| Rodarte 365). 2 U.S. Army William D, Grask Coastal Engineering Research Center TRANSPORTATION OF BED MATERIAL DUE TO WAVE ACTION acorn : Wp mit ’ Ber rg ees + i i: Wf iy Sn Pe vrai oP tO ay rr, Leen uued ibe TECHNICAL MEMORANDUM NO. 2 DEPARTMENT OF THE ARMY CORPS OF ENGINEERS TRANSPORTATION OF BED MATERIAL DUE TO WAVE ACTION by George Kalkanis University of California SS ;— = -— = ——_ == ™ —— — ——— 7) —— —— ——r 1) oS = $20 =i) Pe oS M nl TECHNICAL MEMORANDUM NO. 2 February 1964 (WM Material contained herein is public property and not subject to copyright. Reprint or re-publication of any of this material shall give appropriate credit to U.S.Army Coastal Engineering Research Center LIMITED: FREE DISTRIBUTION OF THIS PUBLICATION WITHIN THE UNITED STATES IS MADE BY THE U.S.ARMY COASTAL. ENGINEERING RESEARCH CENTER 5201 LITTLE FALLS ROAD, N. W., WASHINGTON D.C. 20016 FOREWORD A better understanding of the basic mechanisms of sediment transport by wave action is being sought by the Coastal Engineering Research Center (formerly the Beach Erosion Board) through several different approaches. One of these which has been pursued under contract with the University of California at Berkeley, is reported on herein. This report discusses the mechanisms of sediment trans- port in a layer immediately adjacent to the ocean floor for long waves of small amplitude and relatively deep water. Proceeding on the premise that a bed-load function exists for the assumed type of flow, a bed-load transport equation was developed. Effects of both the unsteady mean flow velocity and the turbulent fluctuations are taken into account. From this equation the rate per unit width at which sediment in the bed layer is shifted by the oscillatory flow is calculated. This report was prepared at the Hydraulic Engineering Labora- tory of the University of California at Berkeley in pursuance of contracts DA-49-055-eng-17 and DA-49-055-CE-63-4 with the Beach Erosion Board (now the Coastal Engineering Research Center). These contracts provided in part for the study of sand movement by wave action. The author of the report, George Kalkanis, was at that time a graduate student and research assistant at the University. This report is published under authority of Public Law 166, 79th Congress, approved July 31, 1945, as modified by Public Law 88-172, approved November 7, 1963. TABLE OF CONTENTS List of Figures - - - - - ------------ List of Tables - ---------------- List of? Symbols, (8-9 —)—) 2-8 IABSTRAGIO (2) 20 202) a0 ree INTRODUCTION - ------------ rrr cre FORMULATION OF -LHE SPROBLEM Gai OSCILLATOR Y-BOUNDARY LAYER FLOW - - - - - --- = a. Theoretical Considerations - - - - ---- (i) Laminar Case - ----------- (ii) Turbulent Case - ---------- b. Experimental Work - - ---------- HYDRODYNAMIC EFFECTS OF THE FLOW ON THE BOUNDARY a. Theories of Sediment Transport in Rivers - b. The Bed-Load Equation - - -------=- DETERMUNAGHON OF Aj By ance Ge oi) thee DETERMENATE TONG OES Coe cei ol tells oy oh Vie ie ee ae A PARTICULAR CASE OF THE SECONDARY DRIFT - -— ~— 7— SUMMARY OF RESULTS - - ------------- DISCUSSION OF RESULTS - - ------------ CONCLUSTIONSH 2S a Sa ACKNOWLEDGEMENTS - - -------------- REFERENCES ee ee ETGURESU Gln = 1) ee APPENDICES A - Stability of Oscillatory Laminar Flow Along a Wall B - Determination of Velocity and Phase Shift Distribution C - Determination of the Parameters Aj, B, and 1/7), D - Tables Figure 1 10 11 Table III-A TTT-B IV LIST OF FIGURES Page Criterion for Transition from Laminar to Turbulent Flow with Oscillatory Motion - - - - - ------- 31 Smooth Plate-Measured Velocity Distribution - - - - - 32 Two-Dimensional Roughness-Measured Velocity Distribution - - -----------+-+-+----- 32 Three-Dimensional Roughness—Measured Velocity Distribution - - ------------------ 33 Smooth Plate-Measured Phase Shift - - - - - - --=--- 34 Two-Dimensional Roughness-Measured Phase Shift - - - 34 Three-Dimensional Roughness-Measured Phase Shift - - 35 Determination of “Ay Bye and | i=) o—8 6 36 dp/® versus Di cunve (iss) a 37 Typical Unsteady Mean Velocity Profiles - - - - - - - 37 Section of Velocity Measuring Instrument - - - - - - 38 LIST OF TABLES Page Smooth Plate-Velocity and Phase Shift Measurement D-1 Two-Dimensional Roughness-Velocity and Phase D-3 Shift Measurement Three-Dimensional Roughness-Velocity Measurement D-8 Three-Dimensional Roughness-Phase Shift Measurement D-17 Experimental Data Used to Determine A,, B, and No D-19 TP: TNO |The L AS A, = AIA wa ae aN * Cp TAy c =-0.65 Sellen 2 c (y) Cc [o} ca d D e f(y), f, (y) g h = 2D H _ But eX a *& L LIST OF SYMBOLS semi-amplitude of displacement at the bottom from irrotational theory. constants of proportionality. constant of proportionality constant determined experimentally constant determined experimentally exponent of f,(y) from original smooth plate measurement. coefficients concentration distribution of solid particles in a state of motion within the bed layer average value of c (y) coefficient of lift depth of water bed material grain diameter base of natural logarithms empirical functions describing the velocity distribution and the phase shift in the boundary layer respectively gravitational acceleration thickness of bed layer wave height (from crest to trough) wave number empirical constants distance traveled by a solid particle in one realization of motion. lift force - total ii cl TRY = £/t Sin t LIST OF SYMBOLS (cont.) lift force due to unsteady mean flow 1ift force due to turbulence subscripts number of particles of size D per unit area of bed surface probability of the lift force exceeding sub- merged weight pressure oscillatory bed-load rate steady mean bed-load rate cylinder radius (Jeffreys model) resultant of all hydrodynamic forces exerted on a solid particle exchange time period of oscillation temperature degrees Fahrenheit instantaneous boundary layer velocity components unsteady mean velocity components in the boundary layer turbulent velocity components in the boundary layer unsteady mean velocity components at the boundary from irrotational theory settling velocity free stream velocity at the outer edge of the boundary layer critical free stream velocity mass—transport velocity speed of particle propagation LIST OF SYMBOLS (cont. ) Ws = A,D'y . dry weight of the solid particle W" = (p,- p¢)8 A,D submerged weight of the solid particle ee Ve semi-amplitudes of displacement from irrotational theory Zz dummy variable of integration iL,¥ zZ== standard normal variable Lio 7/2 ai F a= (c(1-c) coefficient of fo(y) from original smooth plate measurement w 1/2 ee 8 = (3 scale parameter for characteristic length Y associated with f,(y) and f5(y) Vs dry unit weight of bed material 6 thickness of the boundary layer € roughness diameter No normalized standard deviation of turbulent 1ift force f£,(y)sin f,(y) 6 = ean ewe Sse eeu a ye hase angle yy Get uO costa) = r wave length ww = By Longuet-Higgins notation NY) kinematic viscosity of the water € dummy variable of integration Meo Oe density of the solid particles and of the water respectively (oj standard deviation of turbulent lift force LIST OF SYMBOLS (cont. ) shear stress phase increment used in numerical integration dimensionless parameter flow intensity function angular velocity of oscillation vi TRANSPORTATION OF BED MATERIAL DUE TO WAVE ACTION by George Kalkanis, University of California ABSTRACT A practical method has been developed which can be used to determine the rate of sediment transportation in a layer adjacent to the ocean floor. The method is applicable only when the flow in this layer caused by surface waves is unstable. The waves in question should be of small amplitude and great length, permitting the linearization of the equations of motion. The fundamental principle of the supporting theory is that at equilibrium the sub- merged weight of the solid particle resting on the ocean floor is balanced by the vertical component of the resultant of all the hydrodynamic forces exerted on the particle by the flow above. Both effects of the unsteady mean flow velocity as well as of the turbulent fluctuations are taken into account. The distribution of the lift forces associated with the former was determined experimentally while a statistical approach based on the experi- ence with the same phase of the problem in a steady mean flow was used to determine the latter. Proceeding on the premise that a bed-load function exists for the type of flow we are dealing with here the bed-load equation was developed. From this equation it is possible to determine the rate at which sediment in the bed- layer is shifted by the oscillatory flow across a section of unit width, The concentration of bed material in the layer that at any instance is at a state of oscillatory motion can be determined from the above rate. Finally, this concentration in combination with the velocity distribution in the bed-layer associated with any incidental secondary flow is used to calculate the rate of transportation of bed material in the direction of this flow. 1, INTRODUCTION The purpose of this study is to develop a method by means of which low rates of sediment transport due to the action of surface waves may be predicted with sufficient accuracy. More specifically the waves considered are long waves of small amplitudes in relatively deep water. The problem is of great importance to engineers and scientists as can be witnessed from the considerable amount of research devoted to it especially since the beginning of the century. An extensive survey of the literature, how- ever, revealed that so far the subject has been treated only qualitatively. The numerous publications are mostly restricted in presenting observa- tions made in the field or the laboratory. With exception of a few purely empirical equations, which describe only particular measurements, no attempt has been made, at least to the writer's knowledge, to derive basic relationships which will constitute the basis of a quantitative treatment, In this paper we will try to analyze separately the indivi- dual mechanisms that constitute the overall phenomenon of sediment trans- port by ocean waves. After each phase of the problem is well-described and understood, the desired relationships will be developed based on all available theoretical concepts and empirical evidence. A more detailed description of our objective is given in the following section. 2. FORMULATION OF THE PROBLEM It has been observed that loose sediment is moving in considerable quantities near and along the ocean floor even in relatively deep water. This motion cannot be attributed to the action of ocean currents because the flow intensity of these currents is usually very low and the hydro- dynamic forces associated with it are not sufficient to overcome the forces resisting motion. It is evident, therefore, that the mechanism mainly responsible for this motion has its origin in the oscillatory flow near the bed which is caused by the surface waves. It is obvious that in general there is no net transport associated with this motion since the particles oscillate more or less about their mean position. The claim set: forth is that the hydrodynamic effect of this flow is to relieve the part- icles of all or part of their weight so as to bring them to a state of incipient equilibrium. At this state any incidental secondary flow or current, no matter how weak, will be able to set the particle in motion. The problem now may be defined in a more precise form, An oscillatory motion is induced by the surface waves on the boundary layer. It is desired a. To develop an expression that describes the flow field in the boundary layer. -b. To determine the dynamic effect of this field on the solid particles forming the bed. The individual results of these two phases of the problem will be com- bined in a logical fashion to obtain a relationship by means of which it will be possible to predict the pattern of the motion of solid particles near the bed for a given set of wave characteristics and bed composition, We proceed first with the study of the flow in the boundary layer. 3. OSCILLATORY BOUNDARY-LAYER FLOW a. Theoretical Considerations (i) The Laminar Case. The problem of the boundary-layer flow can be treated in the ordinary fashion. According to the fundamental principle the flow field outside the boundary layer can be described by means of the irrotational theory while the complete equation of motion within the layer is simplified with the help of dimensional arguments. The boundary conditions of the simplified equation are set so as to satisfy the nonslip requirement at the solid boundary and the continuity of the velocity components at the outer edge of the boundary layer. Lin (1957)* has presented a solution to the more general problem in which the flow both inside and outside the boundary layer has a mean steady com- ponent as well as an oscillatory one. .In the present case the two flows have oscillatory components only. The equations of motion will be 2 CO om Bay Se Lee y ot (3-1) at Ox oy p ox oy and Qu Ou, Qu, 1 aP s — u eae eee = he Ace Mena =E es Way p ox Ca inside and outside the boundary layer respectively. Continuity is assumed to be satisfied individually gu Ov so that ae + ay = 0 (3-3) and O41 + OM, = 0 (3-4) Ox oy Equations (3-2) and (3-4) describe a two-dimensional inviscid un- steady and incompressible flow. The solution of these two equations under a specific set of boundary conditions will define the velocity field in the fluid as a function of time and space, provided of course, that the effect of viscosity is negligible. In the problem at hand the first boundary condition is a sinusoidal progressive wave at the surface (y = 0) and the second, zero vertical velocity component at the bottom (y = -d). To the first approximation equation (3-2) may be linearized by omitting the quadratic terms; it can be written then as OU) Sora. MenOy (3-5) ot p ax * Numbers in parentheses denote date of reference listed on page 29. This simplification is justifiable only under the assumption that the slope of the surface waves is very small; this is equivalent to k = << 1 where k is the wave number and H the amplitude of the surface wave. When the water depth is not very large (d<1/k) the solution of the simplified equation indicates that the water particles describe elliptical orbits around their mean position with component displace- ments in the x and y directions given by the expressions: (Lamb, 1932 sec. 229). cosh k (y + d) H > eran Tel cos (kx-yt) (3-6) _H sinh k (y +d) _. NOS Crea TT ae ee SESS ed (3-7) It is evident that the vertical component of the displacement be- comes smaller as the distance from the surface increases and that right at the bottom (y = -d) the motion degenerates into a simple harmonic os- cillation along the x direction. The corresponding velocity components are obtained by differentiation of equations (3-5) and (3-6); so that cosh k (y + d) Oe ae a = uy (x,y,t) = Se ). Srean Tal Sin (kx wt) (3-8) A OMe H sinh k (y + d) th ie vy Gey tf) = NEE San Ra ee oe (kx - wt) (3-9) As y — -d vy (-d) — 0, and H : u, (x,-d,t) == w cosech kd sin (kx - wt) (3-10) il or ou, (x,-d,t) = aw sin (kx - wt) = ug sin (kx - yt) : H where obviously a = 5 cosech kd and ug = ay On the basis of dimensional considerations it is reasonable to postu- late that the thickness of the boundary layer is very small <6 = iI Ww compared to 1/k so that for all practical purposes u, (x,t) may be assumed constant within the boundary layer and approximately equal to u,(x,-d,t); as it is customary we will use the notation U, for the free stream velocity at the outer edge, so that U,, = Uy (x,-d,t) = aw sin (kx - wt) (3-11) with surface waves of large wave length (ka< CP a" (4-5) The analysis of the experimental results revealed that Cy; in the above expression had a constant value Cy = 0.178 for a wide range of flow con- ditions provided that the average velocity was measured at a distance 0.35 D from the theoretical bed. Since it is practically impossible to determine the value Cy in an unsteady mean flow, it would be reasonable to assume that it is about the same as in a steady stream. As a matter of fact, as it will be shown shortly, the only assumption that is really necessary is that Cy is constant throughout the entire cycle of oscilla- tion. Before we proceed any further we wish to summarize the statements adopted in this study which have their origin in El-Samni's experiments with steady mean flows. a. The theoretical bed lies at a distance 0.2 D below the top of the grains resting on the fixed bed. b. The 1ift force associated with the mean flow can be calculated by using the velocity at a distance 0.35 D from the theoretical bed, c. The 1ift coefficient associated with the mean flow has a constant value independent of the Reynolds number. If it were not for turbulence it would be rather easy to establish a cri- terion of stability similar to Jeffreys’ in the form of the inequality L > Ww’ (4-6) (W’ is the submerged weight of the particle) Gis Ci Pr > A,D > (ps = (ove) gA,D (4-7) where by u we mean the amplitude of the velocity calculated from equation (3-24) with y = 0.35 D. In a turbulent stream, however, all the local flow parameters, and consequently the local life as well, vary rapidly with time. An accurate description of the temporal variation of the local life by analytical methods would be possible only if sufficient information regarding the structure of turbulence in the boundary layer were available. Considerable _amount of effort is made now by numerous investigators toward developing new theories or toward improving existing ones on the subject. The fact remains that even if we could wait until the theoretical work had been sufficiently advanced to permit practical applications, still some approximations would be required since most of the studies are concerned with steady mean flows and smooth boundaries. In the meantime and due to the lack of more reliable information we are forced to depend again on experimental evidence. A possible criterion of stability in a turbulent stream could be set as i SY ob ie Se Ty? (4-7a) where L' is the turbulent component of the 1ift force. The study of the variation of L' with time constituted the last phase of the experimental work conducted by El-Samni. By measuring the instantaneous values of the lift force exerted by a steady stream of water on the plastic spheres mentioned above, he was able to show that L’ behaves like a random vari- able having a normal distribution with mean zero and standard deviation ~ ae ; i @ = ib is =s-5 - Assuming that the behavior of L"* in an oscillatory mean flow will be similar although with a different numerical value of oO we could proceed as follows: Let p be defined as the probability that a particle resting at a certain location in the bed becomes just ready to move; this implies that ae r w? ; or Bae Te oie ea TMG eae = | (4-8) L No Gis ve 3 2g(P. Pre) A,D ee as or p= IPae { Z > GER: Cae = le, } ( i ) fo) CypgDo wat, p_-Pp Dg tee We ae (4-10) PF u > IN and B, = Z (4-11) Cy NoAt Then p = Pr { hs WAR a } (4-12) ii No ? and since Z = ae has a normal distribution with mean zero and standard Lo deviation go = 1 Cc _ pp 1 e p = oe (4-13) Jat kul 1 ig No where z of course is a dummy variable. The values of No and By, can only be determined experimentally. It is evident now why it is not necessary to determine the exact value Ge Cr. As long as Cy, is constant its effect will be reflected in the value of Bx. We may recall now that u is a periodic function of time which as the ex-— periments indicated can be approximated by the right hand side of equation (3-24). Consequently y is a periodic function too which means that a more appropriate form of equation (4-13) would be 2 2 T/2 Ao if = = — e = p T fam Y 4 dz dwt (4-13a) ° * The geometric representation of the probability p as expressed by equation (4-13a) is shown in the sketch below. §(U) mI. 2 It is evident from the form of equation (4-13a) that the turbulent field was assumed constant during the entire cycle of the oscillatory mean flow. 15 The number Nj of particles of size D per unit area of the bed surface that at any instance become free to move and indeed do move is propor- tional to this probability as well as to the total population of similar particles per unit bed surface, It is evident that NaS 5 (4-14) where Aj is a constant. The rate of transportation qg which is defined as the rate in dry weight at which solid particles move across a section of unit width oriented perpendicular to the flow can be expressed as ST ee LAGU S asi ay A,D 1 where Wj = Ad? bp Ys, Ys being the dry unit weight of the particles and Vi the speed of the particle propagation. It should be remembered that the motion of the particles is not continuous, but that it consists of a sequence of discrete steps. It will be reasonable, therefore, to express this speed of propagation as the ratio of the average distance covered by a particle in a step and of the total time required for the completion of a full cycle of motion. If we denoted the former by £ and the latter by t equation (4-15) after the substitution for Wj becomes 3 4 A a, = =, AD MEVe Pie) tn ee A,D Al 4 PDYs | (4-16) The distance 4 as in the case of a steady mean flow may be considered as a charactsristic length proportional to the grain diameter, so that g = AL D. Moreover by definition t can be written as BS By Pe (4-17) where ty is the part of the cycle during which the particle is at rest (between two consecutive excursions) and t2 is the part during which the particle is in motion. Experiments with light-weight coarse material and steady mean flows in flumes have demonstrated that in general tz is much smaller than ty which led to the conclusion that for all practical pur- poses we may assume t = t1. Einstein (1950) has defined tj, which he called "the exchange time" as a measure of the time required for the replacement of a particle that is just being picked up by the flow at a certain spot of the bed by a similar particle that is being brought to rest at the same spot. Therefore, one may think of tj as a parameter which depends on the properties of the particle and the surrounding fluid only and which, consequently, is independent of the local flow conditions. The simplest such parameter having the right dimensions is the time re- quired for a particle to settle through a distance equal to its diameter in the fluid at rest. If the settling velocity is denoted by Vs then we can write D Ba = a) ae (4-18) cy being a constant of proportionality A basic characteristic of our model is that the criterion of equili- brium is governed mainly by the lift force exerted by the flow on the particle, Since the coefficient of this lift force assumes a constant value even for very small values of the Reynolds number the settling velocity at equilibrium may be written as / Ps — Pp D V = ¢€ ete See (4-19) 2 P£ Combining now (4-18) and (4-19) and making the proper substitutions for 4 and t equation (4-16) becomes qB Piven) Da y| ESE ES) (4-20) al 3 D p¢ which is the "bed-load" equation; after being rearranged (4-20) becomes SB SRD [e(Ps - Pf) Atte NERO yn Eee (4-21) A,A,D D Pp YsA2A,D This is identical with Einstein's (1950) equation (38) provided that the material forming the bed is uniform, The significance of equation (4-21) is that it describes the equality between the rate of deposition (left hand side term) and the rate of erosion (right hand side). The remarkable characteristic of either equation (4-20) or (4-21) is that they indicate that the bed-load rate is only indirectly related to the flow intensity through the probability p. Eliminating this probability between equations (4-13a) and (4-20) we obtain the fundamental relationship between flow intensity and bed-load rate. 1 ats le Before we conclude this section, it would be necessary to define a little better the average distance of travel £ = AyD. A rather small value of p implies that only on a small fraction of the bed surface the lift force is strong enough to remove a grain of a given size at any instance, Therefore a particle that has been lifted by the flow will probably come to rest immediately upon completion of the first step since it is very probable that the conditions locally will favor deposition. In this case £ = A*;D where A'y is the true constant of proportionality between distance of travel in a single step and particle diameter, If, on 17 the other hand p is rather large, there is a good chance that the condi- tions around the spot on the bed where the particle is coming to rest after completing a step of length A';D are not favorable to deposition; the result is that the particle will be forced to take an additional step of length A*;D. This can be repeated a number of times until the particle finally finds a point on the bed where it is permitted to rest. This model suggests that the actual distance travelled will be proportional to the length of each step (A',LD) times the number of consecutive steps made in each realization, This number of steps may be thought of as a discrete random variable having a binomial distribution with parameter p. The probability that the particle will travel a distance At; D is (1-p) which is the probability of failure in the first trial. The probability of covering a distance 2A';D is p(1-p), the probability that failure: will follow one success, In general the probability of covering a distance (n + 1) A';D is p™ (1-p) which is the probability of the first failure occurring after n consecutive successes, The expected value of the dis- tance covered by the particle in a single realization can be expressed then by A;D where Rye A'yD ALD = EP Ge) et TD) (4-22) We introduce now this expression for A,D in equation (4-20) to obtain oe I p 2 |} gPs-P¢) = a fs 2 q Th 5 Vea WD) (4-23) es Pre Solving (4-23) for p we get Ax % (4-24) 1+A, 6 A.A where A, = —- (4-25) a ib q [ and ee —_—- pe (4-26) Vs BCP .-P¢) We equate finally the right hand sides of equations (4-13a) and (4-24) ta obtain the very important relationship between flow intensity and bed-load rate in the form of equation (4-27) 2 1/2 a eZ 2 2 © apelin © Bel (4-27) 12 1 Jp i 1+A, 6 0vBY aT, The practical application of equation (4-27) necessitates the determina- tion of the constants Ax and By and of the standard deviation T|, of the turbulent lift force. The procedure used toward this end will be de- scribed in the following section 5. DETERMINATION OF Ax, By, and TI, Suppose that a rather large set of corresponding values of the para- meters ¥ and 6 is available. These values were calculated from equations (4-10) and (4-26) respectively. The data used in these calculations may have been obtained in the field or in the laboratory but in either case we are confident that they are reasonably accurate. Our task then consists in selecting a set of values of the parameters to be determined so that ¥ and ¢ calculated from equations (4-10) and (4-26) will satisfy as closely as possible equation (4-27). The determination of the constants in the present study was based on values of ¥ and % calculated from experimental data. The description of the equipment and the procedure followed in this phase of the study is given in Appendix C. In these experiments three different sizes of sand were used and for each size nine runs were made with different amplitudes of oscillation and frequencies. The values of q were directly obtained from the experiment and they were next introduced in equation (4-26) to obtain $ while equation (4-10) was used to calculate the corresponding ¥. The latter was then plotted against the former on a log-log paper as shown in Figure 8. On the same paper a family of curves was plotted also representing graphical solutions of equation (4-27) with Noas parameter. The four curves shown on the figure were calculated with 1/No equal to 1.0, 1.5,.2.0, and 2.5. Both Ax and Bx in these calculations were taken equal to unity. The left hand side of equation (4-27) was in- tegrated numerically with the simultaneous use of normal error tables. The three steps of the procedure used to determine the points of the theoret- ical curve were the following. First an arbitrary value of Y was chosen which was divided by (cos ¢;)* where gj is the midpoint of one of the nine equal intervals dividing the quarter of the cycle, so that ¢, = 5°, G2 = 15° etc. The probability p which is approximately equal to the average value of Pp; where 4 eo “fp Dp. = —= e dz (5-1) cel , (cos 61)2 No was next obtained. The rearranged form of equation (4-25) was finally used to calculate the corresponding $; so that $= — (5-2) By examining the shape and relative position of the graphs on Figure 8, one may conclude that the translation of the experimental $, Y curve along the two axes would bring it to a close approximation with the theoretical one constructed on the basis of NA 1.5. The coefficients by which the coordinates of the experimental $, Y curve should be multiplied in order to achieve this approximation express the sought-for values of the para- meters A, and B,. It is evident from the foregoing that ha = 50 Bin 4 (5-3) a abe SS No We may conclude, therefore, that for a given set of wave characteristics and grain size of bed material it is possible to determine the bed-load rate qp through equations (4-10), (4-27) and (4-26). This is an important result, but by no means the final answer to our problem. Because of symmetry the rate in one direction during the first half of the cycle will be exactly equal to the rate in the opposite di- rection during the second half. The net effect of course will be zero steady movement. The question is now as to how can we make a practical use of the results obtained so far. The argument advanced is that al- though the calculated value of dp does not give a direct measure of the amount of sediment that systematically moves in some direction, it can be used to determine the number of solid particles per unit area of bed surface that at any time are exposed to the transportive effort of any incidental flow no matter how weak; this flow is not strong enough to dislocate the particles from their state of rest, but once it finds them in a state of motion produced by the surface wave it is able to move them forward, The rate of sediment transport per unit width of the bed associ- ated with such a secondary flow will be 2D Q@, = | uGyetyay (5-4) fo) where h is the thickness of the layer within which motion occurs, U(y) expresses the velocity distribution of the flow and c(y) the concentration distribution of the solid particles in motion. The magnitude of h may be taken equal to 2D, a customary approximation for flows of this type. Another approximation very common with steady mean flows is that c(y) within the layer may be considered constant. The velocity distribution U(y) of course remains undefined, but for any particular case is presumed known. Equation (5-4) then becomes 2D Q. fers | UCy) dy (5-5) fo) which will provice the desired answer. to our problem provided of course that co is known, This is the phase of the problem where the results ob- tained in this investigation find a direct and rather important application, 20 We have shown already that qp can be determined for a given set of con- ditions, In the following section a method will be described by means of which cg could be calculated from known values of dp - 6. DETERMINATION OF co The rate of flow through a cross section of unit width and height h = 2D is 2D Q, 1, UCy) ay (6-1) (eo) while the rate of sediment transport through the same section at any instance is 2D Qa = cof UCy )dy (6-2) fe) Co is a measure of the concentration of the sediment which at any time is at a state of motion within the bed layer. In this layer the rate of trans- port due to the oscillatory flow is by definition equal to dp. Therefore in a way similar to (6-2) we can write 2D ac = dp 2 | u dy (6-3) (0) 2D The integral = | TZ dy is nothing else but the expression for the mean [e) value of @ which we will denote by 4,. We can write then OG 2D A Th (6-4) u dy (eo) u is obtained from equation (3-24) and naturally is a function of both y and t. Hence the expression for the mean value will be of the form 1 27 a2D The calculation of 1, from (6-5) is a lengthy and tedious operation which has to be performed in each particular case. This renders the method proposed: here practically inapplicable. A simpler approach could be based on the assumption that Um is pro- portional to the amplitude of the velocity at some arbitrary distance from the wall within the bed-layer. In other words we postulate that Mg =A, [u, | (6-6) 2) with A; a constant of proportionality and Up the velocity from equation (3-24) at an arbitrary distance, say y = D, from the theoretical bed, A_ now can be calculated from the equation 5 271 n2D if [ u(y , t)dydwt A = fo) fo) 5 pata ee any aT (6-7) 47 Du | y=D A large number of values could be obtained from (6-7) for a wide range of variation of the independent variables, and then averaged out. This average value may be considered as a universal constant and be used to calculate u, from (6-6) With Up = [ally =n: The work could be substantially reduced by the use of a computer; otherwise the operation is not much easier than the more accurate one mentioned previously. Its main advantage over the latter is that it has to be performed only once. Because computer time was not available when the study reached this point, a simpler but less accurate method was used to calculate u,. This method is adequate at least in establishing an order of magnitude, The method consisted of a numerical integration of equation (3-19) in which the two functions f, (y) and fz (y), as we have seen in section 3, were of the form ? ASSsy fH (y) = .5e apD 2/3 By OY) Sod (By) One may observe that for given values of the parameters a8 and gD both f;(y) and fo(y) are functions of the ratio y/D only. AS a characteristic set of these parameters, the mean values from the twenty-seven runs men- tioned in a previous section were used, The ratio y/D was made to vary in increments of .4, beginning with y/D = .2 and ending with y/D = 1.8. The angle wt was varying between 0 and 27 in increments of 7/12 beginning with wt = 71/24 and ending with wt = 47 7/24. Typical velocity profiles constructed this way and corresponding to different values of the angle wt are shown of Figure 10. The value of the constant A5 obtained from the integration of these profiles was found to be equal to .618, Equation (6-4) can be written now as Co = .618 nee (6-8) This is the final result of our study. When c, from (6-8) is intro- duced in equation (6-2) the desired value of Qp is obtained provided, of 22 course, that the velocity distribution U(y) of the secondary flow is known. In actual cases it is hard to predict the character of such secondary motions because in general they depend upon local conditions. It is, however, possible to deduce reasonable estimates of their behavior by statistical methods based on long time records. Since the wave char- acteristics themselves are usually evaluated by similar methods it be- comes evident that the error in the estimated values of the parameters entering our problem as independent variables has a bivariate distribution. This, of course, reduces the accuracy of the results. There is at least one case, though, in which the steady mean motion U(y) in a body of water is induced by the surface wave itself, This particular case will be described in the following section, 7. A PARTICULAR CASE OF THE SECONDARY DRIFT This type of a second-order drift which is generated by the surface- waves in a direction parallel to the wave propagation has been originally studied by Stokes (1851) and more recently by numerous investigators, The works of Bagnold (1947) and Longuet-Higgins(1953) wil be singled out because in addition to their scientific merits offer a rather convenient application to the problem at hand. Associated with this secondary motion is a steady mean water particle velocity which usually is called '"mass- transport velocity". Stokes' expression for this velocity, which we will denote by U, is of the form a®wk cosh2k(y +d) a?w coth kd WS 2 sinh@kd i 2d S72) where y, is measured from the mean free surface negative downwards, The only necessary and sufficient condition to be satisfied for the derivation of this expression is that the flow is irrotational. Longuet-Higgins pointed out that the requirement of small amplitude surface wave, as was suggested by Stokes, is not necessary. According to Stokes’ theory, the velocity near the bottom is negative (in opposite direction to the wave propagation) and for typical values of the product kd it increases with distance from the bottom attaining its maximum positive value at the free surface. Bagnold (1947) on the other hand has shown experimentally that the actual behavior is quite different. He observed a strong forward velocity near the bottom and a weaker backward velocity at higher levels. This confirmed the belief that Stokes’ theory was not accurate, as it has been evidenced from older experiments in which. forward velocities were observed both near the bottom and the free surface and backward motion in the interior. A more reliable theory which confirms the experimental re- sults has been developed by Longuet-Higgins (1953). The two fundamental assumptions are that the mean motion is periodic in time and that it can be expressed as a perturbation of a state of rest. The theory recognizes the existence of three distinct regions, the first near the free surface, the second in the interior and the third near the fixed boundary. Associ- ated with each of these regions is a particular mode of secondary motion. 23 In the present case our interest is concentrated on the boundary layer near the bottom only. Omitting the lengthy procedure of Longuet-Higgins’ rigorous derivation it will suffice to present the final results for the case of progressive waves in water of uniform depth. The first-order motion of the fluid is described by means of the irrotational theory so that the horizontal component of the velocity at the boundary may be ex- pressed, as we have seen in section 3, as e = awSinwt (7-2) The "mass-transport velocity" can then be obtained from the second-order approximation. Ta eae £ (w) (7-3) 4 4 sinh@kd Pe a where f (p) = 5-8 e Mcosu ra) ena (7-4) These are Longuet-Higgins"* equations (254) and (253) respectively. Consistent with our notation DiS, So that the expression for U may be written as a BR a =2 T= = ws - { 5-8 8’ cosBy +3. Bt (7-5) 4 sinh® kd where y, measures the distance upwards from the effective bed. It is evident therefore that the rate of sediment transport of a certain size can be determined by combining the "sediment-transport equation" (6-2) and the "mass-transport velocity equation" (7-5). A given set of surface wave characteristics, temperature and depth of water and grain size of uniform bed material is sufficient for the calculation of this rate. 8. SUMMARY OF RESULTS The independent variables entering the problem of the sediment trans- port by wave action are the following: The amplitude H, the length ) and the angular velocity w of the surface wave. The depth d and the temperature T° of the water (and consequently its kinematic viscosity y). The grain diameter D and the density pg of the granular material forming the bed which is assumed to be uniform, A method has been described in the preceding sections of this study which can be used to predict the rate of sediment transport associated with any 24 given set of values of the independent variables, The procedure to be followed consists of the following seven steps. a. The necessary condition that has to be satisfied for the method to be applicable is that the flow within the boundary layer caused by the surface wave is unstable. To test whether or not this condition is satis- fied in a particular case the graph in Figure 1 may be used. The roughness size ¢ can be taken equal to D while the velocity ug is calculated from ‘where a= - cosech kd = ; cosech = The point with coordinates D and u,/y is plotted in Figure 1 and accord- ing to its position relative to the experimental curve for the three- dimensional roughness, a prediction can be made about the character of the flow in the boundary layer. b. Assuming that the test in step a proved that the condition of instability is satisfied, we proceed with the determination of the flow field in the boundary layer. It has been shown in section 3 that the mean unsteady velocity in the layer is given by the expression : s te = [2 2 9G) = BenG Cos ey GY | sin(wt + @) where 9 = feoyat fi(y) sinfg (y) 1-f,(y) cosf,(y) It was also shown that _ 133y f(y) a 65. af D and 2/3 ey) = 2) (yy) f where eA = OR Ny ey c. The value of the parameter | y| is next calculated from equation (4-10); i.e. Og. 7 Pe ep) ee cle The velocity u is measured at a distance 0.35D from the theoretical bed, which means that Bet) fi (y) = 5.2 ap and 2/3 Eo(y) = .25 (BD) 25 d, With | Y] known the curve through the experimental points in Figure 8 can be used to obtain %. This curve is very closely approximated by the graphical representation 0 the equation 1/5 =Z 2712 = i in eal oh latyy Sais 2 cos E e. The oscillatory sediment transport rate qp in the bed layer which we may call the "oscillatory bed-load rate" is calculated from $ through equation (4-26), i.e. q fe) = epee Disea OO n73/2 4 B(p.-p,) iS) f, The concentration Co of the solid particles that at any time happen to be in a state of motion within the bed layer (of thickness 2D) is q = 1 ——— Cy .618 aD Gig) where up is the value of u from (3-24) at y = D. g. The rate of sediment transport per unit width in the direction of any incidental secondary flow described by U(y) will be 2D Ory Ree | U(y) dy (@) In the general case U(y) is an additional independent variable which has to be determined by methods similar to the ones used in the determination of the surface wave characteristics. In the absence, however, of any such flow the only possible steady mean motion in the boundary layer is due to the surface wave itself which is called the "second-order drift flow". This steady flow within the boundary layer is in the direction of the wave propagation, The expression describing the velocity distribution associ- ated with this flow is of the form = eee nad NIG -By -2By U = Oy Sine 5-8 e cosBy+3 e } as proposed by Longuet-Higgins. This expression can be substituted for U(y) in the above integral to calculate Qg. This is the sought-for value of the rate of transport of bed material in the direction of the wave propagation. 9. DISCUSSION OF THE RESULTS The procedure outlined in the preceding section leads to a quantita- tive answer of our problem. In trying to apply the method to an actual 26 case some caution is warranted with regard to its applicability and its accuracy. In developing the various relationships it was necessary to simplify the physical model by making certain assumptions. Unless these assumptions are still valid under the prototype conditions the method may need modification. The basic assumption which practically governs the entire study was that the amplitude of the surface wave was small (kH << 1); this permitted the linearization of the equation of motion in the free stream. A second assumption, but of lesser importance was that the depth of the water d was rather large and uniform (small bottom slope.) We wish to emphasize at this point that the proposed method is only meant to predict the motion of the bed material within the bed layer, Consequently it is not applicable in regions where the mixing is violent and where considerable amount of sediment is being carried in suspension as for instance near the surf zone and onshore. Another point that calls for attention is that the criterion of instability of the boundary layer is not well defined. As stated in more detail in Appendix A, there is some uncertainty regarding the slope of the empirical curve serving as the criterion of instability in the region 4 -1 : ; : ; : : : pa <6x 10 ft . (Figure 1). Until this uncertainty is removed, it will Vv : be advisable in this region to use the line with the steeper slope pro- posed by Manohar. The accuracy of the results, as one would expect, depends entirely upon the quality of the approximations made in the course of developing the various relationships. We shall examine these approximations of the procedure step by step and make suggestions regarding their possible improvement. a. This step deals only with the applicability of the method and was covered already in the discussion above. b. The velocity distribution in the boundary layer was determined experimentally in a flume under conditions similar but not identical to those of the prototype. The error introduced this way cannot be very significant and in fact it can be absorbed by the effect of the approxi- mation made in the subsequent step. c. The value of ¥ was calculated from the velocity profile at a distance y = .35D. This distance was so chosen because it has been proven correct in dealing with steady mean flows; yet there is no proof that it holds true in the case of an oscillatory mean flow also. The accuracy of the method may be improved by determining the distance at which ¥ should be calculated more precisely. 27 d. The value of § was obtained from the curve in Figure 8. Its accuracy depends on how closely the theoretical curve approximates the actual one. The exact shape of this curve can be confirmed only after a considerable amount of reliable data from the field becomes available. e. No approximation was involved in this step. f. The determination of the constant in equation (6-8) from the experiment with an oscillating bottom is not quite accurate. The error due to the value of this constant is not serious however, and can easily be reduced as more information from field measurement is obtained. g: In order to determine the Sediment, rate an’ the direction of Gehe flow the concentration was multiplied by the average discharge in the bed layer. This of course is only an approximation of the real mechanism. The concentration in the layer actually remains constant in a statistical sense only. The individual grains are continuously picked up and dropped by the flow. Therefore the solid particleS and the surrounding fluid are subjected to random accelerations and decelerations., The effect of the resulting inertia forces was assumed to be reflected by the value of the empirical constant. However the accuracy of the method may be improved considerably by refining the analysis of the force field in the bed layer. 10. CONCLUSIONS Surface waves are responsible for the formation of an oscillatory boundary layer near the ocean floor. For a certain range of values of the wave parameters the laminar boundary layer becomes unstable. In this case solid particles of bed material are brought to a state of incipient equilibrium, primarily as the result of the instantaneous hydrodynamic lift. The concentration of solid material, in a thin layer adjacent to the bed (the bed layer) which at any instance is thus made free to move can be determined by the method developed here, The rate of the transport of such material in the direction of a secondary flow can be calculated by multiplying this concentration with the average discharge due to the secondary flow in the bed layer. The accuracy of the method should be tested against actual field measurements. 11, ACKNOWLEDGEMENTS The staff of the Hydraulic Laboratory of the University of California and several students contributed in various capacities to the successful completion of this study, and the writer is very grateful for this assist- ance. He particularly expresses his gratitude to Professor Einstein for the encouragement and the invaluable advice throughout the various phases of the work. He also wishes to thank Professor Harder who gave freely of his time to advise with the design and the assembling of the electronic equipment used in the measurements, 28 REFERENCES Bagnold, R. A. (1947). "Sand Movement by Waves: Some Small-Scale Experi- ments with Sand of Very Low Density." Journal of Institution of Civil Engineers. vol. 27, p. 447. Chepil, W. S. (1958). "The Use of Evenly Spaced Hemispheres to Evaluate Aerodynamic Forces on a Soil Surface," Trans, A.G.U. vol, 3, pp. 397-404. Einstein, H. A. (1948). "Movement of Beach Sands by Water Waves." Trans. A;-GoU, wol, 2, NOS 5S; D2, OSS=G55< —-—--- (1950). "Bed-Load Function for Sediment Transportation in Open Channel Flows) Use. Dept nuAguice (S.C. Ss. mechnicals Bultecin No. 1026. Einstein, H. A. and A, El-Samni (1949), "Hydrodynamic Forces on a Rough Wall." Review of Modern Physics, vol. 21, No. 3, pp. 520-524. Jeffreys, H. (1929). "On the Transport of Sediments by Streams." Proc. Cambridge Phil. Soc. vol. 25, pp. 272-276. Kalkanis, G. (1957). "Turbulent Flow Near an Oscillating Wall." Beach Erosion Board Technical Memo No. 97. Karlsson, Sture K. F, (1959) "An Unsteady Turbulent Boundary Layer". Journal of Fluid Mechanics, vol. 5, pp. 622-636. Lamb, H. (1932). "Hydrodynamics." Sixth Edition. New York Dover ' Publications, Lane, E. W. and A. A. Kalinske (1939) "The Relation of Suspended to Bed Material in Rivers," Trans, AviGaUs vol. 20, pp. 637-641. Lhermitte, P. (1961). “Movements des Materiaux de Fond Sous 1'Action de la Houle." Proc. VII Conf. on Coastal Engineering.vol. 1, jo, Allo LGil Li, Huon (1954). "Stability of Oscillatory Laminar Flow Along a Wall." Beach Erosion Board, Technical Memo No. 47 Lin, C. C. (1955). "The Theory of Hydrodynamic Stability." Cambridge University Press, =----- (1957). "Motion in the Boundary Layer with a Rapidly Oscillating External Flow."" IXth International Congress for Applied Mechanics, vol. 4, pp. 155-167. 29 REFERENCES (Continued) Longuet-Higgins (1953). "Mass Transport in Water Waves." Philos. Transactions Royal Society of London. vol. 295, p. 535. Manohar, M, (1955). "Mechanics of Bottom Sediment Movement Due to Wave Action". Beach Erosion Board Technical Memo No. 75. Additional References Boyer, R. H. (1961). "On Some Solutions of Nonlinear Diffusion Equation" Journal of Mathematics and Physics. vol. 40, p. 41. Coles, D. (1956) “The Law of the Wake in the Turbulent Boundary Layer". Journal of Fluid Mechanics. vol. 1, pp. 191-226. Cornish, V. (1954). “Ocean Waves." Cambridge University Press, Eagleson, P. S. and R. G. Dean (1959). "“Wave-Induced Motion of Bottom Sediment Particles.'' Journal of Hydraulics Division, ASCE, vol. 85, No. HY 10, Paper 2202, pp. 53-79. Eagleson, P. S., B. Glenne, and J. A. Dracup, (1963). "Equilibrium Characteristics of Sand Beaches." Journal of Hydraulics Division, ASCE. vol, 89, No. HY 1), Papex) 3387\ppeissqom Grant, U. S. (1943). "Waves as a Sand-Transporting Agent." American Journal of Science. vol. 241, pp. 119-123. Li, Huon (1954). "On the Measurement of Pressure Fluctuations at the Smooth Boundary of an Incompressible Turbulent Flow." Univ. of Calif. IER. Technical Report No. 65. Sverdrup, H. U., M. W. Johnson, R. H. Fleming, (1952). '"'The Oceans" Prentice-Hall, New York Townsend, A. A. (1961). "Nature of Turbulent Motion."’ Handbook of Fluid Dynamics, V. L. Streeter, Editor, 1st Edition. McGraw-Hill, New York Schlichting, H. (1955). "Boundary Layer Theory". Pergamon Press. 30 47 aa Cow ease ["} k \ THREE- DIMENSIONAL 4 ROUGHNESS ro x,o From Li, fixed bed v + Loose bed or 4;@ Fixed bed 3 }Monoha r Smooth boundary ''© Present Study Avis Gume nO: 2 sae Shen Blo" FIGURE | - CRITERION FOR TRANSITION FROM LAMINAR TO TURBULENT FLOW WITH OSCILLATORY MOTION All grain sizes 3| 0.02 0.03 0.05 0.07 0.1 0.2 0.3 0.4 O06 FIGURE 2- SMOOTH PLATE- MEASURED VELOCITY DISTRIBUTION f, (y) O31 .67 1.87 277 O31 1,00 1.26 227 0625 1.00 1.47 262 0625 .504.50 455 104 501.95 290 104 .67 1.65 267 061 0.03 0.06 0.1 0.2 0.3 0.5 1.0 FIGURE 3 - TWO-DIMENSIONAL ROUGHNESS MEASURED VELOCITY DISTRIBUTION fy) 32 (A) 'y NOILAGIYLSIG ALIOONSA GayYNsvaw on) EXO) vO zO ne) 600 Zi i O22 voEe cel 9s" | Ve ZI'1 78° BG) 7 | eel £38" —SSINHONOY TIVNOISNAWIG _ 200 200 G26 G26 Gl'2 2, IS'S IS’S Olx &q V-6EE V-SEEe V-pze v-22¢ V-9IE V-Ele NAY SSYHL-v SYNDIS 10'0 : 33 (A)?} JAIHS SSVHd G3SYNSVAW (A) #7} 14IHS SSVHd GSYNSVSW SSSNHONOY IVNOISNSWIG-OML - 9 JYNols 31V1d HLOOWS -G 3yYnNdIS oo oz os o€ oz (or 20. 036S'0 eee ooe 0:02 oo! 08 09 Ov oz oO! / / ¥ / y / ay =p o'2 YVNIWY1 =i 191 L3YOSHL lee Ip =a oe | (eo) m a", Ov We | 0 ees | |= YVNINY oy Sa 5 WOlL3YO3 o: (SSS 09 OA ZAI ok st 9 aml Ltt 08 Va - — ab IDE kg 4001 a/n 4 f) A +—- Lt 7 V, ek) SO= (4 eee oz Zz —— == i | = la 00+ 8b 29:0 ez a lv1 00°! 222 v ze'l ¢8'0 Elz © 400 ™ 12) NnY 0'00! 34 100.0 70.0 4.0 THEORETICAL FIGURE 7- THREE-DIMENSIONAL ROUGHNESS MEASURED PHASE SHIFT f(y) 35 Ol ool’ el g *g *y OlO” JO NOIVNINYSL3S0 - 8 JYNSIS ® 100’ 1000" 20000" 7 Tat mae I T Teal Ke) ‘tdg O01 G2°6 “WO X GIL WeOlX1G°S O1l Bu = 7: S3ANND TVVOILSYOSHL exe) 100° 1000° ool Ql 0.2 0.4 0.6 1.0 2.0 40 6.0 10.0 200 40 FIGURE 10 - TYPICAL UNSTEADY MEAN VELOCITY PROFILES AT wt = (2n-1) 7/24 37 2 FT. BRASS PIPE, 1" 1.0. RUTISHAUSER HEAD WITH DIAPHRAGM REMOVED BRASS CASING BAKELITE INSULATING DISC ANIMAL MEMBRANE TUBE CENTRAL ELECTRODE MINERAL OIL 0.002" THICK, | EFFECT DIAM. BRASS DIAPHRAGM WATER ~ I.D. BRASS TUBES FLATTENED SECTION yo =—_ N x 4 OPENINGS (FULL SIZE) FIGURE J1- SECTION OF VELOCITY-MEASURING INSTRUMENT 38 APPENDIX A STABILITY OF AN OSCILLATORY BOUNDARY LAYER It has been shown in section 3 that the wave motion at the outer edge of the boundary layer may be approximated by a simple harmonic motion. The stability of this layer and its transition into turbulence plays an im- portant role in the problem associated with the transportation of sediment. In a laminar oscillating stream the predominant force induced by the flow on the particle is a tangential drag. When this force becomes larger than the force resisting motion the particle will begin to move. A criterion of initiation of movement then can be set as TAY K (Oz Pe) g tan 8, (A-1) where @ is the angle of repose of the particle. The shear stress T at any distance from the wall may be determined from equation (3-15). It is evident that the maximum value of T occurs at the wall so that ar = LBu, (A-2) max ™ By ang Therefore the criterion of motion in a laminar boundary layer can be set as K D ee vg = [Eiperve) sD tan 61 J Og This is Manohar's (1955) equation (19). In an unstable boundary layer on the other hand the predominant force, as we have seen in section 4, is the hydrodynamic lift which is associated both with the mean unsteady flow and the turbulent perturbations. The mechanism through which motion is induced is not only different in character for the two cases but also in the unstable case it is more intense by several orders of magnitude, The purpose of Li‘s investigation (Li, Huon 1954) was to determine the factors and relationships governing the transition of an oscillatory laminar layer over smooth and rough beds. The mathematical model being so complex, a theoretical approach was ruled out from the outset. The effort therefore was concentrated to obtain empirical results experimentally. The study in the laboratory of the boundary layer created by a surface wave would re- quire equipment of very large size. It was instead considered expedient to investigate the stability of the boundary layer near an oscillating wall in a body of water at rest. It is apparent of course that the patterns of flow on the prototype and in such a model are by no means identical but the reasoning was that the critical values of the governing parameters will be very closely approximated by each other. Following is a brief description of the apparatus and the procedure used in Li’s study. A glass walled flume 12 feet long, 1 foot wide and 3 feet high was used. About 4 inches above the bottom of the flume a plate, 78" x 11-1/2" was mounted on a series of rollers so that it was free to move in a plane parallel to the bottom. Through a series of steel cables and levers the plate was connected to a 1/2-HP AC motor which was mounted external of the flume. By means of an eccentric arm the rotating motion of the motor was converted into a reciprocating one which very nearly approximated simple harmonic motion. The motor was equipped with a speed reducer which allowed the period of the oscillation to vary from 1 to 150 cpm. By varying the eccentricity of the arm the amplitude of the linear motion of the plate could be varied from 2 to 18 inches. The amplitude and the frequency of oscillation in the experiments were made to vary within ranges so as to simulate prototype conditions associated with waves of 0.4 to 60 seconds period and 0.5 to 10 feet height in water about 60 feet deep. During the testing the water depth was kept at about 2 feet. The shortness of the flume and comparatively shallow depth of water tended to cause the formation of a standing wave. In order to reduce this wave to a minimum a series of vertical baffles together with a set of heavy floats were used, Three series.of tests were made in this flume. The first, designated as the smooth series, was carried out with the surface of the plate waxed, The second, designated as the two-dimensional rough series, was carried out with the use of half-round wooden strips or steel rods as roughness. The third, designated as the three-dimensional rough series, was carried out with the use of sand or gravel as a roughness. In all cases each individual run was carried out with a uniform roughness. That is to say, all roughness elements for any one run were of the same size. The procedure was to fix a particular roughness to the plate, choose an amplitude of oscillation, and then increase gradually the frequency. The type of the flow regime was determined visually by dropping potassium permanganate crystals from the water surface to the bottom. As the crystals dropped to the bottom they left a trail of dye. Once at the bottom they would oscillate with the motion of the platform. Continuing to dissolve the crystals would leave a clear back and forth trail as long as the flow was stable. The flow was defined as unstable when the trail left by thecrystals would break down and mix with layers above. The recorded frequencies and amplitudes at the critical limit together with the size of roughness and the water viscosity were combined to form a critical Reynolds number. The re- sults obtained in these tests as supplemented by Manohar may be summarized as follows: According to its performance the boundary can be classified as hydraulically smooth, as rough and as a transition from smooth to rough. The boundary behaves as hydraulically smooth even with two and three- dimensional roughness provided that the parameter B, where ¢ is the rough- ness diameter is smaller than a certain value; B of course is equal to A - The range of B, for the three classifications determined from the experiments is the following. Two-Dimensional Three-Dimensional Roughness Roughness Smooth boundary Bo S (87S Be ello Transition ONS S. Oa S abot HISSE= Be <0249 Rough boundary Tela) eS Be sGae) SS Bhs (i) For the hydraulically smooth boundary it was found that the critical value of the Reynolds number defined as was constant and equal to 400. (ii) In the transition region the Reynolds number was of the form with constant critical value for the two-dimensional and three~dimensional roughness; namely 640 two-dimensional Z iT] 104 three-dimensional Zz I (iii) In the rough region the governing parameter was not a Reynolds number but the dimensional quantity wee On2 v which was found to have the critical values of 0.2 UpE 4 ars FE BoSA x kO two-dimensional 0.2 UoE tes 4 , = = 1.78 x 10 three-dimensional The summary of the results as reported by Manohar are shown in Figure 1% The explanation given by these two investigators to the rather singular behavior at the rough region was that the transition from laminar to turbulent is mainly due to the instability of the flow along the wake formed between the individual particles. The characteristic length then is the size of the wake which, although a function of the roughness, diameter, is not necessarily proportional to it. This explanation, although reasonable, is not quite convincing. The writer's reservations were confirmed by some observations made during a more recent phase of the investigation. These observations were made while measuring the phase shift on the vertical in the same experimental flume. The method used involved again the use of dye which this time was injected from above through a thin brass tube. It was observed that the flow was definitely unstable at a value of “° equal to i aclon for a two-dimensional rough- v ness with ¢ = .0625'. Moreover unstable flow conditions were observed while experimenting with three-dimensional roughness of ¢ = .017' when the value of Ee) was about equal to Ons These points when plotted in Figure 1 demonstrate that the constant value of the critical Reynolds number as defined for the transition regime may well be extended to cover the rough case too. This implies that in Li*s and Manohar's experiments the flow in this region was already unstable before it could be established as such from observations. There is a possibility that the crystals and the high-density dye solution were confined within the layer of the dead water under the theoretical bed. The fluid in this layer oscillated back and forth with the plate and as long as the velocities were small it never had a chance to spill over the crest of the roughness elements and mix with the flow above. So even if the flow were unstable there was no way of detecting it. On the other hand, one may argue that the more recent observations do not describe the behavior of the real model either, be- cause the observed premature transition might have been triggered by the tube itself. We want to make clear, however, that the tip of the tube was maintained at a level several millimeters above the bed at a distance where the amplitude of the fluid velocity was negligible while the first indication of instability was observed near the bed. In concluding this Appendix we would like to point out that this phase of the problem needs further investigation so that more reliable information will become available. * Figure number refers to figures following main text. APPENDIX B VELOCITY DISTRIBUTION IN OSCILLATORY BOUNDARY LAYER This section contains the description of the experimental method and the results obtained in the phase of the study which had as its objective the derivation of an expression describing the velocity dis- tribution in an oscillatory boundary layer. The linearized equation of motion from classical hydrodynamics may be used to describe the flow field in the interior, Within the boundary layer as long as the flow is laminar the velocity field can be described to the first approximation by an ex- pression of the form u = Uo {sinut = oy sin Cwt-By) } (B-1) where ug is defined as the amplitude of the velocity at the wall from irrotational theory. When the flow in the boundary layer becomes unstable the only way the field can be described is by empirical methods. The experimental work involved may be extremely difficult; however, it becomes much simpler if one postulated that the unsteady mean flow in this case can be described by an expression of the form u = ug 1Sinwt-f,(y) sin | wt-f5(y) (B-2) fof 1 . 2 in analogy with the stable solution. Equation (B-2) may be written as u u UD | = S|) 2) sanat =|= | sin | wt-£0) | (B-3) Uo fe) Uo font a { where ball | =a! and 2 | = f1(y) Uo Uy The purpose therefore of the tests was to determine the functions f(y) and fo(y). The structure of equation (B-2) suggested the possibility of determining these functions by the convenient method of measuring the velocity amplitude and the phase shift in a flume where the bed oscillates with an harmonic motion of the form uy = upsinwt relative to the water at rest. The flume and the rest of the equipment used in these tests has been described in detail elsewhere, Li (1954), Manohar (1955), Kalkanis (1957). In Appendix A we gave a brief description of the main apparatus and in this section we will give the necessary information regarding the opera- tion of the additional equipment used in the tests. The two functions f,(y) and f,(y) can be determined independently and indeed in some cases they were so. However, the device used in the measurement of the velocity amplitude at some distance from the oscillat- ing wall was simultaneously measuring the phase angle of the velocity at this level relative to the wall. In other words the same record could be used to determine both fj(y) and fo(y). This instrument is essentially a modified pitot tube shown diagrammatically in Figure 11. The basic prin- ciple of its operation is that the differential pressure at the two tips causes the diaphragm to deflect; in doing so it modulates a high-frequency signal by altering the capacitance of a Rutishauser pressure pickup. The modulated signal is rectified by a discriminator into a voltage which is proportional to the differential pressure at the tips of the instrument and consequently a function of the incidental velocity. Therefore the excursions of the needle of an oscillograph which is activated by this voltage will give a measure of the local velocity. The instrument was calibrated dynamically by oscillating it with a prescribed simple har- monic motion in water at rest and correlating the amplitude of the needle excursions with the amplitude of the velocity of oscillation. The tests were made on a smooth plate of a high-gloss finish as well as on rough plates with two and three-dimensional roughness fixed on them. The vertical distances recorded during the tests were measured from the crest of the roughness elements. These values were subsequently corrected to account for the distance between this level and the theoretical bed. The correction in all cases was equal to 0.2D. The values of the function fi) resulting from the analysis of the experimental records are listed on Tables I, II, and III. The graphical representation of the results is shown in Figures 2, 3 and 4, The exponential dependence of the velocity on the distance seems to describe the data better than any other simple relationship. A small number of points only, obtained in some character- istic runs, were plotted in order to make the graphs readable. However, for the selection of the most representative relationship the complete data was used, The experiments with smooth wall indicated that the proper character- istic length for the normalization of the argument of f(y) should be the amplitude of the oscillation. In_a previous report (Kalkanis 1957), it was suggested that the parameter = should be used for this purpose in analogy to the laminar case. The original work on which this suggestion was based consisted of three runs only reported here as runs 111, 112, 113, a very inadequate number. As part of the present investigation, which has as its main objective the study of the rough case, four more runs were made with a smooth wall (114, 115, 116, 117) in order to test the reliability of the instrument. The second set of measurements was made more than a year after the first and considering the fact that the equipment in the meantime underwent certain modifications, the agreement of the results between the two sets was remarkable. In any case the adoption of the amplitude as the characteristic length was strongly sup- ported by the experiments with a rough wall performed more recently. By adopting a similar purely empirical approach the analysis of the data obtained in these tests revealed that the more suitable length scale to be used with f,(y) was the parameter a8D. The forms that most closely approximated the experimental data were = 2 sz 108 fi(y) = eee for two-dimensional roughness and _ 133y 181 GY) = soe ep for three-dimensional roughness. It is understood that we do not claim that these expressions describe the real physical model; on the other hand we believe that the flow field described by these equations cannot be very much different than the actual one since a considerable amount of data was used in their deriva- tion. The implication is that the calculated values by means of these functions will be as close to reality as the form of equation (3-15) permits. In the determination of the function fo(y) the characteristic length for all conditions of roughness seemed to be the parameter 4 of the laminar case. The phase angle seemed to increase as a power function with distance. The empirical relationships obtained as the results of these tests are the following: 5S) (ayn Smooth wall f5(y) and 55) (any Two or three-dimensional rough wall. f,(y) The phase angle was measured by two different methods; in the first it was obtained directly from the velocity measurement record. The time at which the flow was changing direction was registered on the time scale of the velocity record while the time of change in direction of the plate motion was recorded by another needle of the oscillograph. The time interval between these two changes of direction after having been aver- aged out for a number of periods gave the desired phase angle. In the second method the record of the change of flow direction was obtained visually; through a thin brass tube a dye streak was introduced into the flow; the change of direction of the streak at any level was observed and it was recorded by means of a push button arrangement; the motion of the plate was recorded as above. This method is more accurate be- cause the equipment it employs is very simple; but on the other hand it is subject more to personal bias. The two methods were checked against each other in tests with smooth wall and the good agreement of the re- sults was a proof that either one may be used with equal confidence. The first method has been used to measure the phase shift in the smooth case and with a wall having a two-dimensional roughness, while with sand as roughness the phase shift was measured by the dye method. With both f1(y) and fa(y) determined equation (B-3) may now be written as 5 1/2 I+ tqGy) n= 2tialGy) cos £2(y) | sin(wt-6) (B-5) As paar [ fi(y) sin fa(y) DS EGNOS LEG) ] APPENDIX C DETERMINATION OF A,, B, AND 1], The method used for the determination of A,, By, and Ne has been outlined in section 5 of the text. What we intend to do here is to describe the experimental procedure used in connection with this method. The same flume which has already been described in previous sec- tions was used, but with some slight modifications. The method called for actual measurement of the quantity qg. By definition qp is the rate at which sediment near the bed crosses a section of unit width positioned perpendicular to the wave propagation. Our experinental procedure was based on the proposition that the flow field and consequently the magnitude of dp when the bed is fixed and the boundary-layer flow is caused by a surface wave are the same as when the bed is oscillating with a simple harmonic motion while the water surface is at rest. The validity of this argument will be discussed later but for the time being let us assume that it is correct. The oscillating plate had to be modified so that measurement of q,g could be made possible. The plan view and longitudinal section of the modified plate is shown in the sketch below The procedure is self-explanatory. The space between the trays was filled with loose sand of a given size. The plate then was set into motion and after the completion of a number of periods it was stopped, and the amount of sand collected in the middle tray was removed and measured, A wire screen was placed on top of the trays so as to be flush with the rest of the sandy bed to achieve uniform roughness conditions. Three sizes of sand were used each with three different amplitudes of oscillation. Thus 27 runs were made altogether. The size range used was psduel cs) 1D) < #10 mesh #10 < OD < # 8 mesh #38 < OD << # 6 mesh The average value of D for each of these ranges were 5.51 x On feet, ToS Se one feet and 9,25 x Oe feet, respectively. As a matter of fact, these were exactly the same sizes used in the measurement of the velocity and phase shift distribution. The size of the sample in each run was measured volumetrically. No distinction was made as to whether the material in the tray came over the rizht or left edge. Each run was repeated a number of times and the measured quantities were averaged out. The bulk volume of the sample was converted into dry weight by multi- plication through a coefficient determined experimentally. Since the width of the flume was equal to 1 foot, the dry weight of the sample when divided by the period and the number of oscillations completed during the testing time gave the value of dg. Next equation (4-25) was used to determine 6. This equation is of the form ate q3 Pe 33/2 =) a joe SENT) Vs &(ps-Pf) qB 32. p For given values of ae the ratio TF is proportional to D Herein Ps bee op 3/2 OE was equal to 2.63 which means that = = 1190 D . the graphical representation of this equation shown in Figure 9 was used to calculate $ from measured values of qp. On the other hand ¥Y was calculated from equation (4-10) which is Ps—P D Wife) aeons + Pr a In fact, the amplitude (icilg was used instead of a because what we were really interested in was the value of |v]. A pair of values of the parameters $ and ¥ were thus obtained for each run, These values were next plotted against each other as shown in Figure 8. Although there is considerable scatter, it seems that the experimental points can be reasonably represented by the empirical curve drawn througn them. We may claim therefore that this curve expresses the functional relationship between 6 and ¥Y. On the same figure a family of theoretical curves was drawn as it has been explained in section 5 of the text. These curves are the graphical representation of equation (4-27) in which the values Ax and Bx were equal to unity. Each curve corresponds to a particular value of ‘Ilo ° One may observe that the theoretical curve with Ss 1.5 is very similar to the experimental one but offset both in the Heretical and the horizontal. A parallel translation of the latter by a factor 30 along the horizontal and by a factor 4 along the vertical makes it prac- tically to coincide with the former. This means that the values of 6 and Y, satisfying equation (4-27) with Ax = 30, Bx = 4 and No SiS under any set of experimental conditions will be very close to the values of the same parameters calculated directly from measured quantities. In conclusion we set forth the claim that a theoretical curve constructed from equation (4-27) with the constants as determined here could be used to describe the relationship between the two functions $ and ¥, Several reviewers in the past and more recently P. Lhermitte (1961) are rather critical of Li's and Manohar's experiments. More specifically they express some reservations about the applicability of the results in an actual case on the ground that in the experimental flume an inertia force is induced on the particle which in the prototype is absent. Their contention is that this force is significant and consequently cannot be ignored. Since the same flume was used in the present study one may anticipate similar criticism especially in connection with the determina- tion of qp from direct measurement. We believe, however, that the criticism is not fully justified be- cause this inertia force under the average experimental conditions is indeed small compared to other forces acting simultaneously on the particle. The maximum value of the angular velocity in the set of runs with a = 1.25 feet was w = 1.86 rad/sec. corresponding to a maximum tan- gential acceleration of aw? = 1.25 x 1.862 = 4,32 ft/sec? which is much smaller than g. The tangential force could have some effect in setting the particle into motion if it were in phase with the lift in which case the conbined effect could not be ignored. Since the two forces are 90 degrees out of phase the instant one reaches its maxi- mum value the other is practically equal to zero. Therefore it is justifiable to base the condition of equilibrium on the balance of the vertical forces whose absolute value is relatively large and ignore the effect of a much smaller horizontal component which is fully out of phase, ; geht ie hy : ay 3 vo BE Us | SEEASG wae £% : Mag a la] ; ae ay ies UT ee APO RY haa! oe ey i "1 Pager a ih ey Niki fl H oF tink sik i i i fi APPENDIX D TABLES a= =e 280° 8°sTt O°T? 0 - BYES 66°83 od sR O°TE =O oS T60° €°Pt 0°6T 68°C £9°9 ae == 0°€ C3°a cO°L 960° 8°OT O° LT oS o8°s LEIS €°€ 0°0e 9L°S GL1’°9 cit’ (So TES O°ST ee SS°S OZT° 0°ST 0°8sT 79°C c3°¢ TéT° 8°6 0° eT g¢9°S €6°R 927° Speilaes O°LT 09°S c6°r 9ET° €°s O°rT 59°C $9°R O€T° L°Ot 0°9T cEe°s SO°r SPT’ 2-2) 0°6 €v°? SER GET” 0°OT O°ST BO° cT’eé TOT * e°s o°L AG are 90°F Ser f° O°rT 80°S ca°a Z6l° LPS o°s | 6E°S LEAS €Pt’ L°3 O° eT | 6E°S 8r°e ZOT° 0°s O°ST | 8t°? 6T°€ €9T° €°L O°Tt gO gOT | Cees 06°2 6LT° L°9 0°OT On x x Gn Z 4s 12 sq = A 4 c0T OT | On x x | Zz z 4F t-33 os = g 98S/2F EL°S = On | 70 ag 2 k K D8S/zF g-Ol ¥ 90°T = A des/per TE°h = | JolL9 = L 37 COT = ®& POT :NNU j {-13 062 = 9 288/13 19°% = On 98S/e4F c-O1 ¥ 90°T = & oeS/PeI Si°T = CO 7 i Iol9 = L 37 OS°T = GOT cs°e ZE° et == = 0°Or x 8r°e ZE° OT == == O°TEe 90°E€ 99°L == == 0° €@ -- -- got’ Ser 0°02 - —— GL°S@ EE59 LIT’ L°@t 0°6T 80°e PG°L ¢co° 0° €% 0° €% == =o SIT’ 0°ST 0°sT 16°? €Z°9 960° 0°6T 0°6T om ee ToT? (SP 101 O°LT | €9°S G6°R TOT’ O°SsTt O°cT 2S 2 ZET° L°OT O°9oT | 09°S 6S°P ZOT° O°rT O°rT c9°s 00°s => 25 O°ST €P°S I9°€ 9ST ° O°Tt O°TT a == €PT° £°6 O°FrI | TEs C6°S CPT’ 0°6 0°6 LY°S €e°F 6hT° L°3 O° €T 60°Z 29°2 ae? o°s o°s aS == OST’ 0°s 0°eT 60°C 0E°s G0d° O°L O°z $O°C 99°¢€ sm Ta O°rt PL°T L6°7 06° 0°9 0°9 co =. WANES L°9 0°OT = OT OT e0T got Bon e e On x Gn oe aise . e ay = 400 kg én kK k 1 4g 74 & {-13 ese = 008/33 IS°E = On t-1F sze=g D8S/4F 8Z°S = On 208/744 g-OT * 90°T = A des/pel FE°Z = ~C deS/24F c-OT X 90°T © & oes/per 86 °% = © IpL9 = L 37 OS°T = #8 €OT :NNU Io49 = L 3¥ 0OO°T = ® TOT T'IVM HLOOWS I FIavVL TTVM HLOOWS I A&TIEVL Zee €Z° TT == a= 0°Lz bS°E ST°6 ¢Lo° 0°%s 0°%e 18°Z LO°L 660° O°LT O°LT OL°S LO°L T60° O°LT O°LT IS°s 66°F 60T° 0°aT 0°ST GSS PL°ES err’ 0°6 0°6 60°% T6°S POT’ O°L O°L 80°% 80°% €6T° o°s o°s on got got én zB 43 40 k¢ = a & t-37 91h =d 088/233 ¢_Ol ¥ PO°T = 4 IodL = L 298/33 €9°€ = On des/per g9°s = 13 0OO°'T = B LOT NNU Zee TS°1t -- -- O°Le TO°e €T°6 620° 8°8T 0°%% s0°e €1°6 T80° 8°8T 0°%z OL°S 90°L 060° S°rT O°LT os*s 86°F 921° €°0T 0°sI Tes PL°S LST° DON, 0°6 9T°S 16°S c8t° 0°9 O°L 86°T 80°% SIs" SP o°s OT OT On & £ Gn zB 4F 30 4g = rN & t — — — <= {-13 Slt =g 2eS/1F OZ°h = On 29S/e4F o_O X FOTT = A des/per 6g°s = © AofL = L alae MEO SG 90T ‘=NNY 6L°E 0% °LT 0F0° b°vP o°Le 99° 9S°ST 9S0° °Ze 0° LZ LS? € €%° OT 190° $°9% 0°2% 80°e T6°L 820° b°0% O°LT L8°S gc°s Z60° PPT O°st tard 6I°F Site 8°0T 0°6 ZE°S 96° 6ZT° °s o°2 So°% S£°S 69T° 0°9 o°s gOT ¢e0T On x x z Bs 1m kg én &k & {-13 sor =g 22S/3F IL°E = On D8S/zIF g-OT X FOTT = A Das/pel LE 'h = Iotl = L qF cs'O = B TTVA HLOOMAS I HIEGVL TIVA HLOOMS I YTAVL az°o+ “A = & peansvow :4A Das e's Le°6 -- c6's Z°rE 0°82 , wu L1°% 60°8 -- 80°sS Z° 6 0°€% oe aorsiney S6°T oL°9 -- aD BK O°8T €6'T st°9 - 98°f 2°22 O°9T ZL°T Zers Todo cul O°sT ce°s 06°83 -— LO°S Z°6E o°se 09°T 60°? mie Z8°S S° OT 0°OT LOG OL°L —s 98°F Z°PvEe 0°82 ZS°T €6°¢E 880° Lv°S? S°RL 0°s coz £9°9 a7 Cr’ 2°62 0° 8% eb't 99°¢ Oot oes Z°s O°L 18°T apes =e BP°E BRS O°stT ZE°T gee got" aat Z°2t 0°9 EL°T #0°S Z80° ST°e Z°@e 0°9T 7a re Eas S6°T ett o's Z9°T 96°F OIT’ Suece cee: O° ET oe oe ovT” 9L'T o* Ot ORY, 6h'T 89°€ TPr° oe°z Z°9T 0°0T ee == Tee o9°T e°6 ove ZE°T Zee SrT° Z0°S Z°rT 0°8 = LUT LL°% Z6l* ew — BB 0°9 ‘ (als gol -- -- e6T° 6ST Z°1T o's i a 3 : ; ; ade a3 13 861 SPT Z°O1 0°? i A re) Zn R i “A 5; a0 e0t et 43 G2°S = ade A n ade ay ay | -13 LLZ =d 9eS/1¥ 9Z°T = On 400 nq én 74 k Ww, | 99S/243 o-ot X ST Sd des/per 18°T = © mis sore | Io09 = L a3 19°00 = @ zz: Nnu 43 vO'L = age q-1F LaZ=g 29S/2F 9Z°T = On = D®S/zg3F g-Ol X ZZ°T = A 2eS/PBI 9Z°T =~ 1009 = L 13 0O°'T = & pIZ :NNU -- €€°ZT -- ST°9 Z°vE 0°82 eh°s pes -- L8°? Z°vE 0°82 Tre eS‘ OT -- GZ" 2°6 0° €% b2°S L6°L -- ol? Z°6% 0° Ez 66°T €L'8 -- Ser Zhe 0°8T P2°2 T9°9 -- Sh°f BS 0°S8T Z0°% 10°s -- 66'S Z°2e 0°9T S6°T 90°9 -- Q9T°S 2°2z 0°9T L6°T £6°9 -- ors 2°61 O° ET 99°T b2°S 180° PLZ Z°61 O°€T 06°T T2°9 690° 26'S Z°9T 0°OT 6S°T faa SZT° Te°Z Z°9T 0°OT Tt TEe's 660° 9c°% Z°PL 0°8 SP°T 88°e LST Z0°% Z°eL 0°s 89°T LL’? 60T* yond Z° EL O°L sEe°T 09°s OLT° 88°T Z°eL O°L TSR leer Zer° 0z'z Z°21 0°9 8Z°T ee°€ T8t° PLT Z°2 0°9 -- -- OvT" 00°% Z°1T o's -- -- est’ 09°T Z°ur o°s Techies Seago = ee ee ees = -- -- Zee" SP'T Z°OL 0°? OT g0T got On x x OT OT OT z age VE a5 * g ey ey 417 sg my i ren Wr =p — . Jade 43 43 cl Sy : : 10 kd a k k f oe Sr | 13 eg°S = ade 43 Z0°'L = Adz q-3F 098 =¢ 298/33 BG°T = On [-1% €12 =d 208/33 IG°T = On 28S/z4F g-Ol X 2Z°T = 4 oes/ped GT's = & 28S/z43 c-OT ¥ Z°T = & deS/pel ZB°T =~ 1009 = L 373 OS°O = ® Tie ‘:Nnu Io09 = L 23 8°00 = & etz :NnU ¥e 2: “43 TEO° = a (Ff) “43 TEO’ = a (F) SSHNHDNOU ‘IVNOISNEWIG-OAL II FIavL SSHNHONOU ‘IVNOISNAWIG-OAL II ATaVL az°o0 + “A =A peaunseow 2K c° tr 0°6¢ OFS 18° OT 60° €S°Z PPS 9S°6 Tét° EZ°S SOE 0°F% 12° SZ°s ZLT° Ze" s*te 0°6T r8°T r6°9 Zoe" Z9°l $°9% O° rT 06°T 91°9 L6Z" a O° IT LS*T LETS 9Z€" SZ°T S$ *02 0's TS*T ITS PLE" EIT $°8T 0°9 Sevl Ze°h Ter" To°T S°9T o'r Oe"T 90°F ZEr c6° S°ST ore g0T OT OT On x x x z= age 43 Vy 400 ag ‘a K & WK 1 ge‘oT = ade q-13 29% =g 998/33 LPT = On 28S/z3F o-0T ¥ SLOT = des/Per LPT = 1069 = L 1 00°T = ® zee: NNU 28° ZO°ST ieee 1Z°s S°SP O° EE ES°% LE" ST EST ° 96°T S*0F 0° 8% TERe ZL°It 902° BI OSS 0° €% ITZ OF OT ZEs" SceTe nce 0°6T Els Ib'°6 Sle" 8E°T S*8Z 0°9T 68°T Zb's 60E" PZ°T $°SZ O° eT 98°T OL°L Zee" ok EPR O°TT 8s°T OTL 69E" Orb Setz 0°6 PS°T LL°9 cee" 66° = S02 0°8 LS°T bPO eer" c6° ss S*6T ay g0T — g OT g0T On x x x = age 13 z 4.02 4q én A A ihe 13 9°02 = age : q-13 oc€ = d 0eS/1F E€°s = On 988/243 ¢_0T ¥ £80°T = & des/Pet E6°S = OO I 6689 = L 1 00°T = Tee: NNU "13 G¢90°0 = a (FT) SSANHDNOU TYNOISNAWIG-OML II aATavL az°o + “A = A peanseou :“_ 9F°% br'6 -- -S6°S Z°PE 0°82 -- 90°83 -- 80°s Z°6Z 0° €2% r6'T 89°9 -- I2°h 2° hs 0°ST -- €1°9 -- 98°e Z°2e 0°9T FATE oe's -- pee 2°61 O°€T SChar LbCP ZL0° Z8°S Z°9T 0°OT PPT Z6°E S60° LE°? Z°PT 0°8 er PO's LOT* oe'2 Z° et O°L Pet LEE s0T* Zz Z°2T 0°9 -- -- Ze" SL°T Z° OT 0°? Ot OT OT oO g hs ey. == tle Bib a3 40 7X n m3 k WK 13 Lig = age t-13 912 =g DeS/3F EST = On 98S/zg1F ¢o-OT X S0Z°T =4 Des/pel Eg't =c? Iol9 = 1 17 190 = ® 912 :=NNU PI's L8°L -- BL°s Z°re 0°82 98°T ZL°9 -- €6°F 2° 6% 0°€% SET LS*S -- 60°F Z°hS 0°ST SET Sh -- bo°E Z°6T O°€T 6I'T BIR 690° PLS Z°9T 0° OT €Z°T Leas 860° Ore SPT 0°s 92°T T3°z O€T* 90°% Z°at 0°9 -- -- PET’ 06°T Z°1t o's -- -- LST €L°T Z° OT o'r Ol OT OT On g ey Sy age + 2 407 ‘qd on ri K by 1 Tes = ade t-13 os = d 29S/33 90°T = On 29S/z3F ¢o_OT X SOS*T =A OeS/Pel BS°T = Io19 = L VF es'0 = 8 cstz :=NNU “33 TeO° = a (Ff) SSANHDNOU 'TWNOISNSWIG-OML II STEVL az°o + “ha & az‘o + “KA mw & peinsvew :uy¢ 00° b6°6T TST° Bus SINAZ O°ZE L8°% OL°LT PET’ 8s°r s°6Ee 0°LZ G8°z 9°ST LLT° b9°T SPE 0°se cS°s GS° EI LIZ° OP*I S*6e O° LT Shs ZEST bos" LEE SoM. 0°ST 91°S 86°OT PLS’ LI°T S°bS 0°2T 80°S €S°OT OTE’ Ziel — Siv6e O°TT OT OT On Be & én ade 45 45 17 hq = & & wy 4F 00°IS] = age [-33 8th = 2 DeS/43 EZ°E = On | 28S/z43 ¢-OT ¥ SLO'T = oeS/Pel OF*h =~ | 1969 = L 43 GL°O0 = B 922 :NOU P2°€ S9°6T = 99° S‘6P O°LE 6I°s 219° LT 620° 6£°S «=S° bP O°ZEe 96°% 89°ST ili ZI°S =S°G6E 0°LZ PL°S 60°FT 6IT° T6°T S°SEe 0° €% 9S°S 06°2T IST’ Chita Gace 0°02 ses TL°Il OLT° 6S°T S°6z O°LT eE°S Z6°OT 8IZ° SP°l S°Le O°ST €1°% ZT° OT Tes" LE°l §«6s°ce O°sT S0°s €L°6 Ths" SEL «= S* BS 0°2T 90°S Se°6 692° 9Z°T S°EZ O°TT 06°T £6°8 0ze* TZ°T G°22 0°OT 18°T tS°8B Ore’ Q9T°T S°Te 0°6 Z8't PLoL OLE" SO°T S°6T O°L 08°T PEe°L ose” 66° S°sT 0°9 ¢0T g0T g0T On x x xX ade 45 13 40 Ag en K & wg 37 09°8T = ade a5 {[-1F LEE =g des/1F LG°Z = Nn 288/233 c-OT ¥ 60°'T = A oes/per EP's = © 1989 = L 33 GL°O = ® cee ¢NNU °3% $¢90°0 = a SSANHONOU TVNOISNAWNIG-OML (FF) II GTavL peansvou :%& zo°e ZS°2Z ZLO° 98°% S°6r O°LE T9°e SZ° 0% 980° LS°% = S* bP O°ZE zo°e L6°LT ZIT’ 8° = S*GE 0° Lz ST°e ST°9T 8ZT° S0°S S*cE 0° Ez 90°s 6L°FT Oot’ L8°T S°ze 0°02 eL°Z Zr ST rst’ OL°T S°6z O°LT EL°S IS°St b0Z° 6S°T S°1Z O°sT 19°2 90°ST 102° Toe G4 O° €T 89°S 69° OT ZSG° Qecrmenciwes, O°Il Cbs b2° OT $8z° Of'I $°2z 0°OT Ze°s 8L°6 rea" Pelee CaS 0°6 60°% 18°8 poe" €I°l S°6T O°L s0°z ars s9e° Lost — Goer 0°9 OT OT OT on § ey g én aqz ay pes 40 4g a k k we 13 be°'LT = age | 13 och =d E 298/33 Gz°s = On | 988/293 Ezot x 60°T = A 2eS/Ppel 0S*h =CO 1089 = L 13 0S°0 = B bee =: NNU 0zZ°e 99°22 - == T2° S°bs 0°Z? ss°e ZP°st -- PR's §«S°bT O°Ze Se°e Ge"oT 6S0° cso°s S°6e 0°Lz 0Z°e OL°FI oot’ We GOS 0° €% c6°2 9b° ST 8ZT° Ts*e S°se 0°02 91% TZ°Zt 92T° 8's SG *6z O°LT co°z 6E°TT 621° Ere GOs O°sT Z9°S 16° OT Ist co°s §6S°9% O°rT SE's 9S°0T ver’ L6°l S°SZ O°€T S0°% £L°6 961° ZB°t 4 =S*Ez O°IT £0°S 06°8 Tre" 99°T S°IZ 0°6 r6°T 10°8 GLz° IS°T S°6T orn es 80°% 99°L 8Lz° eT Ger 0°9 OT OT OT On & ce § age 13 bs 402 re) as i & wk 13 PEST = age E 6 -13 PIP = 20S/3F 98°T = On D9S/z35 eeat X 60°T =A oes/per ZL°E = ~M 1989 = L 43 OS'0 = ® eve =: NNU — “4% Gz90°0 = a (TT) SSHNHDNOU IVNOISNAWIG-OML II FIAVL D-5 az°o + WA = A SSANHONOU TVYNOISNGAIG-OML II TIav. pernseou : 4K Z's €L°8T r60° 6T°% s*Es S°Ze 96°% 86°9T Zot° 66°T S*8y S° LZ £8°% 8S°ST IST* €8°l S'bP S° Es Cie ES° PT €9T° OL°T S*TP S*0z T9°S 8h ET 8LT° ss°T Sse S°LT ss°z 80°ST Z08° Ib°l ‘be S°ET ZE°S £0°TT Or" 6Z°T S°TE S*OT F0°S 86°6 $92" LT°T §6$°82 S°L os*t £6°8 Zee" PO'T §S°Se S°r OT OT OT On £ ey § z as + 23 41-2 kg oS k mK Paes VE p'°he = age Bs y-13 OSe =g 2es/1F Z6°T = 0 DeS/zg3F o_OT X LT°T = 4 Des/pel 18° =~ Ioe9 = L 73 19°0 = @ pee Nau L £9°% OF ET =- ST°E S*Ps S°EE 68°% 86°TT Z80° os*e S‘8r S*Lz bb's PLOT 160° Is°@ ‘eh S°2z 61°S IS°6 TOT* €%°S «= S°BE S°LT L6°T zs°8 Zet° 66°T S°tE S°€T BL°T SL°L IST° zB"t 4 S°TEe S*OT T8°T FO°L O6T° So°T $°8z G°L 6L°T ss°9 8Ez° ES°T $°92 s°s OT OT OT On x te, g Zz 8 13 43 409 aq Se ek WA —— 1 ELT = ade S | [-13 Lv%@ =g 208/23 66°0 = On 208/743 g_OT ¥ LT°T =A Des/P¥l BF°T = © Joe&9 = L 12 19°O = B eee = NNU 2 “33 POT’ = a (TTF) azo + “A = A peansven ;%K c38°s 88°cT SS oo’e c*e9 C°Cr SL°S €9°FT 880° 9L°S ¢c°ss S°Le 89°S 8E° eT LOT° oS°S c°ec S°se oS°S €T°ot BET’ 62°C o°8P C° le =F. €T°tt Lie OT°S C°RP S° es = 8e°OT C2ies 96°T c°Ir o°02 9T°S €9°6 Lie o8°T c°sEe C°LT o8°T €9°8 Tho° €9°T C°PrEe ¢°’ st 6L°T seek 62° 6h °T c°Tte c°OT EL°T Sie Z, 08z° PE°T o°82 C°L 6S°T €9°9 cTe’ So°T o°9% g°¢ | SOT) 8g0T g0T On x x x z age 43 3 409 Xe au 7 k ig iW 21S = F | t-37 08% = 08S/13 ZZ°T = Cn | 988/233 g-OT ¥ LIT = Des/Pedl FT = Iof9 = 43 esto = 8 zee: NOY Zoe EL'ST 960° et o'er €0°¢€ 92° LT SoL* PES s°ss¢ c°Le 06°% 80°9T Sia 8l°S? c°rs c*es $6°S 6T°ST 9ET’ 90°S ¢°ts c°oe 99°C TE°vt LST° P6°T o°SP C°le L9°S €T° eT T6T° 8L°T C°PPT c°& 9E°S PS°ST 00Z° 99°T C°tPp $°02 PES 9E°TT 612° Po’ t S°sEe c°Lt LT°S 88°6 6SZ° PE T c’ee c°Ot €0°S 62°6 OLZ° 92°T c°Ts S°OT 8L°T I?’s T6Z° PIT o°8a cc 99° e8°L SSE° 90°T G°9% g°s OT OT OT On § oe § | @n ade as 43 | 409 sq ae n wg | 43 0°92 = z | {-1% S62 = SeS/IF OL°T = 1 9eS/e4F g-Ol xX LT°T = oes/perI CO°s = © | 1969 = 37 €8°O = B Tée ‘¢NNU [oe es at BE =a | “43 POT’ = Gd (FFT) SSANHONOW TYNOISNSNIG-OML II ITAL az°o + Was A az‘o + “AeA SSUNHONOU TVNOISNINIG-OML II TiIavL peinsveouw 2% L8°% Z9°ET -- SLz°z = ots 0°08 99°Z 8Z°ST -- 8b°S 360° OF 0°SZ o9°Z ¢6°OT 960° 1Z°S O°r 0°02 8h°S Tr’ OT ZIT’ OT°S 0°GE O°8T ce’e 19°6 621" r6°T 0°98 O°ST LE°S T8°8 ISt° 8L°T O°E€€ 0°2t e1°S 82°83 OLT’ z9°T O°TE 0°0T OT°Z T10°s est’ Z9°T 0°08 0°6 OT OT On Be 5 Zz age 43 Zz +0 sg a A wg 43 9°8T = age - {-13 19% =¢ 20S/7F IT°T = On 98S/z3F g-OT ¥ S9T°T =A DeS/peI GQ°T = I 6369 = 1 1F 19°0 = B gee = NNY Z8°% 6L°hT oe Ze°e OTs 0°0E 19°% peel == 66°S 0°9F 0°Sz (Amara 68°TT == 19°% = «O° Tb 0°02 02°% Teste T60° S°% 0°6E 0°8T S°@ PPO 60T° ve’? O°9€ O°ST Z6°T LS°6 voT* ST°Z O°Ee 0°2T b6°T 66°8 OZT° Z0°S)~=s«éO° TE 0°0T 68°T oL’s TSt- S6°T 0°08 0°6 g0T g0T g0T On x x x Zz age 45 z FYee) 4g m & thy 13 g0°ST = age é q-1F 062 =qd 2eS/4F 86°0 = 1 DeS/z3E c-OT X¥ S9T°T =A DeS/PEI S6°T = 46669 = L 33 0S°0 = 8 Lee NOU 43 POT’ = a (TTF) peansvou :%, os’é 8S°6T =- Tie Skies G°ZE are’ GL°LT T2T° T6°T S°8P o° le st°s Z6°ST O&T’ TL°l S°eP C°Ze Ss°% 60°FT e9T° Zoot cee Set 82°2 £9°ZT 86T° 9f°I S°bE ¢° ST TES eo° It T0z° Tomita Cake, S°OT 91'S er OT TSz° ZI°T 9°8z S°L $0°S €£°6 00g" OO'T s*Sz $°P OT OT OT On ae e § z 2 1 BS 10 ag a x k hy 4 aE b°St = age A 13 996 =q D98S/3F OT°S = “Nn DeS/213 cot X LUT SA DeS/Pel PI'S = 7 Io69 = 1 13 19°0 = @ 9ec :NNU 89°S Sb°rT r60° b's S$" SF GON eS°S 96°ST 90T° 8T°S S*eP S°2e cee LE‘ST SPT’ 80°C S°Ir S°0z 1Z°2 LYoTT 9cT° €6°I ) u Vv tal? Y G/T B fa) | ft rad/sec ft/sec ft2/sec ft-1l . gms/period l1bs/sec x | eee SO a Gea «103 103 | SERIES A D = 5.51 x 1073 ft, | A752 5S .94 TOS wa245 .361 800 .65 . 28 {O | 7 Se GO 1.20 1.03 279 579 499 1.94 ROOM hon | N75 e193 1.45 1.02 308 834 .347 8.47 Big) WON | 1,00 1.29 1.29 102) 252 629 .459 Dor) 107 Roar | TOON 158 1.58 1.02 279 .936 .309 11.24 GeO 1S s20) | OOM e 7 1.87 1202) 303 1.312 . 220 17.62 gh PGS) | 1698 “tees 1.59 iow | 248 .912 317 8.92 Joey. | Boles. | 1.25 1.59 1.99 oO | ZS 1,445 200 18.36 1052 Bil | eo La8G 2.33 oO SOA 1.984 146 21.50 1400) 29572) | SERIES B D = 7.15 x 10-3 ft. Rss 95 1.03 248 262 982 .69 FAG) Noe al SUB lg Be 1.18 1602 | Bye 588 638 1.97 WOR tp Bul 508. EOE 1.47 oO) Slo .918 408 10.07 7.35 10.29 TOOMMI eT Teo 1.03 252 .678 553 1.28 59 82 1.00 1.56 1.56 03s) 275 .886 .388 4.07 2.18 3.05 1.00 1.84 1.84 7202) ) S00 1.358 276 12.20 11.00 15.40 Loo ee 1.5 TO2N ee 250 .961 .3S0 9.20 4.20 5.88 Loo Th Bs} 1.98 ToOl 230) 1.517 . 247 18.30 10.05 14.07 628 668 2.29 sO. BOs 2.040 184 22.00 14.10 19.74 | | | | : es TABLE IV DETERMINATION OF A,,B, AND No A oo CAG wa Buc ale: rane ERO Sion ae oe ft rad/sec ft/sec ft?/sec ft~! £t?2/sec2 WU gms/period l1bs/sec x x10° 103 103 SERIES C D = 9.25 x 1073 ft. ST eS 92 108 HA ir ease 18 107 07 75m le 48) TS Tul 1.03 268 551 880 .61 132 “307 | 9B AGES 1.40 108 Sor 884 549 237 1.54 1.47 TOON oes 1.25 1.02 248 651 745 . 66 . 29 28 T00m 152 1.52 1,05. Bs 959 506 2Ail 1.28 1g 2) ele OOM INKS 1.93 1.00 SO 1,529 317 17.87 NOG 2s LO NSO 1528 1,07 1.28 LOB Spy . 680 713 1.45 52 49 1.8 Nowe 1.74 TSO 263 1.241 391 6.24 3.04 2.89 1688 eS 2.16 TON 293 1.890 257 30.55 18.50 17.62 D-19 tf +O. As “MOTJF }8Y} JO UOT}IAITp UT TeTI9a}eUW paq JO a3ezr zyIOdsueI}Y 9}BTNITYD 0} pasn aq ued MOTF ATepUOdIaS Te UapTIUT Aue y}IM pazetoosse TaAeT paq UT UOT}NGTI4STp A}TIOTIA YIM UOT} eUTQUOD UT UOT}PI}ZUIIUOD aYyYL ‘“padofTeaAep aie 9}e4S AIO}VLT[TISO STYy} UT JUSW -Tpas JO uoT}eI}UaIIUOD IJOF uotssaidxa ue pue pa}eTTTISO STF IaheT paq ay} UT JUSdWTpas YITYM 3e 94eI |9y} JOF uotTyenba ue ‘suot}zenz NTF JUatTNqin} Fo STSATeUe TBOTSTILIS pue SADIOF YFTT FO UOTANGTIASTp ay} FO uoT}eUTWIAa}ap Tejuawtisdxa Ag “wntaqt~tnba juatdtout fo 33e4s e& 0} 4YsnoIq aire paq & UT SeTIT}Ied jUaUiTpas pawnsse ST 4F aTayM apnyT{dwe [Tews fo SeAeM adezINS SuOT Y}IM pazyetoosse IaheT Sty} UT MOTJ 9TQe}SuN JO suOoT}TpUuod TOF ATUO atqeottdde st poyyeW “JOOTF uead0 dy} 0} BYUadefpe IadeT & UT UOT ezIOd -sueij jUaWTpas JO 3}eI BJuTUTWIa}ap UT asn JOF padoTeAap st poyyouw y eTIFL I1 daddy ISSVIONN @ “ON WNGNVYOWEW TIVOINHDSL “) ‘stueypTey 1 <¢ “saTqe} u}yIM sootpuadde 4 pue rohet Arepunog °*z “*sntIt If ‘‘dd ge ‘poet Arenigqay ‘stueyTey “5 Aq S}UariInd pue NOILOV SAVM OL ANd IVIYSLVW Gad JO NOILVLYOdSNVUL saaem Aq 10d -sueiI} jUawTpas “T °O°a ‘°HSWM dO ‘UHINTO “SHY “OYONA IVLSVOO AWUV “S*n “MOTF 3®Y} JO UOT}IIITp UT TeTIa},ewW paq FO a3¥I YIOdsUeIY |9jeTNITeD 0} pasn 3aq ued MOTF ATepuodaS [ejUapTIUT Aue YIM pazetoosse TaheT paq UT UOT{NqGTI}STp AYTIOTIA YIM UOT }eUTqQUOD UT UOT} ¥IZUZIUOD SUL ‘“‘pedoTaAsp aie 9}3e4S AIOZETTTISO STYy} UT JUSW -Tpes JO uoT}eIZUSZIUOD IOF uoTSSaIdxa ue pUue Pe}eTTTISO ST AaheT paq ayy UT jUaWTpas YOTYM }e aze1 94} JOF uoT}enba ue ‘suoT}eN}ONTJ jJuatnqin} Fo stsATeue [eIT{ST}eIS pue SadTOF YJTT FO UOT{NqtIyStp ay} FO uot}eutTwIazap Tejueutiadxa Ag ‘wntaqr{tnba yuatdrour jo a,e4s & 0} 4YsnNOIq are paq e& UT saTIT}Ied JUaWTpas pawnsse ST }F atT9yM apnzTTdwe [Tews JO saaem aoezins BUOT YZTM pazeTIOSSe TaheT sty} UT MOTF eTQeysun FO suOoTyIpUoD I0FZ ATUO eTqeottdde st poyyeaW “JOOTF uead0 ay} 0} JUDSDeLfpe IaheT & UT UOT}e}I0d -suei} jUaWTpas FO 34eI Bututwte}ap ut asn OZ padoTaaap st poyjew y eTITL II ddaId ISSVIONN @ “ON WNGNVYOWSW TVOINHOSL ‘) ‘stueyTey I a: “saTqe} YIM Sadtpuadde p pure raket Arepunog °z ““sntIt TT ‘°dd ge ‘poor Aaeniqeay ‘“stueytey “5 Aq sjuarino pue soaem Aq 310d -sueI} }JUaUTpaS NOILOV SAVM OL aNd TVIUSLVW CHa JO NOILVLYOdSNVUL “O°a ‘“HSVM ‘"a9 ‘UHINSD “Su “OYONA IVLSVOD AWUV “S‘n *MOTF 3 BY FO UOT}IAITp UT TeTIa}eW paq FO a}3¥1 IOdSUeI} |}ETNOTeO OF pasn aq ued mMOTF ATepuodaS TeJUapToUT Aue Y}EM pazetoosse JaABT pac UT UOT4yNQTIISTp A}EIOTAA YRTM UOT }eUTqQUIOD UT UOT}eIZUIDUOD ayL ‘“paedoTeAap are 93}¥e}S AIO}YT[TISO STY} UT jUOW -fFpes JO uoT}eI4ZUIaDUOD IOF uoTSsaidxa ue pUe pa},eT[TISO ST IaAeT pac 9auy UT JUaWTpas YITYM 3B 3}3¥I |y} TOF uoT}ienba ue ‘suot}eN} NTF jJUaTnqiny jo STSATeue TVITIST}V}S pUe SaDIOJ YJTT JO uOTyNQTI}Stp ayy JO UOT}eUTWISzap Te}UswTIadxa Ag ‘wntiqt{rnba jyuatdtout Fo ajejs & 0} }YsnoIG are paq ze ut sajTotz1ed JUawTpas pawnsse ST }T 919M apnjyt{Tdwe [Tews JO SsaAem ade yINS SUOT Y}EIM pajyetToosse Iahe~T Sty} UT MOTJ aTqe}sUuN FO suOT}TpUuOD IOFZ ATUO aTqeottdde st poyj,aW “IOOTF uedD0 dy} 0} JUaDefpe TadkeT e UT UOT ,eAIOd -SUvI} JUaWTpas FO a}eI BuTUTWIAa}ap UT 9aSn IOF padoTadAap ST pouvjzow y eTIFL IT adISISSVIONN € “ON WNGNVYOWSW IVOINHOSL “) ‘stueyTey ots “soTqei YIM sadtTpuadde p pue zahetT Arepunog *z “*sntrtt TT ‘‘dd ge ‘poet Areniqey “stueyTey °*5D SjUetind pue 4q NOILOV SAVM OL SNa IVIMSLWW ddd JO NOILVLYOdSNVUL saaem Aq 330d -suei} jUewTpas "TT “O’a ‘*HSVM qO “USLINTO “SHU “OYONA TVLISVOO AWUV °S°n “MOTJF }¥Y} JO UOT}IEITp UT [TeTI9O}eW peq JO a4ez zy10dsuer} a4eTNIDTeD 0} pasn aq ued MOTF ATepUO.AaS [e4UapTOUT Aue y}EM pazetoosse IaheT pag UT UOT4NGTIASTp A}ZTIOTIA YIM UOT}eUTQUOD UT UOT}JeI}ZUIDUOD ay “padoTaAap aie a}e}s AIO}JYTTTISO STYy} UT jUOW -Tpes JO uoT},eI}UaDUOD IOF uOTSSaIdxa ue puUe pa}eT[TISO st IaheT peq ayy UT }UAWTpasS YOTYM }e 9381 ay} IOF uOoT}EeNba ue ‘suoT}eN}INTF JUaTNqinz Fo STSATBUL TVOTST}LIS pue SAadTOFJ YJTT JO uoT4nqtIs4STp ay} FO uoT}eUTWIAa}ap [Tejyuowtiadxa Ag ‘wntaqtttnba yuatdtout fo a}¥4s e& 0} 4yYSnoIq a1e paq ke UT SaTotzied yUawtTpas pawnsse st yt atayM apnyt{dwe [Tews Jo soaAem aoezIns BUOT YRIM pazeToosse TaheT STY} UT MOTJ eTQe}sUN JO suOT}Tpuod TOF ATUO atqeottdde st poy,aW “IOOTF uvas0 ay} 0} JUadefpe Take, we UT UOT e4I0d -sueij }UAaWTpas Jo a4eI Sututwiajzap ut asn oJ padopTaaap st poyjou vy eTIFL II daIdISSVWIONN @ “ON WNGNVYOWSW IVOINHOXL ‘) ‘sfueyTey I aS “saTqe} y}IM Sootpuadde yp» pure rake, Arepunog °Z “*snttt If ‘‘dd ge ‘p96T Azeniqay ‘“stueyTey “5 Aq sjUarino pue soaem Aq 310d -suti} jUaWwTpes NOILOV SAVM OL dnd IVIYALWW aad JO NOILVLYOdSNVUL ‘O°a ‘“HSVM ‘°aO ‘URINE “SHY “OYONA IVLSVOO AWUV “S‘N . te sa - i= " °, (F i F Y ay | ’ D i : ' & i ri age oun 6 gee =