Beach Topography and Beach Cusps

Beach topography and beach cusps

ROGER N. DUBOIS Department of Geography, University of Maryland Baltimore County, Baltimore, Maryland 21228

ABSTRACT The purpose of this paper is to investigate the influence of beach topography on the formation of beach cusps in a marine environ- The development of beach cusps was studied along the northern ment. The association between the breaching of a berm by the shore of Delaware from June 9 through June 30, and from July 28 backwash which cuts closely spaced channels through a berm and through August 10 of 1976. At the study site, the formation of the formation of beach cusps has been reported in the literature beach cusps was dependent upon the existence of a tidal berm in (Palmer, 1834; Jefferson, 1899; Otvos, 1964). Specifically, Sal- the developing stage and of a favorable backshore topography. lenger (1975) repeatedly observed the following conditions which Beach cusps developed as follows: after an erosional event on a led to the development of beach cusps along Parramore Island off sandy beach, a berm developed at low tide. The swash extended the eastern shore of Virginia: (a) following an erosional event on a over the berm and ponded between the berm and the backshore. beach, a berm developed on the foreshore at low tide. The swash The stream flow from the ponded water to the sea cut closely extended over the berm and ponded between the berm and the spaced channels through the berm. As the tide rose, the berm and backshore; stream flow from the ponded water cut closely spaced channels migrated landward. When the tide fell, the swash could no channels through the berm. The spacing of the channels was longer overtop the berm, and no water was ponded landward of the suggested to be controlled by subharmonic edge waves, (b) With a berm. Since no water was returning seaward through the channels, rising tide, the berm and channels migrated landward, (c) As the the channel form could not be maintained, and the swash flared the tide began to fall, the swash could no longer overtop the berm crest, channels into bays. A series of beach cusps appeared on the beach and no water was ponded landward of the berm. With no water as the tide continued to fall. The spacing between cusps was irregu- returning seaward through the channels, the swash reshaped the lar and was attributed to the irregular size of swash salients. At channels into bays, and a system of beach cusps was developed high tide, the horizontal distance from the berm crest to the along the beach. When beach cusps are formed by the backwash backshore (L) was less than 12 m. At high tide where L was greater breaching of a tidal berm, it is here suggested that two critical than 12 m, no beach cusps formed. Although a tidal berm de- beach elements be required: (1) a tidal berm, and (2) a favorable veloped, there was no effective bachwash to cut through the berm. backshore topography. The horizontal distance from the berm crest As the new tidal berm developed and as the swash overtopped the to the backshore should be as important to the formation of cusps berm crest, the swash continued landward and during a short as the presence of a berm. Consider Figure 1, which shows beach period of time flooded a portion of the backshore. No cusps were profiles with different types of beach topography. A tidal berm has observed to develop after a tidal berm had been constructed; berms developed at the foreshore of each of the four profiles. In Figure 1 A, and cusps developed together. the berm has formed near a relatively steep backshore; in IB, near a storm berm; and in 1C, near a beach scarp. The horizontal line INTRODUCTION from the top of the berm to a point intersecting the backshore (L) is

Several hypotheses have been put forth to explain the origin of FOREDUNE beach cusps. All of these hypotheses can be classified into three broad groups. The first group of hypotheses associates some physi- cal characteristics of incident waves with the formation of beach cusps: intersecting wave trains (Branner, 1900; Dalrymple and La- TIDAL nan, 1976), swash length (Longuet-Higgins and Parkin, 1962), BERM wave energy (Russell and Mclntire, 1965), and wash salients (Gorycki, 1973). The second group, which deals with a single hypothesis, suggests that edge waves are the cause for the origin of some beach cusps (Galvin, 1965; Bowen and Inman, 1969, 1971; Komar, 1973; Guza and Inman, 1975). The third group of hypotheses states that cusps are a function of the interaction between incident waves and beach topography: irregular depressions on a beach (Johnson, 1919; Otvos, 1964), irregular indentures along a foreshore (Kuenen, 1948), and breaching of a berm (beach ridge) (Jefferson, 1899; Evans, 1938; Williams, 1973; Sallenger, 1975). An analysis of all data reported by the cited authors shows that beach cusps can form under a variety of beach and wave conditions, and that no single process can account for the formation of Figure 1. Beach profiles with tidal berms. L is the horizontal distance beach cusps in all types of beach environments. from the berm crest to the backshore.

Geological Society of America Bulletin, v. 89, p. 1133-1139, 10 figs., August 1978, Doc. no. 80802.

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BACKSHORE The methodology of this study was relatively simple but time consuming. The beaches in the study area, especially at Key Box Road and at Half Way Road (Fig. 3), were monitored from June 9 through June 30, and from July 28 through August 10 of 1976; the total study time was five weeks. At Key Box Road and at Half Way Road, four beach profiles at intervals of 61 m were surveyed, and photographs of the beach were taken as foreshore conditions changed.

BACKSHORE RESULTS

The results of this study are chronologically presented as beach conditions changed. Beach profiles and photographs are presented as evidence in support of the qualitative description of how beach cusps formed at the study area. Throughout the study period when cusps were present, the spac- B. BACKWASH ing of beach cusps was irregular; for example, the spacing within a Figure 2. (A) Swash in the form of salients advancing over a developing set of cusps ranged from 26.5 to 39.6 m. The average angle of the tidal berm. "Hie dotted area is the tidal berm in the developing stage. (B) seaward slope of horns (8.3° of arc) was steeper than the average Water from adjacent salients merge and cut channels in the tidal berm. angle of bays (5.7°). Also, the angle of the slope immediately shoreward of the crest of the beach cusps varied from a 1.0° of arc relatively short in Figures 1A through 1C. As a berm develops and seaward dip to a 1.0° landward dip; when no cusps were present, as the swash overtops the berm crest, the water collects in the nar- the shoreward angle from the berm crest was dipping landward at row zone between the berm crest and the backshore; the ponded about 2.4° of arc. As cusps developed, waves approached the water returns seaward and cuts closely spaced channels through the shoreline at an oblique angle. berm (Fig. 2). Note that in Figure ID, in contrast, L is relatively long. Except for the foredune, the tidal berm is the highest elevational feature on the beach. Consequently, as the swash overtops the developing tidal berm in Figure ID, the swash energy dies out in the backshore, and the swash percolates into the beach. Once the voids in the backshore sediments are saturated with water, a portion of the backshore becomes flooded. To release the ponded water in the backshore, a few widely spaced channels are cut through the developing tidal berm; no beach cusps consisting of closely spaced bays or troughs are formed on the foreshore. If the assumptions about the behavior of the swash and backwash are correct in Figure 1, then the formation of beach cusps by backwash breaching of a tidal berm is dependent upon the presence of a tidal berm and a relatively short horizontal distance from the tidal berm crest to the backshore (L). The development of a tidal berm depends upon the immediate wave condition; however, the distance of L not only depends upon the immediate wave condition but also upon the beach topography prior to the formation of the tidal berm and consequently of beach cusps. Even if a tidal berm forms and water is ponded between the berm and the backshore, it does not necessarily follow that beach cusps will develop. The as- sumption that an initial beach topography can influence the formation of beach cusps was tested in the field.

STUDY AREA AND METHODOLOGY

The study was conducted along the northern shore of Delaware from the Indian River Inlet to Cape Henlopen (Fig. 3). This study area was selected because beach cusps frequently exist along the shore. The coast of Delaware is composed of a barrier system which includes baymouth barriers and barriers against pre-Holocene highlands (Kraft, 1971). At the study area, the average width of the backshore ranged from about 15 m just north of the Indian River Inlet and of Rehoboth Beach to about 75 m in the area of Key Box Road (Fig. 3). The average diameter of foreshore sediments is in the range of medium sands (0.31 mm), and the sediments are moder- ately well sorted. The average tidal range is approximately 1.2 m. Figure 3. Location of the study area.

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STORM BERM Figure 4. Beach profiles at Half Way Road, Delaware.

JUNE 14 JUNE 15 JUNE 22 • • • • JUNE 27 —• • JUNE 29

DISTANCE (M)

June 9 through June 30 The next major change in the configuration of the beach occurred on June 22 when waves eroded the seaward face of the tidal At the beginning of the study at Key Box and Half Way Road, berm. The erosional processes caused a beach scarp to develop at the main topographic features of the beach consisted of a storm Half Way Road (Fig. 4). No beach scarp was formed at Key Box berm and a tidal berm. The first major change in the configuration Road; instead, the tidal berm was shifted landward by the waves. of the beach occurred on June 13 when a storm from the east After the erosional event, wave processes were slow to reform the caused waves of 1.5 m in height to break on the foreshore; the tidal tidal berm. During the evening of June 26, small plunging waves of berm was destroyed, and the swash overtopped the pre-existing 0.3 to 0.6 m in height were observed; the wave period increased storm berm. On June 14 after the storm, the major topographic from 8 to 12 sec, causing the tidal berm to grow in size. On the feature on the beach was the storm berm (Fig. 4). No beach cusps morning of June 27, beach cusps had developed at Half Way Road; were formed; the foreshore profile was concave skyward. On the the swash was overtopping the tidal berm, partially reflecting off of following day (June 15th), a new tidal berm and beach cusps were the scarp (Fig. 4), and returning seaward by way of channels cut constructed on the foreshore (Fig. 4) in the manner as described by through the berm. L was 0.0 m at high tide; between the scarp and Shallenger (1975): with the rising tide, the berm was breached by the berm crest, the angle of the slope was dipping 1.0° of arc sea- closely spaced channels (27 to 38 m) which were cut by the ward. At Key Box Road, a tidal berm developed without beach backwash; with the falling tide, the swash flared the channels to cusps (Fig. 5). Here at Key Box at high tide, L was approximately produce bays. Also with the falling tide, the horns grew seaward; as 13 m. Beach cusps could not form because there was no effective the swash refracted from the horns, coarse sediments transported backwash to cut closely spaced channels through the tidal berm. by the swash were deposited on the seaward face of the horns. At The swash overtopped the berm crest, collected in the swale be- high tide, the horizontal distance from the crest of the serrated tidal tween the storm berm and the tidal berm, and eventually flooded berm to the backshore (L) was approximately 10.6 m. The sea was the swale area. Widely spaced channels, approximately 92 m apart, calm as the beach cusps developed. Wave height at break-point was were cut through the developing tidal berm to release the ponded about 0.9 m; wave period was 8.5 sec, and waves were of the water in the swale. Deposition occurred on the tidal berm as the plunging type. From June 16 through June 18, the beach cusps swash overtopped the berm crest at high tide. By June 29, the berm were slowly destroyed; the horns were truncated, and the bays were crest at Key Box Road had aggraded by 0.3 m, and prograded by filled in. By June 18, a tidal berm existed without beach cusps. 3.0 m (Fig. 5); L was now approximately 18.3 m at high tide, and

3 I— Figure 5. Beach profiles at Key Box Road, Delaware. STORM BERM

KEY BOX ROAD PROFILE 1 z o

JUNE 22 JUNE 27 JUNE 29

20 30 40 50

DISTANCE (M)

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no beach cusps had formed. By June 29 at Half Way Road, the 10 sec. From August 3 through 8, some beach erosion occurred, but beacn cusps had also aggraded and prograded seaward; the depo- no redeposition of sediments took place on the beach, and no beach sitional processes buried the beach scarp (Fig. 4). Beach cusps ex- cusps formed. tended from Half Way Road to the Indian River Inlet; no beach In the evening of August 9, Hurricane Belle passed by the Del- cusps existed from Key Box Road to Rehoboth Beach. marva Peninsula. There is no doubt that during the hurricane event some beach erosion must have taken place in the area of the tidal July 28 through August 10 berm. On the following morning of August 10, the sea was calm. Wave height at break-point was 0.6 m, and waves were plunging; From July 28 through August 2, the beach topography consisted wave period was 10 sec. At Key Box Road, a large storm berm had of a storm berm and a large tidal berm which extended from Re- been constructed by the hurricane. The storm berm was flanked hoboth Beach to the Indian River Inlet. The profile of a beach at seaward by a tidal berm that was being constructed by the calm this time was very similar to the beach profile at Key Box Road on waves; L was about 33.5 m at high tide. At Half Way Road, only June 29 (Fig. 5). No beach cusps were present. The elevation of the one large berm existed on the beach; L was about 36.6 m at high tidal berm crest was higher than that of the storm berm crest; thus, tide. No beach cusps were present. At both locations, the swale L was ~85 m at Key Box Road and 46 m at Half Way Road. The landward of the berm at the foreshore flooded as the tide rose; to average angle of the foreshore slope was relatively steep at 10.6° of release the ponded water, channels spaced about 106 m apart were arc. The sea was relatively calm; at break-point, wave height was cut through the berm (Fig. 6). However, at the southern part of Cape 0.3 to 0.6 m, and waves were plunging. The wave period was about Henlopen and just north of Rehoboth Beach, beach cusps were

Figure 6. Foreshore at Key Box Road, Delaware. L is ~33 m. Flooding has occurred between the berm crest and the backshore; no beach cusps developed on the foreshore. One channel has been cut through the tidal berm to drain the flooded area.

Figure 7. Foreshore at the southern part of Cape Henlopen. L is —12 m. The backwash is cutting channels through a developing tidal berm.

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Figure 8. Foreshore at the southern part of Cape Henlopen. L is ~12 m. The backwash is cutting channels through a developing tidal berm.

forming with a new developing tidal berm. As the tide was rising, closely spaced channels (23 to 54 m apart) were being cut through the berm by the backwash (Figs. 7 and 8). At low tide, the swash flared the channels to produce bays, and the result was a foreshore topography of horns and bays (Fig. 9). At high tide, L was —12 m where beach cusps formed.

DISCUSSION

The results of this study support the concept that beach cusps can be developed by the seaward return of the backwash which causes closely spaced channels to be cut through a developing tidal berm during a rising tide; as the tide falls, the channels are flared into bays by the swash (Sallenger, 1975). The irregular spacing of the Figure 9. Foreshore north of Rehoboth Beach. L is —10 m. A series of beach cusps is exposed as the tide falls. cusps was associated with the irregular size of the swash salients. After a wave breaks on the foreshore, the swash separates into salients (Gorycki, 1973). As the height of a wave is not totally uni- may develop at positions of antinodes; thus, the distance between form along its breaking length, it follows that the size of the swash channels would be equal to one-half the length of the subharmonic salients will vary across the face of the foreshore. A large swash sa- edge wave (Sallenger, 1975). The length of an edge wave (L) is lient spreads over a relatively large area of a developing tidal berm given by Ursell (1952) as while a small salient spreads over a relatively small area of a berm (Fig. 2A). After the swash salients have advanced over the berm and L = ¿—T2 sin |(2n + l)/3] (1) L v as the backwash starts to return seaward, waters from adjacent salients merge to form a stream flow that carves out a channel in the where g is the acceleration of gravity, T is the wave period, n is the berm (Figs. 7 and 8). Since the water of large salients spreads over a model number, and /3 is the angle of the beach slope. If equation 1 relatively large area of the berm, the distance separating the chan- is solved when T is equal to twice the period of incident waves, /8 is nels is greater in locations of large salients as compared to locations taken as the average angle of the beach cusp slope, and n is equal to of small salients (Fig. 2B). Once the channels have been formed, 0, then L/2 equals 38 m for August 10; the average distance be- they provide paths of least resistance for the backwash. The energy tween channels was 32 m. Although the predicted and the observed of the backwash in the channels is greater than the energy of the distances between bays of cusps are similar, it is difficult to state swash; thus, the channel form is maintained until the tide falls. that a subharmonic edge wave governed the spacing of bays at the Sallenger (1975) has suggested that the spacing between channels study site because the spacing between cusps was highly irregular. or bays of beach cusps may be controlled by a subharmonic edge For example, within a set of six cusps that covered a distance of wave with twice the period of the incident waves. The channels 211m along the southern part of Cape Henlopen on August 10, the

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average distance between cusps was 35 m, but the range varied m at high tide. It should be noted, however, that as wave regimes from 23 to 54 m; the standard deviation was 11m. The irregular change, the maximum value of L below which beach cusps can spacing between cusps may be governed by the irregular height of form may also change. The L value of 12 m should not be consid- incoming waves, which in turn produces irregular sizes of swash ered as a fixed value for this study site nor for any other beach site. salients. Further, if L is measured two or three days after the formation of When a fully developed berm existed at the study site, and the beach cusps, the value of L may be erroneous. Recall from the re- swash salients could not overtop the berm crest, no beach cusps sults that beach cusps formed seaward of a beach scarp, and that L were formed. The swash salients advance up the foreshore slope was zero. After a day of sediments being deposited on and land- (Fig. 10A), and because of the relative steepness of the foreshore ward of the cusps, the scarp was buried, and the elevation of the slope, the backwash returns seaward in the form of a sheet flow cusps increased. If L had been measured after the burial of the scarp (Fig. 10B). As the backwash returns seaward, waters from adjacent and the elevational increase of the cusps, the value would have been salients do not extensively merge, and no stream flow occurs across erroneous and would not have represented the true value of L that the face of the foreshore; thus, no channels are formed, and no was associated with the construction of the cusps. cusps are developed. Although conclusions as to how beach cusps are formed frequently differ from one study to another, many of the recorded observations about beach cusps are similar to the observations re- TRANSVERSE VIEW PLANIMETERIC VIEW ported in this study. Palmer (1834) noted that waves cause slight depressions on a berm, and that these depressions grow to become channels. Jefferson (1899) reported that the important process which leads to the formation of beach cusps is the breaching of a berm by the backwash as the tide rises; a falling tide exposes a series of beach cusps. Longuet-Higgins and Parkin (1962) com- mented thdt bays are formed by the backwash and are exposed after the turn of the high tide..Russel and Mclntire (1965) wrote that cusps are formed in association with the development of a berm. Williams (1973) observed the formation of beach cusps as a result of the breaching of a berm. The process of berm breaching by stream flow from ponded water was also noted during a wave-tank experiment, although no ideal beach cusps were formed (Guza and Inman, 1975). The observations of Sallenger (1975) have been presented above. Figure 10. (A) Two salients advancing up the foreshore slope as indi- Other reliable observations on beach cusps development have vidual units. Since the salients do not overtop the berm crest, they return been reported in the literature; these observations, however, differ seaward as individual units. No channels are cut in the face of the foreshore, from the observations reported in this study. Evans (1938) wrote and no beach cusps are developed. that cusps are formed along the shore of Lake Michigan by the action of the swash which alternately breaches a berm or beach scarp. The results of this study also show that beach topography ap- The combined action of incident waves and edge waves has been pears to be an important variable associated with the development suggested (Galvin, 1965; Bowen and Inman, 1969, 1971; Komar, of the type of cusps that is formed by backwash breaching of a de- 1973) and observed (Guza and Inman, 1975) as the process re- veloping tidal berm. A juvenile berm must be formed adjacent to an sponsible for the development of some beach cusps. Guza and immediate backshore that has a higher elevation than that of the Inman (1975) noted, however, that the processes which produced berm crest. Thus, as the swash overtops the berm crest, the water the cusps in their wave-tank experiment may be duplicated only in can be ponded in a relatively narrow zone between the berm crest special field conditions, and that edge waves may not be responsible and the backshore (Figs. 1A through 1C). Stream flow from the for the formation of cusps on all beaches. ponded water cuts channels through the juvenile berm which in term leads to the development of cusps. At high tide when the hori- CONCLUSIONS zontal distance from the berm crest to the backshore (L) was equal to or less than 12 m, beach cusps were formed as a new tidal berm Beach cusps were studied along the northern shore of Delaware. developed. When L was greater than 12 m, no cusps formed; as a The construction of these beach cusps appears to be dependent tidal berm developed and as the swash overtopped the berm crest, upon the existence of a tidal berm in the developing stage and a the water flowed landward and flooded a portion of the backshore. favorable backshore topography. The following sequential events Although other studies on beach-cusp development have not inves- occurred each time beach cusps formed: tigated variable L, the pictorial evidence presented by some of these 1. Following an erosional event on a sandy beach, a berm de- studies tentatively support the concept that L may be an important veloped at low tide. variable associated with the formation of beach cusps. Photographs 2. The swash extended over the berm and ponded between the of beaches with cusps show that cusps are at a lower elevation than berm and backshore. At high tide, the horizontal distance from the that of the immediate backshore (Jefferson, 1899; Kuenen, 1948; berm crest to the backshore (L) was equal to or less than 12 m. Otvos, 1964). Longuet-Higgins and Parkin (1962) presented a dia- 3. The backwash in the form of stream flow cut closely spaced gram of beach profiles with cusps; in all cases, the cusps are at a channels through the developing berm. lower elevation than that of the immediate backshore. For the pres- 4. As the tide rose, the berm and channels migrated landward. ent study, beach cusps formed when L was equal to or less than 12 5. As the tide began to fall, the swash could no longer overtop

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the berm, and no water was ponded landward of the berm. As no Branner, J. C., 1900, The origin of beach cusps: Jour. Geology, v. 8, water was returning seaward through the channels, the form of the p. 481-484. Dalrymple, R. A., and Lanan, G. A., 1976, Beach cusps formed by intersect- channels was changed to bays by the action of the swash. ing waves: Geol. Soc. America Bull., v. 87, p. 57-60. 6. A series of beach cusps appeared on the beach as the tide con- Evans, O. F., 1938, The classification and origin of beach cusps: Jour. tinued to fall. The irregular spacing between cusps was attributed Geology, v. 46, p. 615-627. to the irregular size of swash salients. The irregular size of swash Galvin, C. J., Jr., 1965, Resonant edge waves on laboratory beaches: EOS salients may be governed by irregular heights of incoming waves. (Am. Geophys. Union Trans.), v. 46, p. 112. Gorycki, M. A., 1973, Sheetflood structure: Mechanism of beach cusp for- Along the length of a beach, the existence of beach cusps termi- mation and related phenomena: Jour. Geology, v. 81, p. 109-117. nated when L was greater than approximately 12 m at high tide. A Guza, R. T., and Inman, D. L., 1975, Edge waves and beach cusps: Jour. tidal berm was formed after an erosional event had occurred on a Geophys. Research, v. 80, p. 2997-3012. beach. Beach cusps did not form, however, because there was no Jefferson, M.S.W., 1899, Beach cusps: Jour. Geology, v. 7, p. 237-246. Johnson, D. W., 1919, Shore processes and shoreline development (fac- effective backwash to cut closely spaced channels through the simile ed., 1972): New York, Hafner, 584 p. berm. As the tidal berm developed and as the swash overtopped the Komar, P. D., 1973, Observations of beach cusps at Mono Lake, Califor- berm crest, the swash continued landward and flooded a portion of nia: Geol. Soc. America Bull., v. 84, p. 3593-3600. the backshore. At intervals of about 106 m along the beach, a Kraft, J. C., 1971, Sedimentary fades patterns and geologic history of the channel was cut through the developing berm to release the ponded Holocene Marine Transgression: Geol. Soc. America Bull., v. 82, p. 2131-2158. water in the backshore. No cusps were observed to develop after a Kuenen, Ph. H., 1948, The formation of beach cusps: Jour. Geology, v. 56, berm had been constructed because L was greater than 12 m or be- p. 34-40. cause the swash could not overtop the berm crest; berms and cusps Longuet-Higgins, M. S., and Parkin, D. W., 1962, Sea waves and beach developed together. Tidal berms, however, did develop without cusps: Geog. Jour., v. 128, p. 194-201. beach cusps. Also, no cusps formed on a beach during an erosional Otvos, E. G., Jr., 1964, Observations of beach cusp and beach ridge formation on the Long Island Sound: Jour. Sed. Petrology, v. 34, p. 554- event or on an erosional, concave beach profile. 560. Although the observations of this study agree with some obser- Palmer, H. R., 1834, Observations on the motion of shingle beaches: Royal vations of other studies, it should not be concluded that all beach Soc. London Philos. Trans., v. 124, p. 567-576. cusps develop in the manner that has here been described. The sum Russel, R. J., and Mclntire, W. G., 1965, Beach cusps: Geol. Soc. America Bull., v. 76, p. 307-320. of reliable observations on the formation of beach cusps suggests Sallenger, A. H., Jr., 1975, Mechanics of beach cusp formation [Ph.D. the- that there may be different methods by which cusps can be con- sis]: Charlottesville, Univ. Virginia, 212 p. structed on a beach. Ursell, F., 1952, Edge waves on a sloping beach: Royal Soc. London Proc., Ser. A, v. 214, p. 79-97. Williams, A. T., 1973, The problem of beach cusp development: Jour. Sed. ACKNOWLEDGMENTS Petrology, v. 43, p. 857-866.

I wish to express my gratitude to A. H. Sallenger, Jr. for review- ing the manuscript.

REFERENCES CITED

Bowen, A. J., and Inman, D. L., 1969, Rip currents, 2, Laboratory and field observations: Jour. Geophys. Research, v. 74, p. 5479-5490. MANUSCRIPT RECEIVED BY THE SOCIETY NOVEMBER 17, 1976 1971, Edge waves and crescentic bars: Jour. Geophys. Research, v. 76, REVISED MANUSCRIPT RECEIVED JULY 25, 1977 p. 8662-8671. MANUSCRIPT ACCEPTED AUGUST 8, 1977

Printed in U.S.A.

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