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6 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS may be seen in figure 2A. if the procedure is carried out far enough downstream, in incre- ments, this step-by-step computed profile asymptotically approaches the normal-depth line. Local effects on profiles The importance of the effects of local channel nonuniformities on the computed profiles is a relative matter. In the first phase of computa- tions along an M 1 or an M2 curve, the profile usually has more slope and the effect of a local disturbance will show in the profile at that point. but it will probably diminish within a short distance upstream. For example. a bridge in the channel through which an Ml or an M2 curve is being computed will cause a localjump in computed elevations as in figure 9. An Ml curve will continue along another, higher Ml curve. An M2 curve could jump up to an Ml curve. or it could continue along another, higher M2 curve. The effect of the bridge is quickly dissipated. If. however. the channel bed has a very small slope or if the bridge is located in the reach where backwater curves have already con- verged and for which the expected water sur- faces are being computed. the local effects will be reflected farther upstream. In such instan- ces, the effects of several bridges in a series would be additive. The resultant Ml curve could require a long distance before converging with the average normal-depth line for the channel. All Ml curves where the computed profile or the channel bed have very little slope should be examined and interpreted with care. Convergence of backwater curves There are several factors to consider in the technique of using converging backwater curves to determine normal depth, The work "converge" is not the proper word to use in describing the relation of a backwater curve to the normal-depth profile. Backwater curves approach the normal-depth profile asymptoti- cally, and the relation between two Ml curves or two M2 curves is an asymptotic convergence. During computation for two adjacent profiles. results may be identical or within an allowable difference or tolerance. Then the profiles are considered to have converged for all practical purposes, If an Ml and an M2 curve are used as a pair and if the two profiles converge. normal depth is assured. It is not always feasible. however, to work with this ideal pair. An Ml curve lies above the normal-depth line (see figure lA). Cross-sectional properties, therefore. must be available for elevations higher than normal depth at all of the cross sections in the reach. To calculate such properties is frequently im- possible. particularly for large discharges where normal depths may be near bankfull stage; there simply may not be any ground points available above bankfull stages to which the cross sections could be vertically extended. This problem is more acute. of course. for the most downstream cross sections. w here the M 1 curve would be at its highest elevations with reference to the normal-depth line, Another characteristic of the Ml curve is that it requires a longer reach to converge with the normal-depth line than does an M2 curve. both having started an equal vertical distance from the normal depth at the starting cross section downstream. A longer reach of channel must, therefore. be surveyed whenever an Ml curve is used and an even longer one if a pair of them are used. The use of two M2 curves to determine nor- mal depth is just as effective. but there are a few precautions to consider. The starting ele- vations preferably should be a foot or more apart. They should not be near the channel bed because the length of reach required before the backwater curve converges with the nor- mal-depth line will be longer. The starting elevation of an M2 curve to be used for conver- gence purposes should never be taken below the critical-depth elevation. The starting ele- vation for computing an M2 profile upstream from a control would. on the other hand. have to start at critical depth. The difference in this instance is that the computations are defining the actual water surface in the reach. rather than a profile to be used to converge with the normal-depth profile. Convergence of any two or more Ml curves or any two or more M2 curves is no guarantee that normal depth has been reached. Although they may seem to converge mathematically. e e e