, or as the square of either. 



The vertical disturbance of a particle whose depth in still water would 

 be h is 



h being the height of the surface-wave. 



* Drawings of the structure of a trochoidal wave will be found in the British Asso- 

 ciation Eeports for 1844, plate 56 ; Trans. I. N. A. vol. i. for 1860, plate 7, vol. iii for 

 1862, plate 3, vol. iv. for 1863, plate 10, and vol. vi. for 1865, plate 10 ; 'Shipbuilding : Theo- 

 retical and Practical,' Eankine, p. 69; Scott Eussell, 'Naval Architecture,' plate 117. 



