ENGINEERING RESEARCH INSTITUTE THE UNIVEFSITY OF MICHIGAN ANN ARBOR RESISTANCE TO UNSTEADY FLOW I. Analysis of Tests with Flat Plate John S. McNown Louis W. Wolf Project 2446 SANDIA CORPORATION ALBUQUERQUE, N. M. June 1956

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABILE OF OOX:NIE:~TS Page LIST OFC FIGURES iii STj1VARJY iv OBJECTIrVE iv INTRODUCTION 1 THEORY 1 Inertial Effects 2 Resistance in Unsteady Flow 4 Characteristics of Standing Waves 5 EXPER IMENTEATION 7 ANALYSIS OF RESULTLS 9 Interpretation of F(t) 9 Analysis of Wave ii C ONCLUS ION 15 APIED[1 -IX 1.4 P F ERENCES 5 ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN LIST OF FIGURES No. Page i Flow patterns. 5 a. Relationship between CD and k. 3 3- Comparison of wave forms. 6 4. Experimental equipment. 7 5. Force-measuring system. 8 6. Typical force and displacement record. 9 7. Comparison of predicted and measured forces. 11 8. Analysis of force variation. 12 9. Effect of k(t) on force prediction. 14 iii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN SUMARY Unsteadiness of flow affects the resistance of a blunt body because of the inertia of the fluid and because the pattern of flow in the wake is altered. Both the drag coefficient and the virtual mass for the body are vari able with time. Provisional evaluations of these changes have been obtained from a study of the time variation of the force exerted on a plate placed normal to the primary flow resulting from a standing wave in a small tank. Because the phases of the inertial (acceleration-dependent) force and the drag (velocity-dependent) force differ by a quarter-period in the resulting motion, the wave system is preferable to unidirectional motions for preliminary studies. Significant increases were found in both parts of the resistance. The virtual mass is high for blunt bodies because of the inevitable separation; the drag is greater because the separation pocket is usually incompletely formed. A theoretical correlation between the two kinds of resistance was formulated on the basis of a free-streamline flow past two plates as proposed by Riabouchinsky. For a properly assumed variation of the virtual mass, the observed results were predicted with acceptable accuracy. OBJECTIVE To measure the effects of unsteadiness of flow on the total resistance of submerged bodies and to interpret their effects so as to provide a basis for predicting resistance in large scale flows. iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN INTROD UCTION Forces exerted on structures by the unsteady flow of a fluid are caused by combinations of effects due to the velocity, the acceleration, and the pressure gradient of the ambient flow. For bluff bodies, a wake is formed memporal changes in the shape of the wake, which forms if the structure is not streamlined, affect both the drag and the virtual mass. These combined effects occur if structures are subjected to large-scale blasts in air or to storm waves in water. Estimates of the magnitudes of accelerative forces exerted by blasts on test blocks in the course of studies conducted by the Sandia Corporation indicated the importance of acceleration. The velocity can diminish so rapidly afte:r the initial shock that the total force on a structure becomes zero or even negative well before the direction of flow changes. So little information on this and on other aspects of the resistance to unsteady flow was available that a researchl program was proposed. An oscillatory wave motion was selected for the first tests so that the effects of acceleration and velocity could be more easily separated. In addition, tests of unidirectional accelerated mor;ion are planned. Thl-e reseavch program, which is sponsored by the Sandia Corporation, was undertaken in the laboratory of the Engineering Mechanics Department at Tle University of Michigan. A tank had already been constructed for the purpose, and preliminary tests had proven the feasibility of the undertaking. Limited support for the preliminary tests had been provided by the Engineering Research. Institute of the University. The present one-year contract was initiated OctobXer 155, 1955, between the University and the corporation The project is supervised by johm S. MicNown. Most of the experiments and analyses have been conducted by L. W. Wolf with the assistance of K. Aoki. THEORY In classical theory only background material is available for the analysis of resistance in unsteady flow. The concept of virtual mass is well

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN understood as it applies to accelerated relative motion of fluids and submerged bodies.1 Also, considerable progress has been made in the development of the theory of wave motion.2 INERTIAL EFFECTS If a body is accelerated in a fluid, the various elements of the fluid are also accelerated. The integrated effect of these is a force and a reaction between the body and the fluid. Thus, fff dxdydz kMt dt in which the integral is taken throughout the fluid. The virtual mass kM' is defined in terms of a coefficient k multiplied by a reference mass M' which is usually the mass of fluid displaced by the body. For thin, flat plates or for disks the displaced mass is zero. Hence, a circular cylinder or a sphere of the same section is used as a basis of reference. Virtual mass can also be introduced from a consideration of kinetic energy. If the velocity of a submerged body is increased, so is the kinetic energy of the fluid around it. Consequently, work is done and a force is exerted on the fluid. For flows describable in terms of velocity potentials, the kinetic energy of the flow. and its time derivative are used in determining k. Virtual-mass coefficients are available for only a few mathematically simple bodies. For all elliptic cylinders with one axis parallel to the direction of flow, the virtual mass is equal to thie mass displaced by a ci:rcula:r cylinder for which the diameter is the transverse axis of the ellipse.' The flat plate in Fig. la is a limiting case. An expression in elliptic functions was developed by Riabouchinsky for cylinders with rectangular cross sections.3 Also available in terms of elliptic integrals are coefficients for any ellipsoid, provided only that one axis is pa.allel to the direction of motion. For all of these cases, irrotational flow without separation is assumed. Because separation occurs in flow past any body which is not well streamlined, the theoretical results are not necessarily useful in analyzing flows past structures of conventional design. A more realistic representation of flow past a flat plate was proposed by Riabouchinsky,3 in which a second plate downstream of the first closes the flow as shown in the left part of Fig. lb. The cavity, bounded by the two plates and the connecting free streamlines, is a mathematical approximation to actual wakes also shown schematically. Birkhoff et al. extended this analysis and presented drag coefficients for various cases. The method was again extended recently to include the evaluation of the virtual-mass coefficient for systems with various spacings of the plates.5 The results are shown in Fig. 2, Other ways of achieving l finite Cavities are the reentrant jet of Gilbarg and the parallel afterbody o

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN vo (a) /o i Vo - 1 (b) Fig. 1. Flow patterns. 12 I 0 CD 7 - 0 I 2 3 4 5 6 7 8 Fig. 2. Relationship between CD and k. 5

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The classical thfeory without separation is appropriate foar motions in which the displacement is small,t has been used for determining the forces on dams during earthquakes, 19 for studies of bodies vibrating in liquidsl0'11 and for computations of gust and vib'ration loadings of sirp':lane l wings. Considerable discrepancy is found between the results of studies of this type and those obtained for flows in wic the amplitu.de of the motion is large enough for a separation pocket to occur.'This had led to some disagreement in the definition of k, as indicated by -the puLiblished discussionl2 of the paper by Stelson and Mavis. RESISTANCE IN TINS'T'AIDY FL W Analysis of the action of accelerated flow past a flat plate is seriously complicated by several consequences of the unsteadiness of flow. Even the drag coefficient, which is usually considered to be dependent on the velocity alone, can be altered because the wake can be larger or smaller than that occurring for steady flow at a given velocity. Also, a longitudinal pressure gradient exists if the velocity is changing even in the absence of a body Finally, the virtual-mass coefficient is in turn dependent on the shape of the pocket, as shown in Fig. 1, and both its magnitude and its time derivative affect the force. TIhese various contributions to the?.. resistance can be represented quantitativel y by the formula ~F = PX dS CD( A P' +, (1) 2 d - s in which px is the x-componernt of the amkiient pr:essire in trhe absence of tlhe body, S an element of -th.e surface area, C% t.ie coefficient of drag, A the cross-sectionai area of the body, and V t;he aient ve.l.ocity. If th:.ie body is accelerated through an ot-hervise quiet fluid, the pressuLre gradient becomes zero and the force required to acce'lerate!.e bMoody must be added. Difficult- t+-1ougjh an ana:,lysi.s based on Fquation.J. may be, the separation of the various pa:rts is less difficult for an osci.:Llatory motion than it is for unidirectional accelerated motion. For the former, the acceleration and velocity are out of p'ar:se by a quarter period, and each is zero at the time the other is a maximum. Hence, some possibility of distinguishing the two effects exists. In fact, this concept has been used by Reid and Bretschneider13 to evaluate k and CD. Each was assumed to have at a:Ll times tnhe constant5 value indicated at the; moments for which. the other force was ze:ro. However, as already explained, these effects are dependent on the time history of the motion. No sirmple system based on constant values for either CD or k can be complete. In laboratory studies of bodies whi.ch are accelerated and moving continuously in one direction, the effects of velocity and acceleration are not easily separable. In such a study of an accelerated disk,:versonl4 proposed an overall coefficient in which the two effects are cormbined. To

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN obtain an understanding which will apply to a variety of flows these effects must be separated. CHARACTERISTICS OF STANDING WAVES The characteristics of the standing wave used in the present study have a direct bearing on the results obtained. The classical theory of irrotational wave motion of very small amplitude is well established.1 A velocity potential can be written and all essential characteristics of the flow are defined. A sine wave is obtained for the free surface, Ix 2ct = a cos-cos —-, (2) in which rl represents the displacement of the free surface from the equilibrium position, a the amplitude of the wave, L the length of the tank (L = k/2), and T the period of the motion. The velocity at the nodal plane is given by the expression aTg cosh [k(z + H)] sinx 2t 2L cosh (k) sinL sinT'(3) in which z is the vertical coordinate taken as positive upward and H is the depth of water. The period of the motion is dependent only on the acceleration of gravity and the dimensions of the system: -) = ~-g tanhf. As the relative amplitude of the wave (a/x) increases, the foregoing theory becomes less exact. Furthermore, no single method of improving the analysis serves for the entire range of relative depths (H/x). For waves in moderately shallow water, as in these experiments, the wave form is called cnoidal because the free surface is described in terms of the cn elliptic function:'= - a2 + (al + a2) [cn2(c - m) + cn2(a - D,m)], (5) in which 2F (m) 2F(m) = D = ct am =X=g[1 + al a t ah r L) 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN to the crests and troughs, respectively, and c represents the wave celerity. The wave length and depth are related: = 2 F(m) H 5(a, + a3) Finally, a3 is defined implicitly by al + a3 E(m) = as F(m) F and E are complete elliptic integrals of the first and second kinds. The details of this theory are presented by Keulegan and Patterson.15 Significant differences between a sinusoidal and a cnoidal wave are apparent in Fig. 3 in that (1) the crests of the latter are-higher and narrower than the troughs, whereas the sine wave is symmetrical, and (2) no true node exists for the cnoidal pattern. -02 2 0a =.038 ft As iStill water level H=.734 ft.02 L=5.6 ft 1/.01.. Sithe imposed motion and its characteristics must be known in detail T.2.3 4.6 7 8 9 1.0 L Cnoidal -.0 I -.02 Fig. 3- Comparison of wave forms. ae is to bring masurements of the available can be assessedtion oif the profile can be the imposed moanation and its characteristicures musof th be occurrence ale noThe defined analytically. Although many features of the occurrence are not yet understood, the available theories provide a strong framework for the interpretation of the results.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN EXPERTMENTATION All measurements have been made for motions resulting from a standing wave in a small rectangular tank. The tank is 5.6 ft long, 2 ft wide, and will hold depths of water up to 1.2 ft (Fig, 4). The tank is made of stainless steel with Lucite side walls. A standing wave is created by means of a constant-speed motor acting through a speed reducer and an eccentric to give a nearly simple harmonic motion in the vertical direction to a wooden plunger at the surface of the water near one end of the tank. g... i...m'- ". i.:.....,.i1!i~ E.:'.:. i i iig:.:,:':-:..,-,:~-'..':::i. i.:'.,'-........,::g:: i:......:,..:t'S-i-:-E i E'f'f'l::f:.,;0...',.MM!fldiS~iS.-:0:EE:: 1;! E............ ~ Fig. 4. Experimental equipmrent. The period of the drive mechanism is 2.37 sec. The depth of the water is adjusted so that the fundamental mode of the tank from Equation 4, is the same as that of the drive. As the amplitude of the wave is increased, it is necessary to increase the water depth slightly to keep the period of the wave equal to that of the drive. A brass plate 1 in- x 24 in. was suspendied from roller bearings at mid-length and at mid-depth of the tank. Later the suspension was changed t7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN from roller bearings to a set of elastic tinges to eliminate any possibility of friction interfering with the measurements (Fig. 5). The system was restrained from significant displacement by a Statham strain bridge (Model GI48_675), and the signal thus created was fed into one channel of a Sanborn recorder (Model 60-1300) giving a measure of the variation with time of the force exerted on the plate by the water. A typical record is shown in the lower half of Fig. 6. The system was statically calibrated by hanging known weights on the arm extending from the suspension. Fig. 5. Force-measuring system. To measure the state of the water motion, a depth gage consisting depth, was placed at the end of the tank away from the plunger. The signal from it was calibrated and supplied to the second channel of the Sanborn recordr Aoio rcdot pidawvfrw tuoaeSider o A typical record of the wave profile is shown in the upper portion of Fig. 6. hook gage near the end of the tank. Reliabe t readings to the nearest O.001 in were obtained. The height (double amplitude) of the wave was varied from 0.050 ft to 0.070 ft. 8

- NGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN..... ". T..:.. Ag +: J'n.42:. Fig 6 ipical force and. dispaceent:record IP~RmPETATION OF F(t) comparisons with the observed force. The entire cycle was considered rather than maxima or approximate constant values for the coefficients of drag and of virtual mass. The approaches used ranged from the empirical to the theoretical. For the latlter calculations were based on th.ie theoretical results for the finite wake of Riabouchinsky (Fig. 2). Perhaps the simplest of comparisons is one in whaich (1) the drag coefficient is assumed to be constant and to have the same value as is obtainis assumed to bee unithy as it would be for irotationa:l flow wiho.t separation tively substantiated by the experiments. Also, no arbitrary c-hoice of larger W.,;: i i te if~~~~~~~~~~ijj-~'. INRFEATONO Ft Seerl etod o prditig nefrc o t, lae e —L tstd b coprsoswthteobevd oc h n —.Or ylewscosdre 1atIe thnmaia raprxiaecosan ale frth offce: o ra n ofvrulms.Teapoc1e,-sdrne rmte miia ot:,te reia.Frth atrcacltosweebsdo ttetertia e —t for the finite wake of Ridbouchiinsky ('Fig. 2).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:~~: Perhaps the simplest of comparisons is one in wlalch (1) the drag~~~~~~~~~~~~~~~~~~~~~~~~~I~f cofiin sasmd ob osatadtohv aevlea is ban ed orstedyflw (ppoxmter,O., nd(2 th vrtal-as ceffcint::':r::::~.::4::- L_ i:::::1-44 - Iz It::iri::::i! -.0-:sl~~~:. i:i-il l ~ i~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN but constant values for these coefficients could be made for which predictions were compatible with the data. An entirely theoretical method was attempted- also witho-ut success. The coefficients were assumed to be related as shown in Fig. 2. Also, the variations of the values were assumed to be controlled so thaet a minimum force was produced at any time. Out of the infinite number of possible variations of k (or CD) with time, it was postulated that the one requiring the least work would be the most likely. No firm. physical basis for this idea exists, nor did it prove to be a workable hypothesis. Despite many and varied attempts, no complete prediction could be formulated. Graphical methods progressed fairly well for a part of a cycle, but a point was invariably reached for which the guiding hypothesis became ineffective. Approximate mathematical formulations were no more successful. The degree of freedom implicit in Equation 1 is such that a thleoretical value of zero is possible for all times unless k is restricted. Although k is known to be restricted to positive values (probably greater than unity), this condition could not be easily imposed. Consequently, this method of approach was dropped. Good predictions were finally achieved by combining the relationship of Fig. 2 with an arbitrarily assumed variation of k with time. Various trends for k(t) were tested against the data by means of a stepwise integratim of Equation 1 described in the Appendix. The achievement of a realistic result imposed a severe restriction on the assumed k(t); a quantitative correspondence with the data was an even more stringent restriction. The results obtained for various assumed values of k(t) are illustrated in the Appendix. Once acceptable results were obtained fora wave height of 0.040 ft, the same k(t) was used for all four amplitudes for which measurements of F(t) had been made. The computed and observed. forces are compared in Fig. 7. Closer correspondence could have been obtained in several ways. However, further efforts in this direction do not seem worthwhile at present because of the several approximations involved in various parts of the process. The extent to which the predictions and observations correlate justifies in a general and qualitative way the method of approach. The two primary contributions to the total force are separated in Fig. 8. Also shown are the variations of velocity and acceleration and the assumed k(t). The extent to which the curves for the acceleration and the acceleration-dependent force are disproportionate or out of phase is a measure of the importance -of the time derivative of k. That is, the last term in Equation 1 can be separated into two parts:. d(k_) = + ka dt dt dt 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN.05 Double // amplitude.04 / iO"\0.06 ft.03.02 -.04t ~ 0.04 ft 0 0~'Y~i~ IIJ 900 180 110 ~~2700 /I PHASE ANGLE -.01 Experimental results 0.05 f t I I I I — Calculated values ".03 -.04-. Fig. 7 Comparison of predicted and measured forces. If k were truly a constant coefficient, the first pa;rQ- would be zero. Instead its contribution is significant. ANALYSIS OF WAVE Among the limitations on the present study is a minor uncertainty as to the exact motion created by the wave. te dependence of the period of the wave on the amplitude is not defined explicitly by theory. Nor is the precise motion in the vicinity of the plate known. A cnoidal wave form was observed except for the portion of the tank near the plunger. H~owever, sinusoidal variations were assu.mied for both velocity and accele-ration0 Thle discrepancies are not likely to be significant. -. 011

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN I I1 I1'~' i~~4e I I NI ITo t force.02 M' I I I 11 1 i L X l I I I I I I IL 00 1 I yl I II I I I I I-I I1 I' 2 L-0PH ANL 700 00 90~ 180~ 270~ PHASE ANGLE Fig.o 8. Analysis of force variationude Various secondary factors may alter the data somewhat. The maximum value for the Reynolds number was 750 The upper liit of t0.04e variations due to viscosity is approximately 103, so that a part of the data may have been affected by viscosity. There may also have been small irregularities at the ends of the plate or at the supports. The force on a one-inch plate in free surface. Finally, the water velocity decreases in accordance with Equaaverage value was used. The proposed theory can be criticized because the assumed pattern for the variable wake is much simpler than that observed. Certainly, the relationship between k and CD presented in Fig. 2 should be considered as representative rather than exact. A cavity bounded by two plates connected by free Instead, cylindrical vortices form and grow into a wake. Once formed, of 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN course, these vortices may persist. It is probably for this reason that the value of k was always greater than unity. A theoretical formulation based on a growing vortex may be possible. Although none of the foregoing discrepancies is ia;rge, some of them may cause appreciable errors. Accordingly, no attempt was made to refine and reduce either the data or the analysis. Some of thie problems will be studied or avoided in future tests. The method of approach used to predict the force is consistent and comparatively simple. CONCLUSI ON Unsteadiness of flow can hlave a profound effect on resistance. Systematic measurements of the variation of force with time have formed the basis for a quantitative interpretation of the force exerted on a body in a timedependent motion of a fluid. For flow past a flat plate, the manner in which the wake forms is vital in determining the resistance. For the nearly simple harmonic motion at a node of a standing wave, the coefficient of virtual mass for the flat plate was found to vary from 1.1 to 1.8, approximately, the coefficient increasing with. the size of thie wake. Simultaneously, the coefficient of drag varied from about 5 to 3. The two coefficients are inversely related in a way whiich has been illustrated by an extension of Riabouchinsky's analysis. Although: much remains to be studied, a means of satisfactorily describing the occurrence:has been formulated. 13

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN APEaIDIX The calculation of the force va: ia-cion with' t ime was acconmplished by first assuming a function k(t). T~.le drag coefficient Cp(t) was then determined from Fig. 2. The velocity variation was assumed -to be sinusoidal with time, and the quantity k sin (at was plotted. From tkhis curve d(kV)/dt was obtained graphically. These values were tl.h.en sdr:sti.tuted in Equation 1 and the force was calculated. Early in the work tLe form of k(t) was not known, except that it was apparent that k should be periodic and not have valu~es less t'han 1. Analysis of the data indicated that at 90~ (thie moment of zero velocity) k would be 1.66 because the only contribution to the force is from thre k dV/dt term. However, no force curve could be obtained that came sufficiently close to the experimental values at other points (curve B in Fig. 9) for which 1.66 was used'02 Trial B E x pe r i me nTiaI',0 00o IIooL 111 80 PHASE ANGLE -. 4. Iff, I \

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - as the minimum value of k. Hence, the requirement; of an exact fit at 900 was abandoned in favor of a closer fit throughout the cycle. Other trials were made including sinusoidal variations of k(t) (c-wrve A in Fig. 9) until one leading to an acceptable correspondence was obtained (curve C in Fig. 9). Even for the rather similar variations assumed for the t.cxee cases shown, large differences are evident in various curves for F(t). Accordingly, the empirical test restricts greatly the range of vaariab1ility of k as defined in the simplified analysis leading to Equation 1. 15

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN REFERENCES 1. Lamb, H. Hydrodynamics. 6th ed. New York: Dover Publ., Inc., 1945. 2. Keulegan, G. H. Chapter XI,Engineerin Hydraulics. Edited by H. Rouse. NeW York. -- John: Wiley.and Sons,'Inc., 1950. 3. Riabouchinsky, D., "Sur la resistance des fluides," Comptes Rendus Congres International des Mathematiciens, Strasbourg, 1920. 4. Birkhoff, G., Plesset, M., and Simmons, N., "'Wall Effects in Cavity Flow - II," Quart. Appl Math., (Jan., 1952). 5. McNown, J. S. "Note on estimation of virtual mass coefficient." (Apr. 17, 1955, unpublished, Sandia Corp.) 6. Gilbarg, D., and Rock, D. H. Naval Ordnance Laboratory Memorandum 8718, 1945. 7. Roshko, A. A New Hodograph for Free-Streamline Theory. NACA TN-3168, 1954. 8. Westergard, H. M., "Water Pressures on Dams During Earthquakes," Trans. ASCE (1933). 9. McNown, J. S., "Hydrodynamic Earthquake Forces on Submerged Structures," Proceedings of the Third Midwestern Conference on Fluid Mechanics (1953). 10* Stelson, J. M., and Mavis, F. T.,rtVirtual Mass and Acceleration in Fluidst Proc. ASCE, Separate No. 670 (1955). 11. Yu, Yee-Tak, "Virtual Masses of R.ectangular Plates and Parallelpipeds in Water," J. Appl. Physics (Nov., 1945); "Virtual Masses and Moments of Inertia of Disks and Cylinders in Various Liquidss," ibid. (1942). 12. Caldwell, J. M. and Silberman, E., Discussion of the paper by Stelson and Mavis (Ref. 10), Proc, ASCE (Sept., 1955); Stelson, T. E., and Mavis, F. T., Discussion J. of the Engineering Mechanics Div.. Proc. ASCE (Apr.,, 1956). 13. Reid, R. 0., and Bretschneider, C. L., iThe Design Wave in Deep Water or Shallow Water, Storm Tide, and Forces on Vertical Piling and Large Submerged Objects," A. and M. College of Texas, Dept. of Oceanography, Tech. Report on Contract NOy-27474, DA-49-005-eng-18, and N7onr-48704, 36 pp., Feb., 1954, unpublished. 16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN EEFERENCES (concluded) 14. Iverson, H. W., and Balent, R., "A Correlating Modulus For Fluid Resistance in Accelerated Motion," J. Appl. Physics, 22 (Mar., 1951). 15. Keulegan, G. H., and Patterson, G. W., "Mathematical Theory of Irrotation al Translation Waves," J. Res. Nati. Bur. Std. (Jan., 1940). 17 — __