34 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



The factors were then plotted in various 

 combinations on logarithmic section paper, and 

 certain approximate numerical relations were 

 thus discovered. The first critical point is 

 reached when d = 0.016 F m 2 ' 3 , or when d = 

 0.0045 i,' 85 . The second critical point is 

 reached when d = 0.004 F m 3 ' 3 , or when <Z = 

 0.0003 L, 1 ' 15 . The coefficients and exponents 

 are not well defined by the data, but the 

 general indications are (1) that each change in 

 phase occurs when the depth of water bears 

 a certain numerical relation to a power of the 

 mean velocity near the cube, and (2) that the 

 changes occur when the depth bears a certain 

 numerical relation to the amount of load 

 carried in each unit of width of current. 



At the bottom the stream is limited and 

 restricted by the bed of debris; at the top by 

 the water surface. To the space between these 

 bounds, a space measured by the depth, the 

 eddies or convolutions of the current are con- 

 fined. Within the range of conditions covered 

 by the experiments the normal mode of flow 

 involves sinuosity of the filaments of current, 

 and the tendency toward diversity of internal 

 movement is strong in proportion as the veloc- 

 ity is high. A particular relation between 

 depth and velocity corresponds to a sort of 

 equilibrium between the factors of turbulence 

 and restraint, in accordance with which the 

 sinuosity of the lines of flow is reduced to a 

 minimum and the water surface and channel 

 bed are approximately plane. This gives the 

 smooth phase of traction. When the depth is 

 increased without increase of velocity, the 

 reduction of restraint permits the develop- 

 ment of internal diversity, and this carries with 

 it diversity of the plastic bed, giving the 

 dune phase of traction. When the velocity 

 is increased without increase of depth, the 

 restraint is overpowered, and a diversified but 

 systematic arrangement of flow lines develops, 

 which carries with it systematic diversity of 

 both water surface and channel bed and gives 

 the antidune phase. 



There will be occasion to speak of these 

 relations in another connection in Chapter 

 XIV. 



It may be noted, as a possible contribution 

 toward an explanatory analysis, that gravity 

 opposes the current on the upstream side of 

 the antidune (fig. 11) and assists the current on 



the downstream side. It is where gravity ac- 

 celerates that load is increased by erosion, and 

 where gravity retards that load is reduced by 

 deposition. In the case of the dune, however, 

 erosion occurs where gravity is a retarding 

 force. But here the descending current, which 

 is accelerated by gravity and which is observed 

 to have gained speed, is free from the bed and 

 bears no load (fig. 10). In order to transport 

 when it resumes contact with the bed it must 

 take debris from the bed, and by so taking it 



erodes. 



UNITS. 



The system of units to which the laboratory 

 data have been reduced and which will bo em- 

 ployed in their discussion is hybrid in that it 

 includes the foot and the gram. The foot is 

 made the fundamental unit for length, area, 

 and volume because it is the unit employed gen- 

 erally by English-speaking engineers. The 

 gram is made the unit of mass, primarily be- 

 cause it is of convenient magnitude, but also 

 because of the manifest advantage of intro- 

 ducing the metric system wherever no practical 

 difficulties interfere. It happens that the two 

 measures which are given in grams are of cate- 

 gories unfamiliar alike to the engineer and the 

 general reader, so that the gram unit encounters 

 no conflicting habit of thought. One measure 

 is the mass of a grain of sand or a pebble, the 

 other the mass of the debris carried by a stream 

 hi a second. 



The unit of tune, for the indication of rates, 

 is one second. Velocity is given hi distance per 

 second, ft./sec. ; discharge in volume per second, 

 f t. 3 /sec. ; and load in mass per second, gm./sec. 



In hydraulic and hydrodynamic treatises 

 slope of streams is measured by the quotient of 

 fall by distance, or the tangent of its angle, and 

 the unit slope is taken as 45. For practical 

 purposes this unit is inconvenient because it 

 transcends experience, and engineers commonly 

 avoid it by speaking of slope in percentage or 

 in fractions of 1 per cent, thus making 1 per 

 cent the actual unit. For most purposes I 

 find the smaller unit most convenient, but I 

 have occasional use for the larger unit and shall 

 accordingly use both. To avoid confusion the 

 symbol 8 will be used with the smaller unit, 

 and where discrimination is important it will 

 be called per cent slope, while the symbol s will 

 'be used with the larger unit. S 100 s. 



