The theoretical investigations of Prandtl and Karman, and the experimental work of Nikuradse, have led to rational formulas for velocity distribution and hydraulic resistance for turbulent flow in circular pipes. With certain assumptions regarding the effects of secondary currents and of the free surface, and with the adoption of the hydraulic radius as the characteristic length, similar rational formulas are deduced for open channels. The validity and the applications of these formulas are illustrated by a study of Bazin's experiments. In this study the equivalent sand roughnesses of the channels used by Bazin are determined. The criterion for determining the conditions under which a channel with woodenplank surfaces is to be considered hydrodynamically wavy or hydrodynamically rough is also evaluated. The rational formulas with constants determined from Bazin's experiments are expressed in the form of power laws. It is shown that Manning's empirical formula is a good approximation to the rational formula for rough channels when the relative roughness is large.

The inertia and drag coeffi cien ts o f cylinders and plates in simple sinusoidal c urrents are investigated. The midsection of a rectangu lar basin wi t h standing wa ves s urg ing ill it is selected as t he locale of curren ts. The cyl inde rs and plates are fi xed hori zontally a nd below t h e water surface. The average values of t he inertia and d rag coefficients over a wave cycle sh ow variations wh e n the intensity of t he current and t he s ize of th e cylinders or plates a re ch anged. These variation s, h owever, can be correlated wi t h t h e p eriod parameter Um TI D, where U m is th e maximum intensity of t he sinusoidal c urrent, T is the p eriod of t he wave a nd D is t he di am eter of t he cylinder or t he width of the plate. For t he cylinders UmTI D eq ua ling 15 is a c ri tical rond ition y ielding t he lowest va lu e of t he inertia coefficient a nd th e largest valu e of t he drag coe ffic ie11t. For t he plates t he higher values of t he drag coefficie nt are assoc iated with t he small er va. lu es of Um TI D a nd t he higher valu es of t he mass coe ffici e nt wi t h t he larger valu es of Um TI D . The variatio n of th e coefficients with t he phase of t h e wa ve is examin ed a nd the bearing of t hi s o n t he formul a for the fo rces is disc ussed. The flow patterns aro und t he cylinde rs a nd plates a re examined photographically, a nd a s uggestion is advan ced as to t he physical m eaning of the parameter Um TI D.

The modulus of decay of standing waves of finite height is derived by assuming that the attenuation of the waves is due to viscous losses in boundary layers close to the solid walls. Dampings are observed in six basins of varying sizes. The basins are duplicated using glass and lucite for the wall materials. With liquids wetting the walls, the losses due to viscosity are slightly increased from causes presumably related to surface tension. With a liquid not wetting the walls (distilled water and lucite), losses from surface activity, of some obscure origin, outweigh many times the losses due to viscosity in the basins of smaller sizes. For moderately large basins, for which surface activity may be neglected, the agreement between the observed and computed rates of decay is found to be satisfactory.

In addition to the generation of waves, a wind produces a mass transport in a body of water resulting in t h e lowering of the level at t h e windward ~ide an~ rise at t he leeward side which is called wind tide or set-up. Two eff ects of the wll1d are Involved: the s urface t raction on the water, and the form resistance of the waves. This paper presents the t heoretical background of t he subject and expcriment al r es ults. It was di scovered t hat formation of waves in an experimental cha nnel could be inhibited by addin g small amounts of soap or detergent to t he water. !his made it P?ssible to study the sl!rface tracti?n effect separately. The eff ect was studied for both lamlIlar and t urbulent motIOn of the dnft and gravi ty currents produced by the wind. The set-up computed from me~s ured wind and water surface veloci ties agreed with theory. Regardless of t he flow regime, the set-up was unaffected by viscosity and independent of t he depth-length ratio of t he channel. The additional set-up due to presence of waves could be correlated only b y int roducin g a characteristic velocity. This additional set-up appears to vary as t h e square root of the d ep th-length rat io. The relation of t he characteristic velocity to t h e criti cal velocity for wind generation is di scussed. The derived empirical formula for set-up is compared with obser vations in Lake Erie.

This paper treats the problem of the damping by viscous action of translation waves. A shcrt exposition is given of Boussinesq's boundary layer theory for wave motion , and expressions for the damping of rectangular and solitary waves are derived . Scott Russell's experimental results for solitary waves are compared with the theory, and satisfactory agreement is found to exist. This fac t makes it legitimate to apply t he formu las developed to correct in model tests on harbors or in other tests of a li ke nature for the di ssipative effects t hat ocellI' in shallow-water waves. I. List of SymbolsA = whd fl . B = breadth of chan ne!. e= base of natural logarithm s. dEddt= rate of di ssipat ion of energy of the wave. dE2/dt= rate of decrease of energy of t he \\ave. IiEl = energy in column of unit length and of width dx.g= acceleration of g ravi ty. h or hI = height of wave above undisturbed surface.h OI = init ial height of wave above undisturbcd surface. H = depth of undisturbed water. k = constant defined by eq 35. K = constant defin ed b y eq 44 .• L = length of wave. n = lrx. N = an integral defined by eq 41. p = the press ure in t he liquid. q=a variable. s = a variable, also the distance travelled by the wave. t= time. x= coordinate parallel to solid surface in direction of motion. y= coordinate parallel to solid surface at right angles to direction of moLion. z= coordinate perpendicular to solid s urface. 1., v, W= velocity components of particles in the bo undary layer parallel to x, y, z, respectively. 1'0, Vo, Wo = velocity components of particles in the potential region parallel to x, y, z, respectively. ll= particle velocity in the potential region. X = x-component of body force per unit mass due to gravity. Y= y-component of body force per unit mass due to gravity. . Z = z-component of bod y force per unit· mass due to gravity. Gradual Damping of Solitary WavesY ( ) = function of. a = a variable, also a damping coefficient. {3 = a variable. K = a constant defined by eq 17. I-' = viscosity of the water. v= kinematic viscosity of t he water. p= density of t he water. q, = a dissipation function . w= velocity of wave propagat ion. II. IntroductionIn the propagation of translation waves in still water, the main source of dissipation of the energy is the thin layer of liquid next to the solid boundaries of the channel; that is, in the boundary layer. The motion in the layer is laminar, and the determination of the velo cities in it can be made by resorting to the ordinary equations of viscous motion. The motion of the liquid outside of th(} layer is potential, and there the values of the velocities are exactly those that would be obtained from the ordinary theory, assuming that the laminar layer is absent. The errors involved in the use of this method can be ignored, since the variation of pressure in a cross section of the laminar layer is negligible. In fact, the gradual decrease of height of a solitary wave, the only wave of translation, which under ideal conditions, travels without change of form, ca...