354 REPORT — 1844. 



The time occupied by the largest class of wave is o'5 seconds, and the cor- 

 responding mean velocity is 3*09 feet per second ; this is the velocity due to 

 a depth of 3*6 inches, but the depth total at the one end of the channel is nearly 

 double this quantity, diminishing to at the end. The time in which the wave 

 in a shelving channel passes along the whole length, is therefore nearly equal 

 to the time in which a wave would travel the same distance if the channel 

 were uniformly of a depth equal to the mean depth of the channel, reckoning 

 in both cases from the top of the wave. In these cases the height of the wave 

 is large. Let us take a small height of wave as Ex. XIV. ; there we have 

 also in this case the mean depth reckoned from the top of the wave =2*2, the 

 velocity in a channel of that uniform depth =2*4, and the time 7^*08. These 

 experiments are sufficiently accurately represented if we take for the velocity 

 of the wave in the sloping channel that of a wave in a channel having a uni- 

 form depth equal to the mean depth of the channel, reckoned as usual from 

 the top of the wave. 



If therefore we are to calculate the time in which a wave will traverse a 

 given distance q, to the limit of the standing water-line, after it has begun to 

 break on a sloping beach, we have, the height at breaking being h = the 

 standing depth of the water at the breaking-point. 





-^^^and.= ^^(,+,). 



Ex. A wave 3 feet high bi-eaking in water 3 feet deep, on a sloping shore 



at a distance of 60 feet from the edge of the water, would traverse that space 



in about 6 seconds, for 



/_ 60 60 , , 



* r^=7=;r^ = 6 seconds nearly. 



32-3 ^/ 9-82 ' 



By repeated observations I have ascertained that waves break whenever 

 their height above the level of repose becomes equal very nearly to the depth 

 of the water. 



The gradual retardation of the velocity of waves breaking on a sloping 

 beach, as they come into shallower water, is rendered manifest in the closer 

 approximation of the waves to each other as they come near the margin of 

 the water. Vide et seq. 



It may be observed also that the hdght of the wave does increase, but very 

 slowly (before breaking), as the depth diminishes ; thus in VII., a height of 

 1'8 in a depth of 4- inches becomes 2'2 in 2 inches depth, and in XII. a height 

 of 1 inch in a depth of 4 inches becomes a depth of 1*2 inch only 1*2 inch 

 high. The increase of height is therefore very much slower than the inverse 

 ratio of the depth, or than the inverse ratio of the square of the depth. 



Form of Transverse Section of Channel. — We have seen that in a given 

 rectangular channel, the volume of the wave, its height and the depth being 

 given, no peculiarity of origin or other condition sensibly affects its actual 

 phcenomena. But it becomes of importance to know whether the form of a 

 given channel, its volume being given, will affect the phaenomena of thcAvave 

 of the first order; for example, whether in a channel which is semicircular 

 on the bottom, or triangular, but holding a given quantity of water, the wave 

 would be affected by the form of the channel, the volume or cross section 

 remaining unchanged. 



Considering this question a priori, we might form various anticipations. 

 We might expect in a channel in which the depth of transverse section varies, 

 that as its depth is greatest at one point, suppose the middle, and less at the 

 sides, the wave might move with the velocity due to the middle or greatest 

 depth ; or we might expect that it would move with the velocity simply due 



I 



