anything out, was too long, though correct to the nearest quarter of a day, which was as close a determination of the interval as was then attempted. Inspection of these plates can leave no doubt that flood movement in the lower Mississippi River up to the bank full stage follows this very simple law. Every point in any flood wave moves down at the same rate, being found with its same value of discharge at all gauges below after given constant intervals of time; excepting, of course, that these values of discharge may be increased from point to point by tributary increments. Or to state this briefly, we may say, that flood waves in the main river are in a permanent phase. It should be stated, however, that there are some instances which make it still a question whether the extreme front of a rise, as well as a small hollow in a double rise, are always in a permanent phase; an instance of the double rise is seen on Plate II, Group 1, early in November, 1888. These are points where we know that changes of plane generally occur, and here also interpolations between daily gauge readings are especially subject to error, which with tributary increments may serve to account for the conditions sometimes found. But for the present these must be made possible exceptions to the former general law. In stating the above as a law of flood movement it may be well to call attention to the fact that while the true interval is determined to a certain degree of precision, and is, therefore, subject to a probable error, the determination of this as the law of movement stands on another footing. Variations from it are altogether imperceptible, and this, though we may test it at times through seven gauges from Cairo to Helena, covering 306.5 miles of river, we must, therefore, accept it as a positive law until some clear evidence of divergence from it is found. The principal influences that at first would cause us not to suspect so simple a law of flood movement have been already mentioned. It is well, however, to consider them in more detail, with the effects they would have on the gauge relations if operative. First, the existence of reservoir capacity in the channel. This has never been considered more than a possibility below the bank full stage, though very apparent above it. Its effect would be to decrease volume and stage on the rising river and increase them on the falling, each in proportion to the rapidity of the change of stage and the distance between gauges. The interval for a slow rise or fall would thus appear too short for a fast rise or fall, and this discrepancy would increase with distance. Second, the plus or minus changes of normal slope caused by the rapidly rising or falling river, causing the velocities and movement of the front of the wave to be accelerated and of the rear to be retarded. This change is by no means insignificant in amount. There are times when the slope between gauges may have a difference of some 15 per cent from this cause, or by the slope formula a difference in the velocities of about 8 per cent. Also the differences in observed mean velocities between the rising and falling stages is a well-known feature of discharge observations. Such an influence, it would be thought, should perceptibly lengthen and change the shape of the wave, while causing both the rapidly rising and the rapidly falling portions to diverge toward larger lower-gauge values than the normal; the rising stage, because the travel is faster than the assumed interval, and hence the equivalent taken is greater; the falling, because the travel is slower and the equivalent taken has not fallen sufficiently; or all rapid changes of stage would indicate an excess of lower-gauge equivalent when compared with slower or stationary periods, which excess would increase with the distance between gauges. But the cause which more than all others would at first lead us not to expect so simple a law of flood movement is the fact that the mean velocity of the river increases as the river rises. This is a well-known feature of all discharge stations. An instance of it is shown on Plate VI, where the mean velocities of the Fulton observations in 1880 are plotted to their respective gauge readings. This, however, only gives the range of mean velocity in a selected section, and it is apparent that in some of the wide sections the range must be much smaller and may in some instances even be negative. To value the range of these aggregate mean velocities it is necessary V to determine the mean velocity of a long reach at different stages, or This for a R stationary river is simply discharge divided by mean water area of the reach; or V QL where Q equals the discharge, L the length, R = S and Vol. the water volume of the reach. The 1879 survey of the Plum Point Reach offers a good opportunity for determinV ing this range in the value of This reach has a length of 38.7 miles, commencing R' from about 6.6 miles above Ashport and extending down to about 2.7 miles below Randolph. In this survey two periods of about a stationary river were selected, the first, March 28-31, 1879, with a stage of 25.4 on the Fulton gauge; the second, June 15-22, 1879, with a stage of 12.2. In or close to each of these periods the sections had been sounded, and the series of discharge observations taken above at Columbus and below at Memphis showed that the discharge during these periods was uniform along the whole reach and gave accurately its value for each period. From these data the mean velocity of the reach of the two stages was determined, and their values are shown plotted to their Fulton gauge heights on Plate vi. The straight line through these points is taken as giving the total range in the mean velocity of the Plum Point Reach, since on the Lower Mississippi, in general, the variation of mean velocity with gauge height has been found to be satisfactorily expressed by a straight line. The high and low water areas for each section in this Plum Point Reach, with their corresponding mean velocities are also shown on Plate VI as longitudinal variations. The range in mean velocity of the reach is seen to be much less than that of the Fulton section, as was expected; but it still has a considerable range from about 3.1 feet per second at low water to about 5.1 at the bank-full stage, and from the nature of the reach it is probable that this is about as small a range as would be found in any long reach on the Lower Mississippi. With such a range it is perfectly apparent that, if each point of the flood was moving down with the mean velocity of its stage, any such relation between gauge readings as is found would be impossible; small floods at low water would have an interval nearly twice as long as at high water, and, omitting the changes of shape, which would be large, the best trial interval that could be obtained for floods of any considerable range would be one showing an interval too short at the low stages and too long at the high stages. We are brought therefore to the fact that these floods, at least from Cairo down, have a travel that, whether made up of complex properties or no, actually corresponds to a plane-wave movement. Whether this is a true wave movement, that is, does pressure acting through the mass cause a travel by transmission; or whether the phenomena may be accomplished by simple flow or translation alone, requires further consideration. To define these questions more specifically we may say that, in translation all pressures acting through the mass are considered as expended simply in the acceleration of velocities of flow, or in a normal slope effect; while in transmission we consider that an element of the pressure may cause a vertical motion which changes the form of the free surface, as in the movement of a plane wave. It may be stated at the outset that from Baton Rouge down the travel is unquestionably one of transmission, for not only does it clearly exceed the known bounds of possible translation, but it lies very fairly within the range of tide movements up an estuary, which are known to be cases of transmission. (See Coast Survey Report for 1881, page 469, for movement of tides up the Delaware River.) Also above Baton Rouge, as at low water the pools present much the same features that hold below, we might naturally infer that the travel across them at the lower stages is also a transmission; but before forming an opinion on this subject it will be necessary to consider briefly the limits of translation or the relation of the travel of discharge to the velocities of flow where no pressure or transmission is acting. Before a definite relation of the travel of discharge with velocities of flow can be reached it is necessary to make a definite assumption in regard to the conditions of flow; and as one extreme we will first assume that the individual masses of water retain permanently their same relative velocities in the cross sections; that is, the mass that has maximum velocity in one section when it reaches any section below still has the maximum velocity of that section, and so on through all the range of velocities to the minimum. This is called "perfect flow," and it is seen that this a condition in which the actual particles of water, forming the discharge at one place, are separated to their greatest possible extent in their movement down the river, and it may be shown that in this movement down the discharge of a changing stage is constantly tending towards a movement with the mean of the maximum velocities of the river. As the opposite of this condition of flow we have the condition of "perfect circulation," or where the individual masses of water pass in turn through all the velocities of the river in such a way that, in the aggregate, they are only temporarily separated during the period of a cycle. In this as the actual particles forming a discharge in one place are, as a whole, held together in their motion down the river, it is apparent that the travel of discharge is with the combined mean velocities; or with the mean velocity of the R reach. Between these two extremes must be found all actual conditions of flow in the river, and the travel of discharge consequently may vary from mean towards maxi mum velocity as we pass from "perfect circulation" to the condition of perfect flow. It should be remembered, however, that from all observation, as well as from theoretical conceptions, we know that perfect flow of water is physically impossible, except, perhaps, for very slow motions in smooth pipes. On the other hand, besides the knowledge that circulation is a prominent feature of river flow, we have in the study of sediment between St. Charles on the Missouri, and Columbus, Ky., on the Lower Mississippi, the suggestion that it is at least close to perfect circulation. (See Report Chief of Engineers, 1887, page 3092, which is found the only data collected that bears directly on this question.) So that from the above we may only conclude that possibly the travel of discharge may have some variation from mean toward maximum velocity. If this were accepted and further assumption made that circulation was far from complete at the low stages and approached completion at the higher stages in such a way as to give the uniform rate of travel found, the actual movement of floods would then be explained without the assumption of a partial transmission. On the other hand we have the unquestionable fact that transmission acts below Baton Rouge and the strong inference that it would act in the same way across the pools at low water to accelerate there the travel of discharge, which might also fairly cause the uniform rate of travel found. Between these two explanations we can not as yet say whether the one or the other or both combined cause the simple law found for flood movements, but it is thought the weight of evidence is decidedly in favor of the latter, or that there is in part a transmission or true wave movement acting in the travel of these floods from Cairo down, which becomes altogether the controlling element below Baton Rouge, and this notwithstanding the fact that they are often so long that the lower end of the front phase is in the gulf while the crest is still falling as rain in the basins of the upper tributaries. Tabulation I gives the intervals and rates of travel of floods as determined by this study. The "full reaches" show the limiting distances through which floods were progressively studied on account of the interference of tributary increments. In using the printed gauge records it may be noted that the difference in time between the Benyaurd gauge readings at 8 a. m. and the Commission gauge readings, with a mean time of 12 m., must be allowed for the intervals prior to February 1, 1887. It is seen by the last column in the tabulation, or travel in miles per day, that besides the marked increase in the rate of travel below Red River Landing, before noted, there is also a decided irregularity in the different parts of the river. It must be remembered that this irregularity above Red River Landing may have been materially exaggerated in the short distances between gauges by the limits of accuracy to which the interval was determined and by possible differences in the actual time of reading the gauges. But allowing for this, there still remains no doubt that there are distinct differences in the rates of travel through different reaches down to Red River Landing. These differences in the travel have as yet not been connected with any physical cause, and beyond the supposition that they depend in some way on the relative lengths of pools and bars, possibly combined with slope and mean depth, there is no explanation for them. The curve of interval to distance from Cairo is shown plotted on Pl. VII, and presents more clearly to the eye these irregularities of travel. On this plate, following out the original assumption that the interval was determined to within 0.05 of a day, dotted lines have been drawn above and below the curve, diverging by this amount for each full reach. They represent on this assumption the extreme possible error of the total interval from Cairo down to any point. The possible error of interval between intermediate points is of course less, and can easily be made up by this allowance of 0.05 of a day for each full reach, or any part of a full reach; for while 0.05 is taken as the limiting error, say, from Cairo to Helena, it is also taken as the error from Cairo to New Madrid, or any other part of this Cairo-Helena reach, which was studied as a whole. The limiting gauges to the full reaches are marked with an asterisk on Pl. vII; also on the plate the curves of limiting error have not been extended below Red River Landing since, as before considered, the determinations of the intervals there may not be so exact. In the annual report of the Mississippi River Commission for 1890 there is a tabulation giving in long reaches the mean water area from Cairo down for the bank full stage. (See Report of Chief of Engineers, 1890, page 3129.) From this, by taking the discharge for this stage as 1,000,000 cubic feet per secV ond, we may determine approximately the mean velocity of these reaches, or for R the bank full stage from Cairo to Carrollton. This is given in Tabulation II, in comparison with the velocity of flood travel for the same reaches, deduced from the time intervals. *The reaches in the Commission's report. New Madrid to Plum Point and Helena to Arkansas River, are respectively taken as New Madrid to Fulton and Helena to Arkansas City to correspond with reaches for which the intervals have been determined. If the deduced mean velocities of the reaches are to be relied on, there seems to be no definite connection between this and the flood travel; and beyond the further evidence that the travel of floods below Baton Rouge is altogether out of the range of a velocity movement, there is little to be gained from this comparison. It should, however, be stated that while there are marked differences in the reaches, the mean values down to Red River Landing of the flood travel and of the mean velocity are very close together; the mean flood travel being 5.03 and the mean of the mean velocities at the bank-full stage being 4.96 per second. It is thought that this may be simply a coincidence, or it may point to a general tendency towards flood travel above Red River Landing as an average being developed, so as to have the general mean velocity of the bank-full stage. This might either mean perfect circulation at stages whose averages for the whole river correspond to the mean velocity of the bank-full stage, or might as well mean the disappearance of the transmission element of the flood's motion at these stiges. But whatever uncertainty there may be in the theoretical consideration of the causes of this law of flood movement, I think we may safely say that there is in hy draulics no law which more accurately expresses so broad a generalization of observed facts and whose field of application gives so great a promise in the study of the flow of water in natural channels. Very respectfully, your obedient servant, Lient. Col. CHAS. R. SUTER, Corps of Engineers, U. S. A. APPENDIX 3. JAMES A. SEDDON, REPORT OF CAPTAIN S. W. ROESSLER, CORPS OF ENGINEERS, UPON SURVEY OF NONCONNAH ROCKS. UNITED STATES ENGINEER OFFICE, SIR: I have the honor to submit herewith the map of a survey of Nonconnah Rocks made pursuant to a resolution of the Commission dated November 18, 1890. The survey included soundings in the vicinity of the rock, examination by a submarine diver and the obtaining of specimens of the stone. The rock is located about 5 miles below the city of Memphis, opposite and above the mouth of Nonconnah Creek, and is about 700 feet out from the Tennessee shore and 2,000 feet from Presidents Island shore. A previous examination of the rock was made by Capt, Leach, September 19 and 20, 1889, and a report made by him under date of December 13, 1889. At that time the main channel was between the rock and Presidents Island, and a narrow chan-· nel, rarely used, existed between the rock and the Tennessee shore. Around the rock and partly covering it was a large flat gravel bar, sloping upwards toward the rock where the general level of the bottom was about 1 foot below the water surface, corresponding to zero on the Memphis gauge. Since that survey the channel has moved from Presidents Island to the Tennessee shore and there has been a deep scour in the gravel bar in the vicinity of the rock, on all sides of it, and in the channel between it and the Tennessee shore. About 150 feet east of the rock the scour was about 27 feet, and 200 feet farther in the same direction, 16 feet; 200 feet above the rock, 10 feet; 350 feet west of the rock toward Presidents Island shore, 4 feet, and in the immediate vicinity of the rock, the sconr varied 17 to 27 feet. The rock itself shows the effect of erosion in the interval, the height at the last survey (2.80 above zero stage) being 3 feet below that indicated by the survey of 1889. The section of the rock by a horizontal plane 8 feet below zero stage has approximately the shape of an ellipse with longer and shorter diameters respectively 190 and 130 feet. Above this section, the shape of the rock, as well as can be inferred from the soundings taken, is something like an obliqué frustum of a cone with a steep slant on its east side and a flat slope on the opposite side toward Presidents Island, and contains approximately 3,000 cubic yards. Since the change in the position of the channel to the Tennessee shore, the rock has become a more serious obstruction to navigation than it had been theretofore, and its removal made more urgent. It is difficult to estimate the cost of excavating it. The material is a soft, ferruginous sandstone or pudding stone easily broken up and pulverized under a hammer, which would be pulverized to a large extent by blasting and washed away by the current. A large part of the highest projection at the eastern end of the ledge can also be blasted off into deep water, thus reducing materially the amount that would have to be dredged up and towed away. In the absence of any similar work as basis for an estimate, I place the cost at $2 per cubic yard, or $6,000 for excavating the rock to a depth of 8 feet below zero stage. The method of removal proposed is to do the work at low water, to break the rock by surface blasting, to proportion the size of blasts so as to pulverize the rock as much as possible and remove by dredging the coarser material which the current will not move. It is proper to add that the difference between my estimate and Capt. Leach's of the quantity of rock to be removed, mine being less than one-half of his, is due to the fact that a larger part of the rock at the time of his examination was covered by the gravel bar, a portion of which, since scoured away, was probably included in his estimate as rock. Very respectfully, your obedient servant, Gen. C. B. COMSTOCK, President Mississippi River Commission. S. W. ROESSLER, |