Field Observations of Edge Waves and Their Effect on Beach Material

Field observations of edge waves and their effect on beach material

D. A. HUNTLEY & A. J. BOWEN

SUMMARY Observations of wave motion are described, length of 34 4- 6 m. The implications of these particularly of the horizontal particle velocities, observations are discussed in the light of recent up to a maximum of 25 m from the shoreline laboratory experiments and theories of edge on Slapton Beach, Start Bay. For the first time wave generation, and a possible swash inter- on a natural beach, the observed velocities action mechanism for generating subharmonic reveal directly the presence of short period wave motion on Slapton beach is described. edge waves, modes of wave motion trapped to A steady longshore current prevented these the shoreline by refraction. These edge waves subharmonic edge waves from forming cuspate occurred at the first subharmonic (a/2) of the or crescentic features in the nearshore beach incident wave frequency a, and their expon- material. Small shoreline cusps observed on ential decay in amplitude with distance from Slapton beach on other occasions were con- the shoreline was consistent with edge waves of sistent with edge waves at the frequency of the mode number zero with a longshore wave- incident waves themselves.

SMOOTH SANDY COASTLINES commonly form large regular rhythmic sedimentary features at the shoreline and in the nearshore subaqueous region, and recent work has shown that many of these features can be related to the action of water waves. Edge waves provide a satisfactory explanation for the formation of sedimentary features which have a rhythmic pattern in the longshore direction, such as submerged crescentic sand bars and regular cuspate shorelines (Bowen & Inman x97I ). Edge waves are free modes of water motion trapped against a shoaling beach by refraction. Their amplitude varies sinusoidally along the shore and diminishes rapidly seawards from the shoreline. Surface waves incident on the beach, on the other hand, can explain the formation of long sand bars with crests parallel to the shore (Carter et al. 1973, Lau & Travis I973). A common aspect of these theoretical ideas is the need for standing rather than progressive waves, in order to provide the spatial differentiation in the hydro- dynamic regime necessary for the formation of rhythmic topography. However Sonu (I973) has questioned the possibility of standing edge waves on beaches which are very long compared to the edge wave wavelengths, since there is no obvious reason why edge waves should not migrate along the coast on such beaches. We have therefore conducted field experiments on a long beach with two primary aims" (i) to make direct measurements of the nearshore velocity field in an attempt to identify edge wave motion directly. Short period edge waves have been directly observed on laboratory beaches (Bowen & Inman I969), but their existence in the field has previously been inferred only from the success of edge wave interaction theories in explaining observed periodic ,sedimentary features and nearshore circulation patterns (Bowen & Inman I969, I97I). Jl geol. Soc. Lond. vol. x3x , t975, pp. 69-8I , 6 figs. PHnted in Northern Ireland.

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(ii) to associate measurements of the nearshore hydrodynamics with the formation of regular sedimentary features. The first aim has been achieved, with the first direct observation of edge waves on a natural beach. This has important implications for the possible generation mechanisms of edge waves. The second aim has not yet been achieved however. Some rhythmic sedimentary features observed are in satisfactory agreement with theoretical predictions, but no features were observed associated with the measured edge waves, due to the presence of a steady longshore current close to the shore, caused by obliquely incident wind waves.

I. Edge waves The theory of edge waves is well established (Stokes 1846, Lamb 1932 para. 260, Eekart t95t , Ursell t952 , Ball i967). Using the shallow water equations, Eckart (195I) found edge wave solutions for a beach of linear slope, h -- x tan/~, and Ball (i967) for a beach of exponential slope h = h0(t --e--~'), where h is the local water depth, x is the distance from the shoreline and/~ is the angle of the beach face to the horizontal. Their solutions predict a family of edge wave modes whose amplitudes decay in the offshore direction and vary sinusoidally in the longshore direction. The lowest order mode, with mode number n --o, decays exponentially offshore, while for higher order modes the mode number n gives the number of zero crossings of the amplitude in the offshore direction. Fig. t shows the offshore decay of the onshore, u, and longshore, v, edge wave velocities for modes n = I and 2, using Eckart's solutions. Another feature of the solutions is the existence of a dispersion relation linking the edge wave period T to the longshore wavelength L. Fig. 2 shows the dispersion curves for an exponential beach (Ball 1967) , the exponential parameters 0~ and h0 being chosen to fit the beach profile at Slapton, South Devon. This shows the existence of a family of dispersion curves, one for each mode number, each mode except n =-o ceasing to exist above a cut-off period. For a given period each mode has a predicted wavelength, but several modes with different longshore wavelengths may occur with the same wave period. Some possible mechanisms for generating edge waves have been studied in a number of recent papers but none has yet gained general acceptance. In fact, several different mechanisms may be able to generate edge waves on a natural coastline, since the effective trapping of the energy against the shore means that a relatively weak coupling between a driving force and the edge wave modes should induce the edge wave motion. Munk et al. (I956) have suggested that edge waves with periods of several hours may be generated by sudden changes in the wind and atmospheric pressure, and show results from the Californian coast to support this idea. Much shorter periods appear to be generated by the wind and swell waves incident on the shore, however. Gallagher (i971) has attempted to explain 'surf-beat' energy, in the range of periods 25-i oo seconds, in terms of edge wave motion generated by non- linear interaction between two incident wave trains. When the beat frequency Act between the two incident wave trains and the difference A;t between their

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longshore wave numbers are related by an edge wave dispersion relation, the incident waves and the edge waves can form a resonant triad, transferring energy from the incident waves to the edge waves. Interactions of this kind however cannot explain the existence of edge waves when only one incident wave train is present. Bowen & Inman (1969) for example found that edge waves could be generated on a laboratory beach from normally incident waves of a single frequency. Recent theoretical work by Birchfield & Galvin (in press) and Guza & Davis (1974) has attempted to show how normally incident waves reflected from the beach may be unstable to longshore perturbations. Guza & Davis show that non-linear shallow water wave theory can be used to estimate the strength of a resonant interaction between the totally reflected, standing wave component of the incident waves and a pair of free edge wave modes, whereby the edge wave modes can grow through energy transfer from the incident waves. Their theory applies only to the generation of edge waves of longer periods than the incident waves, and cannot therefore explain the existence of the important edge waves at the incident wave frequency itself. Nevertheless, they show that the zero-order subharmonic (frequency a/2) edge wave tends to grow most rapidly, in general agreement with the observations of Bowen & Inman (I 969). 501 4"0

3-0 3"0-

2-0 n=l 2-0- n-2 (x) v (x) 1"0 X) 1.0~

1 ___ I i J 0-50 0"75 1.00 0.25 0-50 0-75 1.00 -1.o°1 ) -1.0- x/L x/L FIG. I. Offshore dependence of the onshore u(x) and longshore v(x) currents for standing edge waves of modes n = I and n = 2. The amplitudes are normalized to v(x) = x atx =o.

150.0- BEACH PROFILE h =ho (1-e -(xx) 1 o~ ho=7"O5m /

100-0- FIG. 2 Edge wave dispersion curves n= 3 for modes n = 0-6 on ~ 50-0- Slapton Beach.

0 ! ~ I I ] 5"0 1 .0 15-0 20"0 25.0 T Secs

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Edge wave and near shorefeatures Edge wave motion provides a semi-quantitative explanation for many nearshore features which exhibit rhythmical longshore patterns. Both cuspate and crescentic features of nearshore sediment topography have been studied in relation to edge wave motion. Bowen & Inman (1971) showed that the existence of submerged crescentic bars can be explained by the pattern of drift velocities at the bottom boundary layer of a standing edge wave. Here, incident waves were assumed to be of small amplitude, contributing only to the threshold velocity required to set the beach material in motion. Observations of crescentic bars on natural coastlines suggest that edge waves with periods of 3o-6o sees and significant amplitudes may occur extensively on real beaches. Cuspate features on the other hand are usually formed at the shoreline, with the points of the cusps seaward. It seems likely that these features are caused by the finite amplitude of an edge wave at the shoreline, which results in a sinusoidal variation in the run-up amplitude along the shore. For an edge wave alone there is no motion at the nodes but an excursion of approximately a/tan/~ up and down the beach at the antinodes, where a is the edge wave amplitude at the shoreline. Since the theoretical drift velocity is always offshore near the coastline, erosion is expected to occur at the antinodes, thus forming a set of cuspate features with a wavelength of L[2, where L is the wavelength of the edge wave. This suggests that where both crescentic and cuspate features occur their wavelengths will be equal, with cusp points opposite horns of the crescentic bars, and this is in fact observed on natural beaches (Hom-ma & Sonu 1963). The presence of incoming waves complicates this picture. Bowen (1973) has shown that linear superposition of the run-up due to an edge wave and an incident wave of the same frequency and comparable amplitude should produce cuspate features of wavelength L while edge waves of a different frequency than the incident waves result in cusps of wavelength ½L. Bowen (I 969) has shown that second order interaction between incident waves and edge waves of the same frequency can generate cells of nearshore circulation containing strong seaward-flowing currents, known as rip currents, with the cell width along the shore equal to the longshore wavelength of the edge wave. Komar (1972) has studied the formation of shoreline cusps by nearshore circulation cells and finds that for given edge wave and incident wave amplitudes, cusps with the wavelength of the edge wave grow to a certain size with their points opposite the rip currents and then cease to grow as the forces driving the nearshore circulation cell become balanced by counter forces due to the oblique incidence of the incoming waves on the cuspate shoreline. Changes in the incident wave field or edge wave amplitude can then produce new circulation cells with rip currents which may be displaced from the point of the cusps. Komar's observations suggest that rip currents are important features in forming the shoreline topography on a wide range of scales but it has not been possible to show un- ambiguously that edge waves are responsible for longshore periodicity because it is difficult to relate the present theories for simple beach profiles to the observed complex cuspate beach shapes. Periodic nearshore sediment features are therefore expected to occur with a

longshore wavelength equal to the edge wave wavelength, when edge waves and incident waves are of the same frequency, or equal to half the edge wave wavelength, when edge waves are at a different frequency or when they alone dominate the velocity field. The regularity often observed in these nearshore features suggests that a single edge wave mode is generally present on natural beaches, although a whole set of modes is theoretically possible.

2. The field experiment Slapton Beach in Start Bay, South Devon, was chosen for a preliminary experiment to measure the nearshore velocity field because it has a relatively simple topography along a fairly straight coastline and extended about 2"5 km on either side of the point where measurements were made. The two-component electromagnetic flow-meter (Tucker et al., I97O ) was chosen to measure the horizontal velocity field in the turbulent nearshore water. This is a rugged and compact instrument with no moving parts and a relatively fast response time. For the field experiment, meters were mounted on tripods standing freely on the sea bed to measure the two components of the horizontal velocity field, and the instruments were connected by cable to recording systems on the shore. The tripods were placed close to the shoreline at low water and the tide then covered the instruments for a period of about four hours spanning high water, allowing measurements to be made from near o to up to 25 metres from the shoreline as the tide came in. This relatively simple technique has proved both flexible and reliable on beaches ranging from the steep shingle beach at Slapton to a very shallow (slope ~o.o I) sand beach, and under a variety of wave conditions including full storm waves on beaches of intermediate slope (~o.o4) ; experiments have not yet been conducted under storm conditions on the steepest beaches. Position and orientation changes of the tripods could be checked at each low water. No lateral movement of the tripods was detected on any of the beaches and, when the 1 m side triangular base of the tripods was dug into the beach to a depth of about t o cms, tilt of the heads out of the horizontal plane was at most a few degrees. Horizontal currents were recorded digitally, with a sampling period of 0. 5 secs, for separate runs of 15 min duration. The resulting time series were analysed both by band pass filtering to produce time series of velocity components in a chosen frequency band and by spectral analysis to produce spectra of the velocity field in the range o-I.O Hz.

Observation of edge waves The main features of the velocity field observed at Slapton beach are discussed in Huntley & Bowen (1975). In the present paper interest is focused on a set of measurements made on August xoth x972. On this day the breakers were small, about 25-3o cms from peak to trough at the breakpoint, and predominantly surging, with a surf zone about 6 m wide. Measurements were made with a single flowmeter, held 15 eros above the bed, which was covered for a four hour period as, with the incoming and receding tide, its distance from the shoreline varied

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from o m to z z m and back to o m. In analysing the data from these measurements it has been assumed that the velocity field remained quasi-stationary as the shoreline moved, so that the single fixed flowmeter measured the true variation of the velocity field along a line perpendicular to the shore; the consistency of the results supports this assumption. A typical onshore current spectrum, Fig. 3, shows a broad peak at about o.2 Hz, corresponding to the observed breaker frequency, and a steady fall in energy towards higher frequencies, typical of offshore, wind wave spectra. An unusual feature of the spectrum however is the rather sharp peak at about o.z Hz, about one half the predominant wind wave frequency. This peak was found in both longshore and onshore spectra and, surprisingly, was found to increase in size as the shoreline was approached, even within the surf zone (Fig. 4). In contrast, other peaks, including the wind wave peak at o.2 Hz and smaller peaks in the region of o.o6--o.o8 Hz, showed a slight increase towards the breaker line and then became indistinguishable inside the surf zone. This be- haviour has led us to identify the energy at o.z Hz with a zero order edge wave (Huntley & Bowen t973). Ball's theory (i967) for an exponential beach slope gives an edge wave dispersion relation of the form

0-2 = ½{(2n + z)(~ + 42") ½ --~(2n' + 2n + z)} (i) g~ho and, for zero order edge waves, the velocities decay offshore as (u, v) ,~ exp (--p0~x) (2)

Spectrum 2.8.5. 10.8.72 N = 1536 A t = 0-5 secs A f. 0.025 HZ (n 10~ =E

ol o,. 10 -2 c uJ

10-31 I I I I I 0 0.2 0"4 0.6 ~8 1-0 Frequency HZ F x o. 3. A typical onshore current spectrum, showing the wind wave peak at 0.2 Hz and an additional peak at o.x Hz. The spectrum was measured 9.o m from the shoreline.

where p(p + z) = ,~2[~; ~, a are the edge wave longshore wave number (2~r/L) and frequency (2~r[ T) respectively and ~, h0 are the parameters in the beach slope equation h = h0(z -- e-**). For Slapton beach, the beach profile in the intertidal zone could be reasonably approximated by the exponential equation, with h0 = 7"o5 m and 0c = 3"z4 × z o m. Equation z then predicts that a zero order edge wave of frequency o.I Hz has a wavelength L of 32 m (Fig. 2). We have therefore plotted in Fig. 4 the exponential decay predicted by equation 2 for an edge wave of wavelength 32 m. The measured values fit the expected wave curve to within experimental uncertainty. In fact the least squares best fit to the data gives L = 34 4-5 m for the onshore current and L = 34 4- 6 m for the longshore current. The presence of an edge wave provides the only possible explanation for the observed velocity field at o.z Hz. Although progressive gravity waves incident on a sloping beach will also experience a slight rise on approaching the breaker line, the predicted rise is much too small to explain the rapid increase found at o.z Hz; furthermore, incident progressive waves undergo a rapid decay in amplitude within the surf zone, while the observed amplitude continues to

0.50- 0-10-

x ONSHORE

0"40- \ ~ ~> LONGSHORE \ =~ x \ ~o.os- \\ ~. ,, ,,

ONSHORE IIE 0.30- \ ~ x I " "0 0 5 10 15 20 25 : Distance from shoreline M .m 0.20- E -i ~ ®° \ ! 0.10- LONGSHORE %, -

o-.- .T>T -T-~--T-- o 1o . io 2s Distance from shoreline M F zo. 4- Graphs of longshore and onshore current amplitude plotted against distance from the shoreline. The points marked with a vertical cross are amplitudes at o. zo 4- o.o2 Hz and the pecked lines passing through them show the expected decays for an edge wave of the predicted wave length of 32.o m. The vertical error bars corre- spond to 8o % confidence limits in the spectral analysis; horizontal errors are a consequence of averaging amplitudes during an incoming tide. Inset diagram shows amplitudes of smaller peak at o.o 9 Hz measured on a subsequent tide. Diagonal crosses onshore, diamond symbols longshore currents. Solid line for onshore points shows the calculated increase for an incident swell wave outside the breakpoint and a schematic decay inside the surf zone.

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increase well inside the breakpoint, which was about 6 m from the shore. On other occasions measurements of incident swell waves were made at Slapton beach when no edge waves were present. Fig. 4 also shows the onshore variation in amplitude of a much smaller peak at about 0.09 Hz on a subsequent tide, contrasted with the edge wave velocities. The smooth curve through this data represents the predicted increase for an incident wave and is in marked contrast with the exponential increase for the edge wave; the longshore component remains approximately constant for this incident wave. It may also be significant that the ratio of onshore to longshore current amplitudes for the data fitting the exponential is approximately constant, independent of offshore position, at 4.6 -J- I.O. For an obliquely incident wave of xo sees period, we might expect this ratio to increase towards the shore over the distance measured, as the wave refracts towards the beach; this increase is in fact seen for the swell wave at 0.09 Hz. On the other hand the presence of standing waves, formed by reflection of incident waves at the shoreline, would predict a rather more rapid increase in amplitude towards the shore than progressive waves (Carrier & Greenspan x958 ). However, calculations for I O second waves at Slapton show a predicted zero crossing of reflected wave amplitude at about 14 m from the shore and this cannot be satisfactorily reconciled with the observed velocities particularly at I I-O m offshore. Moreover the onshore and longshore currents at o-I Hz were observed to be in anfiphase (Fig. 5), in direct contrast to the phase quadrature expected for a standing gravity wave formed by reflection at the shore. The observed phase and amplitude ratio between onshore and longshore current components at o.x Hz both indicate that the edge wave was a standing wave, since a progressive edge wave would have components in quadrature and an average amplitude ratio of unity. Trapping of the edge wave to form a standing wave has therefore occurred despite the very small edge wave wavelength of 32 m in comparison to the length of the beach, which extended about 2. 5 km on either side of the flowmeter. The possibility of standing edge waves on a long beach was specifically rejected by Sonu (I973) , but our observations confirm that they can exist even when there is no obvious topographical trapping of the wave motion. This result means that edge waves can explain crescentic bar

iV~llT!717V-\I-1r-l'lI!'IU-UVV,,V~/VV~i;FIfI~-';-lI-~I':. ' ",.;;~:li Jff ~ ~"

0.50 Time (seconds) 4) 0 100 20o 300 400 0.25.n LONGSHORE ~

0"25 ~ FIo. 5. Filtered time series showing the variation with time of the horizontal velocity components of the o.I Hz edge wave.

formation on long beaches (Shepard I952; Hom-ma & Sonu 1963; Sonu 1973) as well as in confined bays. It is possible that the mechanism responsible for generating edge waves favours the formation of standing rather than progressive edge waves. The existence of zero-order standing edge waves at the subharmonic frequency a/2 of the incident wave frequency a seems to support recently proposed mechanisms of edge wave generation. The fairly steep beach at Slapton caused appreciable reflection of the incident waves and presumably set up a significant standing wave component. The theories of Birchfield & Galvin and Guza & Davis consider generation of edge waves by interaction with standing gravity waves, and, in particular, Guza & Davis show that zero-order subharmonic waves are likely to be the most significant edge wave components generated. However our observations suggest that the edge waves may be formed by a stronger interaction with the incident waves than the rather weak second order effect predicted by these theories. By filtering the analogue record of the currents through a bandpass filter centred at o.i Hz, it is possible to remove velocity components at the incident wave frequency and study the time dependence of the edge wave amplitude alone; Fig. 5 shows an example of filtered onshore and longshore currents. The amplitude of these filtered records commonly increases at least ten times more rapidly than the growth rate predicted by Guza & Davis. Close study of the surf-zone, with the aid of a film taken concurrently with the velocity measurements, suggests a strong interaction mechanism which may be responsible for the rapid changes observed. A common sequence of events in the surf zone is as follows. A steep breaker plunges into the very shallow water in the surf zone and surges rapidly up the beach. Before the backwash has begun a second wave arrives and travels forward into the deep shoreward-moving water as an unbroken surge, finally collapsing at the shoreline rather like a surging breaker. A strong backwash follows which interacts with the third wave to form a steep breaker and leaves only a shallow covering of water in the surf zone. The cycle is then repeated, resulting in a sequence of high breakers at the first subharmonic of the incident waves. Similar interference between successive swashes is described by Emery & Gale (195 I) and their observations, on beaches of different slopes, suggest that, for a given beach slope, there exists a natural frequency of run-up. An approximate calculation of this natural period can be made if we assume that the run-up is caused by a reflected wave of finite amplitude at the shoreline. Using linear wave theory the amplitude of the reflected wave on a beach of linear slope varies as a zero order Bessel function of argument 2a(x/g tan t)1/~ where x is offshore distance and tan fl is the beach slope. The work of Carrier & Greenspan (1958) suggests that we might take the first zero crossing of the Bessel function approximately as the breakpoint, with the waves surging up and down the beach from this point. Equating the argument, with x -- xb, the surf zone width, to the argument of the first zero crossing of the Bessel function leads to a natural frequency a for the given beach slope and the surf zone width. Using values for Slapton beach, the natural frequency is close to o.I Hz, in agreement with an expected subharmonic resonance. This value is also in good agreement with the empirical observations

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of Emery & Gale (195 I) for a beach with the slope of Slapton beach. Agreement between this hypothesis and their results for beaches of smaller slope is less satisfactory, however, probably because other mechanisms are responsible for run-up period on shallower beaches (Huntley & Bowen I975). The important implication of the hypothesis for edge wave generation is that if, as is generally assumed, the breakpoint is determined primarily by the height of the incident waves relative to the local water depth, the relationship between the wave height and the frequency of the incident waves must be fight before a subharmonic resonance will become significant in the surf zone. The need to match these independent conditions may explain why edge waves were not observed on other occasions at Slapton. Similarly, the rapid changes of amplitude shown by the o. I Hz currents (Fig. 5) may be due to the incident waves passing in and out of the necessary frequency and amplitude conditions for nearshore resonance. Unfortunately it is not possible to check this hypothesis in the absence of synoptic measurements of the incident waves further offshore where the o. I Hz component did not dominate the velocity field. An alternative explanation for the strong interaction between edge waves and incident waves may be that the non-linear terms which lead to the theoretical, weak interaction for fully reflected waves (Guza & Davis) are greatly increased in magnitude (relative to the first order, linear terms) when the wave breaks. Undoubtedly, large swash amplitudes are related to large amplitude reflected waves caused by high reflection coefficients on steep beaches, although the details of the relationship are not known. Hence, although it is not clear how the strong swash interaction hypothesis is related to reflected wave interaction theories, both rely, in some way, on the presence of a reflected standing wave component for the generation of a subharmonic edge wave. The importance of the reflected wave component was also suggested from the laboratory experiments of Bowen & Inman (t 969) and may explain why subharmonic edge waves were observed at Slapton but not on a much shallower beach (Hunfley & Bowen t 975). Wave-generated ripples During the field experiment a series of nearshore ripples, with crests running parallel to the shoreline, were observed seaward of the breaker fine. These tipples migrated up and down the beach face with the fide so that the first trough was always just behind the breaker line (Fig. 6). Inman (t957) also studied nearshore tipples in several coastal regions of North America, and found that the wavelength of the tipples was approximately equal to the orbital diameter of the wave motion up to a critical orbital diameter which depended upon the sand size. Above the critical diameter the ripple wavelength decreased again, apparently because of the increase in maximum orbital velocity of the waves. We have calculated the orbital diameter for the breakers observed while the tipples were present, and find that at the breakpoint this diameter is about t .o m in fair agreement with the observed spacing between the tipples. The high mobility of these tipples suggests that the wave motion was readily able to move the beach material up to about 9 metres from the shoreline. The

measured orbital velocities near the bed under the steepening waves had average amplitudes of o.28 m/s, o.41 m/s, and o.7o m/s respectively at Ix m, 9 m, and 5 m from the shoreline, when the breakpoint was approximately 6 m from the shore. These values of orbital velocity appear to be reasonably consistent with estimates of the threshold velocity of the Slapton beach material. Shoreline cuspate features The peak onshore velocity of the subharmonic edge wave at the shoreline can be found by extrapolation from the observed velocities (Fig. 4). At an antinode of the edge wave this velocity is calculated as about o. 7 m/s, which corresponds to an elevation amplitude a of o'27 m and gives an expected excursion up the beach of approximately a/tan fl ~ 2"7 m. This excursion and its predicted longshore wavelength of 32 m agree well with the observed run-up variability. No shoreline cusps were observed while the subharmonie edge wave was present however, despite this run-up pattern. Within the surf zone, where the incident wave velocities were sufficient to set the shingle in motion, a steady longshore current of about I o cm/see, caused by the oblique incidence of the wind waves, must have dominated the mean flow and prevented the formation of any longshore cusps. Further offshore the formation of crescentic bars was undoubtedly inhibited by the low amplitude of the edge wave and the large tidal range. Shoreline beach cusps of much shorter wavelength than expected for subharmonic edge waves were observed at other times on Slapton beach. From their observed longshore wavelengths (~2 m), the dispersion relation for edge waves (Fig. 2) suggests that these cusps were formed by zero order edge waves with a period of about 3 seconds, approximately the period of the incident waves at the

I 1-2 .I. 0"9 _I_ 0"9 _1 Sea Level }~ -I- -I- -I

~ii::.."~-~::-i~i~.:.v:...... 0.3

0920 hrs BP" :-;i

•~::i----.?~-.:.~:::-!!~::;!~ !.:. _.::...... 0-1 ~,~ BP" '~:!~~

..... t_ 1-2 _t [: o., -I

BP" '"::~~ BP ..... Break Point -..-.-:z-:..-::-:..2.!:4i.:..i¢~:...... Lengths in Metres t_ 0-75 _t I - - I

F Io. 6. Tidal migration of ripples observed ~~ on Slapton Beach.

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time that these cusps appeared. The result is not unexpected since, as we have seen, edge waves of the same period as the incident waves can generate nearshore circulation cells with convergence towards strong seaward flowing rip currents, and this nearshore circulation tends to produce beach cusps of the same wavelength as the edge wave, with their points opposite the rip currents. Although no velocity measurements were made while beach cusps were present, the mean flow fields measured with the flowmeter at other times confirm that weak nearshore circulation systems of this kind can exist at Slapton; a small dye diffusion experiment close to the flowmeter on one occasion also revealed a poorly developed rip current flowing seawards through the breakpoint (Huntley & Bowen 1975). Unfortunately, resolution of the velocity field of an edge wave at the same frequency as the incident waves has not been possible with the sensors used at Slapton. The very narrow surf zone made it impossible to distinguish edge wave motion from the orbital velocities of the incident waves using a single flowmeter. These results confirm that edge waves provide a successful quantitative explanation of periodic longshore sedimentary features in nearshore beach material, but definitive results linking these features directly with a measured edge wave velocity field are still missing. Further experiments are therefore needed: (i) to separate edge waves from incoming waves of the same frequency (ii) to resolve different edge wave modes (iii) to associate directly edge wave motion with the generation of sedimentary features (iv) to study the influence of rhythmical topography on incoming waves, particularly the effect of pre-existing topography formed by a different wave regime. 3. References BALL, F. K. 1967. Edge waves in an ocean of finite depth. Deep Sea Res. 74, 79--88. BmCHraXLD, G. E. &GALVlN, C. J. (in press). Generation of edge waves through non-linear subharmonie resonance. Geol. Soc. Amer. BOWEN, A.J. 1969. Rip currents, r, Theoretical investigations. J. Geophys. Res. 74, 5467-78. 1973- Edge waves and the littoral environment. Proc. 13 intern. Conf. Coastal Enging. Vancouver. New York: American Society of Civil Engineers. 1313-2o. ---- & IrCUAN, D. L. I969. Rip currents. 2, Laboratory and field observations. J. Geophys. Res. 74, 5479--9o- I97I. Edge waves and crescentic bars. J. Geophys. Res. 76, 8662-7 I. CARRIER, G. F. & GREENSPAN, H. P. I958. Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97-x 09. CARTER, T. G., L~, P. L.-F. & ME1, C. C. x973. Mass transport by waves and offshore sand bedforms. J. WatWays Harb. Cstl. engng. Div. Am. Soc. civ. Engrs. 99, x65-x84. Ecx~atr, C. x95x. Surface waves in water of variable depth, Wave report Scripps Instn Oceanogr. S xo-ref. 5 x- x2. xoo. 99 PP- EMERY, K. O. & GALE, J. F. x95x. Swash and swash marks. Trans. Am. geophys. Un. 32, 3I--6. GALLAGHER, B. x97t. Generation of surf beat by non-linear wave interactions. J. Fluid Mech. 49, 1--20. GUZA, R. T. & DAVIS, R. E. I974. Excitation of edge waves by waves incident on a beach. d. geophys. Res. 79, x285-9x. HOM-MA, M. & Sores, C.J. 1963. Rhythrnie pattern of longshore bars related to sediment charac- teristics, Proc. 8 intern. Conf. Coastal Enging. Mexico City. New York: American Society of Civil Engineers. 248--78 .

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HUNTLEY, D. A. & BOWEN, A.J. x973. Field observations of edge waves. Nature Lend. 243, x60-2. ------1975- Comparison of the hydrodynamics of steep and shallow beaches. In Nearshore sediment dynamics and sedimentation. (ed. J. R. Hails & A. Carr) London. I~, D. L. x957. Wave generated ripples in nearshore sands, Tech. Memo. Beach Eros. Bd U.S. xoo, 66 pp. KOMa~, P. D. I972. Nearshore currents and the equilibrium cuspate shoreline. Tech. Rep. Dept Oceanogr. Oregon State Univ. 239, 26 pp. LAMB, H. t932. Hydrodynamics (6th ed.) Cambridge. LAu, J. & TaAws, B. x973. Slowly varying Stokes waves and submarine longshore bars. J. Geophys. Res. 78, 4489-97. MUNK, W., SNODORASS, F., & CARm~R, G. x956. Edge waves on the continental shelf. Science x23, I27-32. St-mPAaD, F. P. I952. Revised nomenclature for depositional coastal features. Bull. Am. Ass. Petrol. Geol. 36, x9 ° x-I 2. SONU, C.J. t 973. Three-dimemional beach changes..1'. Geol. 8x, 42-64. STOKES, G. G. t 846. Report of the British Assodation, Part t. Also, x88o, Mathematical and Physical Papers (Collected) x, x57-87. TUCKEa, M. J., SMITH, N. D., PmRC~, F. E., & COLL~S, E. P. x97o. A two-component electromagnetic ship's log. J. Inst. Navig. 23, 3o2-t6. URS~LL, F. x952. Edgewaves on a sloping beach. Proc. R. Soe. A~x4, 79-97.

Received 24 January t 974; revised typescript received 6 June x974.

DAVID ARTHUR HUNTLEY & ANTHONY JOHN BOWEN Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Cheshire L43 7RA.

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