USS: Gs Bag Kee CTR TP 77-9 Calculating a Yearly Limit Depth to the Active Beach Profile by Robert J. Hallermeier TECHNICAL PAPER NO. 77-9 SEPTEMBER 1977 Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to tne U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. oaade? 4 3 WMA BL/WHOI M (NN Oo 0301 0 SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE T. REPORT NUMBER 2. GOVT ACCESSION NO,| 3. RECIPIENT'S CATALOG NUMBER WW? 7/V=E 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED CALCULATING A YEARLY LIMIT DEPTH TO THE Technical Paper ACTIVE BEACH PROFILE 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Robert J. Hallermeier 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELE OJECT, TASK Department of the Army ata: nee Coastal Engineering Research Center (CERRE-CP) D31193 Kingman Building, Fort Belvoir, Virginia 22060 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Coastal Engineering Research Center 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release, distribution unlimited. . DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Beach profiles Sediment entrainment Coastal engineering Sediment transport Coastal processes Wave-sediment interaction . ABSTRACT (Continue om reverse side if necessary and identify by block number) A sediment entrainment parameter is used to calculate the maximum water depth for intense agitation of a sand bed by shoaling waves with given height and period. Calculated depths agree with measured water depths over a terrace cut into a fine sand slope by constant laboratory waves. For high wave con- ditions expected 12 hours per year on exposed U.S. coasts, the calculated depth is about twice the wave height and agrees with published conclusions on the yearly seaward limit to the active beach profile. DD ens, 1473 EvITION oF 1 Nov 65 1s OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) hiseu if 1b) A) a As haa zn ts Uae (ena ran) Sid hk eae Me : fleece, paves USN a PREFACE This report provides coastal engineers with a procedure for calculat- ing a limiting depth to appreciable sand level changes seaward of a beach, using an estimate of extreme waves occurring at the locality. The re- ported results are from an ongoing study of the seaward limit of effective sediment transport, carried out under the sediment-hydraulic interaction program of the U.S. Army Coastal Engineering Research Center (CERC). The report was written by Dr. Robert J. Hallermeier, Oceanographer, under the general supervision of Dr. Cyril J. Galvin, Jr., Chief, Coastal Processes Branch. The author appreciates the comments and cooperation of Dr. Galvin, C.B. Chesnutt, and Dr. J.R. Weggel, CERC, and J.T. Jarrett, U.S. Army Engineer District, Wilmington. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. JOHN H. COUSINS Colonel, Corps of Engineers Commander and Director CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . I INTRODUCTION Il DEVELOPMENT OF THE CALCULATION PROCEDURE. Til COMPARISON OF CALCULATED RESULTS WITH LABORATORY PROFILES. IV APPLICATION TO NATURAL BEACHES. .. . V CONCLUSION. LITERATURE CITED. APPENDIX A CHOICE OF LENGTH SCALE IN SEDIMENT ENTRAINMENT PARAMETER . obo 0 Oo 6 4 0 6 6 616,00 B LABORATORY STUDIES OF PROFILE DEVELOPMENT . TABLES Dimensionless water depth, Ee Aan ae Sa ey and (4) for four values of ec . suie as : Calculated limit depths for 10 design wave conditions . FIGURES Solutions of equations (5) and (7). Definition of cut depth, i in mccain profile development. slitirenaite : eines Macattel oars Measured de ey with calculated dg for 27 laboratory tests. HOR ae tay ReMi naa Metra GUT SU a teh tT UT ei nee eal Unpublished profiles developed in Saville's (1957) large-scale PSS num bea (Sei tei edie ie alleles an Bis ates ERO EMU AMIDA Offshore survey data from 6 June 1972 to 30 Lae 1974 at Torrey Pines Beach, California . 3 Bilis Page 25 27 16 iil 11 13 14 18 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.8532 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars 1.0197 X 10°? kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! MPy obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (If! — 32). To obtain Kelvin (K) readings, use formula: K.= (5/9) (F — 32) + 273.15. — = 2mdg/L oO ® ii} Up*/(egy'd) o; = Up2/(gy'D) HS Des SYMBOLS AND DEFINITIONS wave-induced water orbit diameter near the bed sediment grain diameter water depth water depth at wave-caused cut into plane slope water depth at seaward limit of intense bed agitation maximum water depth in wave tank acceleration due to gravity wave height wave height in deep water wavelength wavelength in deep water maximum wavelength in wave tank wave period maximum wave-induced horizontal velocity near the bed ratio of density difference between sediment and fluid to fluid density number less than one kinematic fluid viscosity dimensionless limit depth transcendental number 3.14 ... dimensionless sediment entrainment parameter dimensionless sediment mobility parameter wave frequency CALCULATING A YEARLY LIMIT DEPTH TO THE ACTIVE BEACH PROFILE by Robert J. Hallermeter I. INTRODUCTION Certain coastal zone activities require setting a seaward limit to the very active (littoral) zone of a sand beach; e.g., sediment budget calculations, submarine placement of beach fill, and design of coastal structures such as jetties. In principle, repetitive measurements of waves and bathymetry can define the seaward limit at a site by establish- ing the water depth beyond which bottom changes caused by wave action are negligible. However, these data are costly and rare, so an estimate of the limit depth must often be obtained from other evidence or experience. Silvester (1974) discussed physical data showing that extreme surface waves occasionally move bottom sands in water depths up to about 100 meters; i.e., over much of the continental shelves. However, reviewing geomorphological evidence, Dietz and Fairbridge (1968) concluded that waves intensively work fine, nearshore sands into a somewhat concave equilibrium profile extending to a water depth of 10 to 20 meters. This depth is the seaward limit to the active profile, supposedly related to wave climate for a particular site. Offshore of this depth lies a zone of less important sand transport by waves. For these coastal zone activities, a calculation procedure using known wave climate might provide a useful estimate of this limit depth, in case definitive data are lacking. This report proposes a procedure giving a minimum value for the limit depth to the seasonal onshore- offshore transport cycle on sand beaches. The development is based on dimensional reasoning of the energetics of wave agitation of a sand bed, without detailed examination of sand transport processes. Although some- what speculative, this approach arrives at a simple calculation procedure giving a limit depth for a shoaling two-dimensional wave with a certain height and period. Calculated depths agree with the published laboratory and field evidence considered in Sections III and IV. II. DEVELOPMENT OF THE CALCULATION PROCEDURE This development considers wave energy density in terms of two dimen- sionless parameters. The first parameter (eq. 1) has documented perti- nence to the onset of sediment movement as a bedload. A second parameter (eq. 2), Similar in form but numerically smaller, is hypothesized to describe the onset of intense bed agitation, characterized by sediment entrainment into the wave flow beyond a thin near-bed layer. Silvester (1974) presented the empirical expressions derived by var- ious investigators of the motion threshold on a flat sediment bed. An important factor in each expression is the sediment mobility parameter: o> (ey'D) where Un = the maximum wave-induced horizontal velocity near the bed the acceleration due to gravity ite} I oO I the sediment diameter the ratio of the density difference between sediment and fluid to the fluid density Equation (1) gives the peak near-bottom wave energy per unit sediment grain volume, divided by the energy required to raise an immersed grain against gravity a distance equal to half its diameter (Nayak, 1970). Although the motion threshold cannot be described solely in terms of 6;, sediment generally begins to move when this number becomes larger than one (Komar and Miller, 1973). Lofquist (1975) observed that ¢; measured the intensity of sand motion in his tests with a rippled bed; the motion threshold was near %; = 2.4, the ripple crests became less distinct for ®; > 10, approximately, and, for larger %;, vortices of fluid and en- trained sediment became increasingly evident, indicating more intense bed agitation. These results show that 4; \ 1 indicates the beginning of sediment mobility by describing energetics in a thin layer near the bed. After sediment grains begin to move, wavy bed features form. These features are caused by grain flow, but liquid flow of high inertia can scour these features from the bed (Bagnold, 1956). Because a plane (intensely agitated) bed usually occurs in and near the surf zone (Inman and Bagnold, 1963), a criterion for the seaward limit to the active beach might be based on a parameter measuring the energetics of intense bed agitation. Available research results (reviewed briefly in App. A) indi- cate that grain diameter has a weak influence in intense agitation of fine sand beds, suggesting that another Froude number, like equation (1), be used with a larger length scale replacing D. It is hypothesized that intense bed agitation is described by the sediment entrainment parameter: U 2 b *e = (egy'd) ie where ce is a number less than one and d is water depth. (The choice of the length scale replacing D is discussed in App. A.) When 9%, reaches one, the wave energy density is sufficient to raise a grain a distance ed/2, taken to be appreciable fraction of the water depth and much greater than D/2 for fine sands (D<0.4 millimeter). Equation (2) and a coupled pair of quantitative assumptions permit calculating a maximum water depth for intense bed agitation by shoaling waves. Taking ® = 1 and using linear wave theory, equation (2) may be written as 2 2 as) me 2nd, ‘ ae 4 T Fl : L Si L cos L = Se Li (3) where H is wave height, L is wavelength, and d, is the limit depth. To assist selecting a reasonable magnitude for e€, the case of maximum wave steepness (eq. 100 in Madsen, 1976) is examined: (7) = 0.14 tanh (7) (4) MAX Table 1 lists the d,/L solving equations (3) and (4) for four values of é€ with y' = 1.6 (quartz sand in freshwater or saltwater). For the present purpose, e = 0.03 seems the proper magnitude. The limit depth then remains well beyond the breaker depth for a steep wave, and the limit to the active profile generally lies seaward of the breakers. This € is also large enough that the previous assumption of ed>>D can be easily satisfied for fine sands. Based on these considerations, %, = 1 with e = 0.03 is taken to indicate the limit depth of intense bed agitation. Table 1. Dimensionless water depth, dg/L, solving equations (3) and (4) for four values of 1Calculated using Table C-2 in U.S. Army, Corps of Engineers, Coastal Engineering Research Center (1975). *Breaking wave, if bed slope is small (see eq. 102 in Madsen, 1976). With these assumptions, equation (3) can be rewritten as 2 H — sinh? & tanh? €& (2 + sar) = 205.6 (72) (5) where € = (21d,/L), Ho is wave height in deep water (d/L>0.5), and lo = (gT2/2m) is wavelength in deep water. The dashed curve in Figure 1 solves equation (5). For wave steepness greater than 0.015, the difference between H and Hp is less than +10 percent, according to linear wave theory. Ignoring this shoaling wave height change, equation (5) takes the simpler form: E sinh? & tanh & = 205.6 uh (6) O Table 1 shows & remains less than 27(0.179)*9/8, so Maclaurin expansions for the transcendental functions in equation (6) yield the accurate alge- braic approximation uth (Ny Bia RA eet yasans i al \e (7) ae ee 205.6 (#) where the omitted terms are on the order of (0.002 €!%). The solid straight line in Figure 1 shows the solution of equation (7) if only the first term on the left-hand side is retained; the dotted curve shows the solution of equation (7) where the other terms are appreciable; the solu- tion of equation (6) lies between the dotted and solid curves. According to this development, the maximum water depth for intense bed agitation may be obtained for a certain wave condition by reading the corresponding (2nd,/L) from Figure 1, and multiplying by {(L,/2m) tanh (21d,/L)} to obtain d,. This development has considered the ideal two-dimensional situation of a monochromatic wave shoaling according to linear wave theory. Grace and Rocheleau (1973) concluded that available field measurements indicate linear wave theory ' ... provides an excellent prediction of the sample: mean, near-bottom water velocity beneath the crest of long design-type waves....'' LeMehaute, Divoky, and Lin (1968) reported laboratory measure- ments showing linear wave theory accurately gives Up, in near-breaking waves. Thus, linear wave theory is a convenient and accurate first approximation in the problem considered. III. COMPARISON OF CALCULATED RESULTS WITH LABORATORY PROFILES The results from a laboratory test of profile development with a constant wave condition incident on an initially plane slope of fine sand are shown in Figure 2. Changes in profile shape become minor after a large number of waves, and many "equilibrium profiles" have been published for various test conditions (see App. B). Frequently, these profiles exhibit a long, slightly sloping terrace (as in Fig. 2), caused by off- shore deposition of sand eroded from the initial slope above a certain depth. Raman and Earattupuzha (1972) pointed out the existence of stable points occurring in wave development of a sand bed profile. If the waves 10 2md./L Dimensionless Limit Depth, € 0.1 0.00! 0.002 0.004 0,006 0.008 00! 0.02 0.04 0.06 0.08 O11 Wave Steepness, H)/L, or H/L, Figure 1. Solutions of equations (5) (dashed curve) and (7) (dotted curve). Figure 2. ----— Initial Slope ———— Anwrer IS©) hr ——= > - ANraArR- Sr hr Elevation (ft) -|14 —7 O t 14 2| 28 35 Horizontal Distance (ft) Definition of cut depth, de, in laboratory profile development; T = 1.9 seconds,H = 0.36 foot, dg = 0.73 foot (profiles from Fig. 7 in Chesnutt and Galvin, 1974). cut a terrace, the well-defined water depth, d,, at the seawardmost stable point indicates the elevation of the terrace (Fig. 2). Measured values of de from published profiles will be compared to d,, calcu- lated using Figure 1 (eq. 5), because d, has been hypothesized to indicate the limit depth of intense bed agitation, revealed by the nearly horizontal profile feature developed at the cut depth. Figure 3 plots the measured d, against the calculated d, for 27 published profiles from 7 laboratory studies including tests with fine sands (D<0.4 millimeter). (Twenty other profiles obtained in these studies did not show an ideal terrace, although test conditions were similar; see App. B.) There is some scatter, but the 27 pairs of cal- culated and measured depths have a correlation coefficient of 0.936, im- plying a linear relationship between de and dg is certain (Freund, 1962). On the average, the calculated limit depth is 3.6 percent greater than the water depth at the cut into the initial slope. This good agree- ment between calculation and measurement occurs for a wide range of wave conditions, with 0.0039 SH < 0.071 (see App. B for test conditions). The reduced scale of these laboratory tests, compared to natural pro- totypes, should not be important for the present comparison, because the threshold of intense bed agitation is evidently crossed in the terrace zone of the laboratory tests. Also, the calculation procedure is based on wave energy per unit sediment grain volume, and fine sand occurs both in the laboratory tests and in the offshore region of natural beaches. Limited verification of the unimportance of scale effects on the cut depth in a plane sand slope is provided by several profiles obtained by Saville (1957) in large-scale laboratory tests. Saville's test number 5 had the largest ratio of water depth to wavelength, with waves 4.8 feet high in a 15-foot stillwater depth and a 3.75-second wave period; d, = 8.3 feet is obtained for this condition. Figure 4 shows the unpublished profiles developed with sand of 0.22- and 0.46-millimeter diameter. Al- though there is no clear terrace development, the initial slope is eroded onshore of water depths from 7.0 to 8.8 feet, with offshore deposition in both cases. Judging from this evidence, the calculation procedure provides an accurate estimate for the limit depth of intense agitation of a fine sand bed by waves. IV. APPLICATION TO NATURAL BEACHES The natural beach profile adjusts seasonally in response to wave climate with relatively little loss of sand from the nearshore over an average yearly cycle. Subaerial profile erosion by storm waves can be notable, but most eroded sand remains in relatively shallow water and is carried onshore by less steep waves. Bruun (1954) and Winant, Inman, and Nordstrom (1975) analyzed southern California data showing that the sub- marine profile beyond the 5- to 10-meter water depth is accurately described throughout the year by a constant function. *s3soq AzoyeroGeT {Zz 10f “p poqernotes yiIM pezedwod 7p paanseay -¢ oan3ty c (wo) Sp ‘yydaq fiw peyDjno|D9 OS Ov o¢ Oz Ol @) (p261) OMDYU0H pud DJINnwWDUNS (pS6l) 404994 (2261) DYUZNdny4D4DQg pud uDWOY (2261) 4e9uqa4g pup ‘sinydwoy ‘jnod (6961) a041u0W (1961) dnd01q pup ‘auuaj9 ‘uose|boy (226| ‘uoljos0daid ul) 44nusayd O| +x JOObS O02 O¢ (wo) 2p tyydag 4nd paunsve;, Ov 13 Elevation (f+) (ft) Elevation Figure 4, \. NN ape T=3.75 s, H=48ft A 0.22mm_ sand of-SWL AY RK -—-- Initial slope : ——o-— ree SO) Ine After 60 hr —5 {¢) 50 100 150 200 250 Horizontal Distance (ft) NX N S S NS NX NS ss T=3.75 s, H=48ft WL HK 0.46mm sand Initial slope ——:—After 30 hr After 60 hr 50 100 150 200 250 Horizontal Distance (ft) Unpublished profiles number 5. developed in Saville's (1957) large-scale test The limit depth of interest is the offshore edge of the region par- ticipating in the yearly onshore-offshore cycle of sand transport. With the rationale that extreme storm waves will carry sand beyond the limit depth, removing it from the cycling system, this yearly limit depth can be estimated using the previous development with a yearly extreme wave condition typical of a specific site. A representative extreme might be the high waves expected to give extreme bottom velocity for 12 hours per year. Considering nearshore wave measurements in Thompson (1977), the 10 design wave conditions in Table 2 were selected as a representative range for exposed U.S. coasts. The table shows that the limit depths calculated, using equation (6), are all roughly twice the input wave height. This direct dependence on wave height and the fine trends of the calculated results are in agreement with an approximate form of equation (6). Re- taining only the dominant first term on the left-hand side of equation (7) leads to the expression: d, = 2.28 H (1 - 4.78 H/Lo). (8) For the 10 conditions in Table 2, at a given wave height, the calculated limit depth increases above twice the wave height as wave period or Ly increases; at a given wave period, the calculated depth drops below twice the decreasing wave height. Equation (8) gives a depth within 3 percent of that obtained using equation (6) for these conditions. The calculated results (Table 2) show general agreement with other published conclusions concerning the limit depth to the active beach (Bruun, 1954; Dietz and Fairbridge, 1968; Winant, Inman, and Nordstrom, 1975). Some unpublished bathymetry studies at the Coastal Engineering Research Center (CERC) resulted in conclusions summarized by Duane (1976). Repetitive profiles superposed, revealing little bed change beyond these approximate depths: 15 feet for the Great Lakes, 20 feet for the U.S. Atlantic coast, and 25 feet for the southern California Pacific coast. Design wave conditions 2, 5, and 8 in Table 2 would be reasonable first choices for these three locations, and the calculated depths agree with the stated limit depths, so it appears the proposed calculation procedure may give an accurate estimate of the yearly sea- ward limit to the active profile. Further evaluation of the proposed calculation procedure can be made using bottom changes offshore of La Jolla, California, which have been intensively investigated by the Scripps Institution of Oceanography. Shepard and Inman (1951) reported bathymetric survey data establishing Significant bed cut and fill to water depths of at least 100 feet; how- ever, ''On the outer shelf the movement takes place more parallel to the shore than normal to it."" Inman and Rusnak (1956) measured ranges in sand level over a 3-year period. In mean water depths of 18, 30, 52, and 70 feet, ranges were (>2), 0.29, 0.16, and 0.15 foot, respectively, and a seasonal trend in sand level was detected at the 30-foot depth, but not at the two deeper stations. 69° 6£ SS°6f TZ20°0 cl 0c Ot OL°7E L6°TS L1Z0°0 as oT 6 BE IG 96° V2 £9T0°0 cl cl 8 cole 60°TES ¢TZ0°O OL oT L 62° bZ COMZG vez0°0 OL cL 9 L9°0¢ 67°02 S610°0 OL OT S LV 6 cv oT S0£0°0 8 OL v IT 9T T6°ST vvz0°0 8 8 £ Sv vl c8 Vl vev0°o 8) 8 c SSaeE SSeee S7Z0°0 9 9 Tt (3) (8) “be G5) (9) “be °71/H (s) potsed (4F) 2y8Tey ovem uoT} Tpuod wozz y3deq *ssoudseqs SAC M ZUBITFTUSTS oACM *(Ze20Ul SQS*Q = OOF [) SUOTITPpuOD sAeM USTSep QT TOF syqdop YIWTT pe¢wetnoqTe)D °*Z eTqeL 16 Nordstrom and Inman (1975) conducted a study of wave climate and beach response at Torrey Pines Beach, north of La Jolla; beach profiles were measured at approximate monthly intervals during June 1972 to April 1974. Pawka, et al. (1976) reported the wave climate recorded by offshore pres- sure sensors up to four times daily from February 1973 to May 1974. Dur- ing the study, Torrey Pines Beach underwent seasonal changes in config- uration related to changes in wave regime. High-energy winter waves re- moved sand from the subaerial beach and deposited it offshore at depths less than 30 feet below mean sea level (MSL). Low-energy summer waves returned sand to the subaerial beach from depths less than 20 feet be- low MSL. Figure 5 shows the offshore survey data along the three ranges dur- ing the Torrey Pines Beach study (App. B in Nordstrom and Inman, 1975). Measured sand level changes were less than 2 feet beyond a water depth of about 21 feet relative to MSL, and less than 1 foot beyond a water depth of about 26 feet relative to MSL. These fathometer profile surveys have a stated accuracy of + 1 foot, so recorded changes less than 1 foot indicate profile closure. The reference rod arrays at 24 feet below MSL showed ranges in sand level of 1.2 to 1.5 feet during the study; arrays at 33 feet below MSL showed ranges of 0.2 to 0.4 foot, with a stated measurement accuracy of 0.3 foot. These data show that 26 feet below MSL is the limit depth to the active profile at the study site. A characteristic yearly design wave should be chosen considering both the breaker observations tabulated in Appendix D of Nordstrom and Inman (1975), and the offshore wave measurements tabulated in Appendix D of Pawka, et al. (1976). Among the 470 daily observations during 23 months, 10-foot-high breakers were noted three times, with a 14-second period on 12 February 1973, an 18-second period on 28 February 1973, and a 10-second period on 18 April 1973. Among the 657 wave measurements dur- ing 16 months, the most energetic condition was reported as 3,840 square centimeters, centered at a period of 14.2 seconds on 12 February 1973; this energy level corresponds to a significant deepwater height of 7.2 feet, and a significant height of 8.0 feet in a 33-foot depth. The den- sity of either wave data set is not sufficient to make the extreme tab- ulated condition represent a 12-hour per year duration, so a higher wave should be chosen as the design condition. A nearshore height of 10 to 11 feet and a period of 10 to 18 seconds seem appropriate design wave conditions for this locality. For these conditions, equation (6) gives depths of 20.3 to 23.9 feet. At Torrey Pines Beach, mean lower low water is =-2.7 feet relative to MSL; adding this possible tidal effect to) the calculated depth results in a limit depth estimate of 23.0 to 26.6 feet relative to MSL. Thus, the proposed calculation procedure gives an ac- curate estimate of the actual limit depth to the active profile in this case. V. CONCLUSION Using the number introduced in equation (2) and several quantitative assumptions, the water depth at a hypothetical seaward limit of intense 17 soutg AdIIO] 3% pL6T Ttady oF 02 ZZ6T eunr 9 Woz eIep ABAANS 9IOYSFFIO OOv 0oOo8 aS “(SL6T ‘UeWUT PUe WOTSPLON WOTF) eLuLOFITe) ‘ydeog (+5) 002'! *g oansTy YADW YOU9G WoO1y aoUDIsSIG 00s‘! 000‘2 oore oos'2 ooze 009'¢e 09- “sy fo) ¥ UO01l,DA|3 ° if (+3) O2- aBuoy unos OI- SW 18 ponutjuo)--*(S/6, ‘ueWU, pue WOTSpION WOLF) BIULOFT[e) ‘Yyoeag soutq AVILO], 18 pL6T Ttady 0¢ 02 ZZ6T 9uNC 9 WorF eIEp ABAINS oLOYSFFO 00v 008 4p = (43) Y4DW Yyoueg wooly 002'l 009 000‘2 aUDISIG oor2e oos‘2 ooe‘¢e aBbuoy uokundg upipuy, ‘Gg omn3sTJ ooge O))= Os- Ores 0 2- (0) [|= SW (44) Uoljone|3 19 penutjuo)--* (S76, ‘UueWwU, pue WOTJSpION WOLF) eTUOFTTeD ‘Yydeog soutg AdIIO]L 1& pL6T [tidy oF 02 ZL6T OUNL 9 WOIF eJep ABAINS oLOYSFFO *S OANBTY (43) WADW YdUaG WOIY adUDIsSIG OO0b 008 00z'! 0o9'l 0002 oor'2 0082 ood'¢ abuoy uUs4oN oo9g'¢ 109- fo} + | (43) Uorjones3 O2g- Olle ISW 20 bed agitation can be calculated using Figure 1 or equation (8) with a given wave height and period. Factors ignored in the presented develop- ment, such as tidal currents, sand size, and bottom slope, can clearly be pertinent to the limit depth of the active beach, but wave height and period have long been recognized as dominant factors in beach processes. Crucial and somewhat arbitrary choices are taking 9%, = 1 and e = 0.03 in using equation (2). Since the calculated limit depths for natural beaches are about 30 feet, (ed,/2) is about 0.5 foot and the wave energy can raise bed sand above large, natural ripples with heights up to 0.5 foot (Inman, 1957). Thus, the criterion used for intense bed agitation is internally consistent with calculated results. Calculated depths agree with measured water depths over the terrace frequently cut into plane slope of fine sand by constant laboratory wave action. For high waves expected 12 hours per year, calculated depths agree with previously published conclusions on the yearly limit depth to the active beach profile. With a more extreme wave condition as input, the calculation procedure could provide an estimate of the seaward limit over a longer period; however, field data are not available to check such an estimate. The present results encourage further evaluation of the proposed calculation procedure. 2| LITERATURE CITED BAGNOLD, R.A., "The Flow of Cohesionless Grains in Fluids", Phtlosophtcal Transactions of the Royal Society of London, Ser. A, Vol. 249, 1956, pp. 235-297. BRUUN, P., ''Coast Erosion and the Development of Beach Profiles," TM-44, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., June 1954, CHAN, K.W., BAIRD, M.H.I., and ROUND, G.F., "Behaviour of Beds of Dense Particles in a Horizontally Oscillating Fluid," Proceedings of the Royal Soetety of London, Vol. A 330, 1972, pp. 537-559. CHESNUTT, C.B., “Analysis of Results from 10 Movable-Bed Experiments,"' Vol. VIII, MR 77-7, Laboratory Effects tn Beach Studies, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., in preparation, 1977. CHESNUTT, C.B., and GALVIN, C.J., Jr., "Lab Profile and Reflection Changes for Ho/Lo = 0.02," Proceedings of the 14th Coastal Engineering Conference, 1974, pp. 958-977. DIETZ, R.S., and FAIRBRIDGE, R.W., "Wave Base," The Encyclopedia of Geo- morphology, Reinhold, New York, 1968, pp. 1224-1228. DUANE, D.B., "Sedimentation and Coastal Engineering: Beaches and Harbors," Martine Sediment Transport and Envtronmental Management, Wiley, New York, 1976, pp. 493-517. EAGLESON, P.S., GLENNE, B., and DRACUP, J.A., "Equilibrium Characteristics of Sand Beaches in the Offshore Zone," TM-126, U.S. Army, Corps of Engi- neers, Beach Erosion Board, Washington, D.C., July 1961. FREUND, J.E., Mathematteal Stattsttes, Prentice-Hall, Englewood Cliffs, Nedon IOG62Z, GRACE, R.A., and ROCHELEAU, R.Y., ''Near-Bottom Velocities under Waikiki Swell,'' Rep. 73-31, University of Hawaii, Honolulu, 1973. INMAN, D.L., "Wave-Generated Ripples in Nearshore Sands,' TM-100, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Oct. 1957. INMAN, D.L., and BAGNOLD, R.A., "Littoral Processes," The Sea, Vol. 3, Interscience, New York, 1963, pp. 523-549. INMAN, D.L., and RUSNAK, G.S. "Changes in Sand Level on the Beach and Shelf at La Jolla, California,'' TM-82, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Sept. 1956. 22 KOMAR, P.D., and MILLER, M.C., ''The Threshold of Sediment Movement under Oscillatory Water Waves," Journal of Sedimentary Petrology, Vol. 43, No. 4, Dec. 1973, pp. 1101-1110. LeMEHAUTE, B., DIVOKY, D., and LIN, A., "Shallow-Water Waves: A Compari- son of Theories and Experiments,'' Proceedings of the 11th Coastal Engineering Conference, 1968, pp. 86-107. LOFQUIST, K.E.B., "An Effect of Permeability on Sand Transport by Waves," TM-62, U.S. Army, Corps of Engineers, Coastal Engineering Research Cemiecir, Wore Beivoiie, Wao 5 Wacs 197S> MADSEN, O.S., ''Wave Climate of the Continental Margin: Elements of its Mathematical Description," Marine Sediment Transport and Environmental Management, Wiley, New York, 1976, pp. 65-87. MOGRIDGE, G.R., and KAMPHUIS, J.W., "Experiments on Bed Form Generation by Waves,"' Proceedings of the 13th Coastal Engineering Conference, 1972, pp. 1123-1142. MONROE, F.F., "Oolitic Aragonite and Quartz Sand: Laboratory Comparison under Wave Action," MP 1-69, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Washington, D.C., Apr. 1969. NAYAK, I.V., "Equilibrium Profiles of Model Beaches," Report HEL 2-25, Hydraulic Engineering Laboratory, University of California, Berkeley, Gelli 5 USYO- NORDSTROM, C.E., and INMAN, D.L., ''Sand-Level Changes on Torrey Pines Beach, California," MP 11-75, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Dec. 1975. PAUL, M.J., KAMPHUIS, J.W., and BREBNER, A., "Similarity of Equilibrium Beach Profiles," Proceedings of the 13th Coastal Engineering Confer- ence, 1972, pp. 1217-1236. PAWKA, S.S., et al., "Wave Climate at Torrey Pines Beach, California," TP 76-5, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., May 1976. RAMAN, H., and EARATTUPUZHA, J.J., "Equilibrium Conditions in Beach Wave Interaction," Proceedings of the 13th Coastal Engineering Conference, 1972, pp. 1237-1256. RECTOR, R.L., "Laboratory Study of Equilibrium Beach Profiles," TM-41, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Aug. 1954. 23 SAVILLE, T., Jr., "Scale Effects in Two Dimensional Beach Studies," Proceedings of the Seventh General Meeting, Internattonal Association of Hydraulte Research, 1957, pp. A3-1 - A3-10. SHEPARD, F.P., and INMAN, D.L., "Sand Movement on the Shallow Inter- Canyon Shelf at La Jolla, California,"’ TM-26, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Nov. 1951. SILVESTER, R., Coastal Engineering,Vol. 2, Elsevier, Amsterdam, 1974, Ds SHO. SUNAMURA, T., and HORIKAWA, T., ''Two-Dimensional Beach Changes due to Waves ,'"' Proceedings of the 14th Coastal Engineering Conference, 1974, pp. 920-938. THOMPSON, E.F., “Wave Climate at Selected Locations along U.S. Coasts," TR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Jan. 1977. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protectton Manual, 2d ed., Vols. I, II, and III, Stock No. 008-022-00077-1, U.S. Government Printing Office, Washington, D.C., 1975, 1,160 pp. WATTS, G.M., ''Laboratory Study of Effect of Varying Wave Periods on Beach Profiles ,"' TM-53, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Sept. 1954. WINANT, C.D., INMAN, D.L., and NORDSTROM, C.E., "Description of Seasonal Beach Changes using Empirical Eigenfunctions," Journal of Geophysical Research, Vol. 80, No. 15, May 1975, pp. 1979-1986. 24 APPENDIX A CHOICE OF LENGTH SCALE IN SEDIMENT ENTRAINMENT PARAMETER Published results for oscillatory flow over horizontal sediment beds indicate grain diameter, D, has a weak influence on the transition from a gently agitated, rippled bed to an intensely agitated bed with lesser bed forms. Mogridge and Kamphuis (1972) reported wave tank and water tunnel tests showing that bed-form height reaches a maximum and then steadily decreases with increasing A/D, where A is near-bottom horizontal fluid orbit diameter; the maximum bed-form height depends primarily on A and only slightly on D. Chan, Baird, and Round (1972) reported tests in a small-diameter pipe showing a well-defined transition from a rippled bed to a regime of intense bed agitation with surface particles in motion throughout the oscillation cycle. The empirical expression for this transition is Melee Gg Ospepeed yon2 = 00g (Al) where W is oscillation frequency, v is kinematic fluid viscosity, and D occurs as a relatively weak factor. The sediment mobility parameter (eq. 1) is a function of wave height and period, water depth, and grain diameter. In writing the sediment entrainment parameter (eq. 2), replacing D by ed constitutes a direct approach to clarifying the algebraic dependence of the limit depth, d,, on wave height and period, combined in (H)/Lo). Because definitive data on sediment entrainment are lacking, any choice for the length scale in the denominator of this Froude number must be tentative on physical grounds. Four lengths naturally occur in a shoaling wave: H, d, L, and A. For the four possible expressions, Table A-1 shows the resulting functional dependence of limit depth on wave steepness (corresponding to eq. 6), and the value of each expression at € = 0.9 divided by its val- ue at € = 0.3, the important range of limit depth. _Using ed for the length scale gives calculated results in agreement with experimental ev- idence over a wide range of — or H,/L, (Fig. 3 and App. B), so the other possible choices cannot, because the other values in the last column of Table A-1 differ greatly from the first. 72s) fable A-1. Comparison of results with four possible length scales. Length Resulting dependence (Bes at —€ = He) scale on (H/Lo)* \Value at & = 0. Water depth, d — sinh? & tanh & 83.82 Wave height, H sinht & WZE) U2 Wavelength, L sinh? & tanh — 2 oe! Near-bottom orbit sinh? & avi; Diameter, A = H/sinh & 26 APPENDIX B LABORATORY STUDIES OF PROFILE DEVELOPMENT The data base for the comparison in Figure 3 is 47 published profiles from 7 laboratory studies with constant wave action on an initially plane slope of fine sand. Table B-1 lists these profiles from seven references. Column 1 gives the original test identification. Column 2 lists the slope of the initial plane beach; values in parentheses indicate the slope was truncated by a steeper offshore slope. Colum 3 gives the sand diameter, and colum 4 gives the time elapsed in forming the profile. Columns 5 and 6 list the wave condition; H,/L, was calculated using linear wave theory and the reported condition, where necessary. Column 7 lists the maximum water depth in the test tank in the dimensionless form d+/L+, where L+ is the maximum wavelength in the tank. Column 8 gives the limit depth calculated from equation (5) for the listed Ho/ Lo; value in parentheses indicates the calculated depth is equal to or greater than the column 7 depth. Column 9 gives the calculated limit depth, -dg 5 unless parentheses occur in colum 8. Column 10 gives the cut depth, dg; value in parentheses indicates the cut depth is uncertain, and a dash indicates no clear cut occurred. Column 11 gives the percentage differ- ence between d, and d,; the 27 values not in parentheses are the tests included in Figure 3. Colum 12 lists subjective profile classification. 27 Table B-1. Test conditions and final profile characteristics in seven laboratory studies. Pe 4. bg 6. Te 8. 11, 1 Running Wave condition Water Equation (5) Final Test Initial diameter, time, ‘ depth! 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