Evaluation of ʃcot x sin 2mx In(sin x/sina)dx, Etc.

For m = 1,2,... the following indefinite integrals are evaluated ⌠cot x sin 2mx dx, ⌠tan x sin 2mx dx, ⌠cot 2x sin 2mx dx, ⌠2 csc 2x sin 2mx dx, ⌠cot x sin 2mx ln(sin x/sin a)dx, ⌠tan x sin 2mx ln(sin x/sin a)dx, 2⌠cot 2x sin 2mx ln (sin x/ sin a) dx, 2 ⌠csc 2x sin 2mx ln (sin x/ sin a) dx. Also, formulas are given for the last four expressions where f(x) replaces ln(sin x/sin a). Further, procedures for evaluating the above expressions are outlined when cos 2mx replaces sin 2mx. The need of the integrals arose in connection with Fourier series where singularities in the function to be developed had been removed

Advance of irrigation water on the soil surface in relation to soil infiltration rate: A mathematical and laboratory model study

Mathematical equations describing the horizontal advance of an irrigation stream on a soil surface are derived and discussed for different types of infiltration equations corresponding to different known field conditions. Complex variable theory is applied to transform certain complicated forms of infiltration equation solutions to algebraic forms. An irrigation model having a visible Plexiglas photographic front was constructed and operated to test the theory and obtain data not covered by the theory. Glass beads or soil aggregates constitute the porous medium; water is used as the seepage fluid. Potassium dichromate dye is injected into the porous medium to trace the direction and velocity of the stream lines when the water moves within the body of the porous medium. The model data are recorded by photography and show a good agreement between theory and experiment for both the calculated position of the “irrigation” front on the porous media and the “wetted” front below the surface. Comparisons were made between experimental data and theory for two slopes of land, for five porous media, for two irrigation rates, and for two surface conditions, rough and smooth. Dimensionless functions are developed to present the model data

Solving tile drainage problems by using model data

Our purpose in this bulletin is to report, to analyze, and to use in problem solving, extensive model data of tile drainage of land. The data were obtained with a glassbead-glycerol model (Grover et al., 1960; Grover and Kirkham, 1961) and inc1ude: (a) values of depths and of corresponding times of fall of the surface of saturation to these depths at various distances from the drain tubes and (b) values of the drain tube discharge rates. The zero reference time for the fall of the surface of saturation and also for the discharge rate is the instant at which the surface of saturation passes through the simulated soil surface from a ponded condition. Models were made of 109 different combinations of drain depth, drain spacing and soil stratification. For each of these 109 model conditions, the surfaces of saturation were photographed at about eight different depths through the transparent front face of the model. Photographs were read under a magnifying glass to obtain distances and times of fall. Times were obtained from a clock that was started at the zero reference time and photographed with the water tables

The stream function of potential theory for a dual-pipe subirrigation-drainage system

An exact mathematical solution to Laplace\u27s equation is presented for appropriate boundary conditions associated with the problem of dual-pipe subirrigation and drainage. The solution can be used to determine a flow net within the groundwater flow region and the associated water table shape. The solution is general. The effects of several hydraulic and geometrical parameters on the groundwater system, such as thickness of saturated zone, position of subirrigation and drainage pipes, heads in the subirrigation and drainage pipes, crop evapotranspiration, fraction of inflowing subirrigation water that exits through the drains, and the aquifer hydraulic conductivity system are evaluated. Calculations are presented showing how pipe spacing affects the shape of the water table. For example, with hydraulic conductivity of 10 m/d, evapotranspiration of 0.01 m/d, drainpipe radius of 0.05 m, and subirrigation pipe radius of 0.0375 m, calculations show that the maximum water table elevation for a pipe spacing of 40 m is 0.64 m greater than for a pipe spacing of 16 m when 40% of the input subirrigation water volume is being removed from the system by drainage. Finally, the general mathematical solution can be used to predict chemical movement as well as water flow through the system

Physical and mathematical theories of tile and ditch drainage and their usefulness in design

A number of theories for tile and ditch drainage have been proposed in recent years which, if valid, would enable the rational design of many drainage systems. Nevertheless, most drainage systems are still designed by rule of thumb based largely upon the observations of technicians with experience in certain restricted areas. To develop a theoretically sound and practically valuable method of designing subsurface drainage systems, the various approaches which have been made should be critically evaluated and compared, mutually, as well as with field data. However, no such analysis has been found in the literature. The object of this publication is to provide this type of appraisal. The assumptions underlying a number of methods of analysis will be scrutinized in detail, and various applications of these methods to field results will be tested. It is hoped that this evaluation of the status quo will be useful in determining to what extent present theories lend themselves to field applications and what phases of drainage design need further study. In general, this discussion will be restricted to problems of saturated flow, while recognizing that flow in the unsaturated zone above the water table often may be important. Little progress has been made in formulating quantitative theories regarding flow in the unsaturated zone

The Movement of Chloride and Nitrate through Certain Iowa Soils

Miscible displacement experiments were conducted to determine the influence of soil type on the movement of chloride and nitrate. The movement of NO - is important in plant nutrition; Cl-, in soil salinity. In these experiments, 100 ml of an aqueous solution containing 0.55 g of CaC12 and 0.35 g of Ca(NO3)24H2O was displaced through 30-cm long soil columns with 0.01N CaSO4. Breakthrough curves, plots of the chloride or nitrate concentrations found in the effluent against the volume of effluent collected, were made. Breakthrough curves from columns of Webster, Ida, and Edina surface soils and a muck soil indicated that the velocity of chloride was greater than that of nitrate during displacement through these soils. On the other hand, breakthrough curves obtained from columns of Webster, Ida, Edina, and Clarion subsoils showed no separation of chloride and nitrate. The breakthrough curves for all soils studied differed in shape. The dispersion coefficient for chloride, calculated from the breakthrough curves, varied from 1.533 cm2/hr for the muck soil to 0.094 cm2/hr for the Ida, C horizon, soil. The experimentally determined dispersion coefficients were used to calculate theoretical chloride distributions for the muck, Ida, and Webster soils

Potential theory for dual-depth subsurface drainage of ponded land

Dual-depth subsurface drainage is considered to be more effective in removing excess water from soil than single-depth drainage, but this problem has not been analyzed in detail. Therefore, assuming that uniform, water-saturated soil covered by ponded water and overlying an impervious barrier is drained by equally spaced, alternating deep circular drain tubes, existing potential flow theory for a single-depth drainage system was extended. Sample calculations with the newly derived equations show that a dual-depth subsurface drainage system can be highly effective to remove excess water from soil. For example, a relative drain discharge of 160% is calculated when new drain tubes, added at the 0.60 m depth, are placed midway between the original drain tubes, which are 25 m apart and at the 1.20 m depth. In this calculation we have assumed that the impervious layer is at the 3.0 m depth, the radius of the tubes is 0.05 m, the soil hydraulic conductivity is 1 m/d, and the thickness of the ponded water is 0.0 m. For the same conditions, but with the additional tubes at the 1.20 m depth (same depth as original tubes), the relative drain discharge becomes nearly 200%, and with the additional tubes at the 2.40 depth (1.20 m below original tubes) it is more than 250%. When the impervious layer is at a greater depth and when the original drain spacing is more than 25.0 m, the relative drain discharge becomes even larger. The effectiveness of the dual-depth tube system becomes particularly large, if the second tube system is placed below the level of the first one

Groundwater Flow Patterns of the Ames Aquifer

Extensive field studies have indicated that the Ames aquifer (at Ames, Iowa) approximates a confined horizontal aquifer of uniform thickness and height. The aquifer is partly fed through the channel beds of two surface streams, the Skunk River and Squaw Creek. Part of the boundary of the Ames Aquifer is formed by a till layer that can be considered impervious. Pumping tests have indicated that the Skunk River and Squaw Creek maintain nearly a constant head distribution along the boundaries of the aquifer while the aquifer is pumped. A theoretical solution is presented that yields the well discharge, the hydraulic head at every point in the flow region, and the stream function when the aquifer is pumped by one completely penetrating well

Development and Testing of High Energy Layer Rations for Use in Oklahoma

The Oklahoma Agricultural Experiment Station periodically issues revisions to its publications. The most current edition is made available. For access to an earlier edition, if available for this title, please contact the Oklahoma State University Library Archives by email at [email protected] or by phone at 405-744-6311