Longshore Structure of Infragravity Wave Motions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 89, NO. C4, PAGES , JULY 20, 984 Longshore Structure of Infragravty Wave Motons R. A. HOLMAN School of Oceanography, Oregon State Unversty A. J. BOWEN Department of Oceanography, Dalhouse Unversty Synchronous runup records were collected from 4 locatons spaced rregularly over a 7 km stretch of a low-slope beach. The sgnfcant runup heght, heren defned as the sgnfcant vertcal ecurson of water level at the shorelne, was typcally 2 m, 60% of the ncdent sgnfcant wave heght at the breakpont. The runup spectra were domnated by the energy at low frequences, wth 99.9% of the varance n motons wth perods longer than 20 s and 83% for perods longer than 50 s. A peak between and 0.00 Hz was present n all spectra. Analyss of ths band showed the moton to be a standng edge wave wth wavelength between 5 and 0 km and sgnfcant edge wave heght at the antnode of.3 m, accountng for half of the local sgnfcant runup heght. The edge wave perod was appromately 40 s and the mode number n the range 3-7. An apparently lnear sand bar was present 75 m offshore, roughly the same locaton as the frst node of cross-shore velocty for the edge wave. Whle the edge wave could not have been the cause of the bar (a standng edge wave does not create a lnear bar), the opposte may be true, wth the bar potentally provdng a topographc resonance. As the local headlands etend only a small fracton of the offshore dstance of the edge wave, they are unlkely to be sgnfcant reflectors. The sngle rp current observed occurred at the edge wave node. INTRODUCTION Much of the domnant morphology on natural beaches s of large scale compared to the wave length of the ncdent waves. Lnear sand bars may have typcal offshore length scales of m, crescentc bars can have longshore wavelengths of several hundred meters, and oblque or welded bars hundreds to thousands of meters. If we are to beleve that these mor- phologes are a response to wave motons, then the correspondng wave motons must also be long, often fallng n the nfragravty frequency band (frequences 0.03 >f> Hz). Quanttatve models have now been publshed whch eplan the generaton of a number of common bar morphologes n terms of the drft veloctes assocated wth edge waves. Crescentc bars were lnked to a standng edge wave [Bowen and Inman, 97], lnear bars to a progressve edge wave [Bowen, 98], and welded bars to two progressve edge wave modes of the same frequency [Holman and Bowen, In fact, feld measurement on low-slope and barred beaches often fnd nfragravty motons to be domnant [Huntley, 976; Sasak and Horkawa, 975; Wrght et al., 979; Holman, 98; Huntley et al., 98]. It would be of consderable nterest f these motons proved to nclude a sgnfcant edge wave component. Provdng defntve feld evdence for or aganst the presence of edge waves has proved dffcult. Ths s prmarly a result of the smlarty of the alternatve wave motons when vewed n a cross-shore transect, the normal drecton to nstrument nearshore feld eperments. However, several recent eperments show proof of edge waves through analyss of the longshore behavor of the wave motons. Katoh [98] used aeral photography of wave runup over a 200 m length of beach, termnated by a breakwater. Hs photographs were at Copyrght 984 by the Amercan Geophyscal Unon. Paper number 4C /84/004C random tme ntervals, so he lacked frequency control. However, treatng each photograph as a realzaton of a random process, he performed an emprcal orthogonal functon analyss (EOF) to determne the longshore covarance structure of the runup. The resultng sequence of egenfunctons were snusodal n the longshore wth antnodes at the break water. Huntley et al. [98] used velocty data from a 500 m long lnear curent meter array to prove the estence of low-mode nfragravty edge waves on a low-slope beach. Ths paper wll dscuss data from a much longer longshore array (6 km) whch show the estence of very long-standng edge wave motons on the Oregon coast. The net secton of the paper wll dscuss edge wave knematcs, partcularly as appled to the recognton of edge wave modes n the feld. Net we wll dscuss the eperment and then the results. EDGE WAVE KINEMATICS If depth s not a functon of the longshore coordnate, y, then the velocty potental (l)(, y, t) for a progressvedge wave s (I)(, y, t) = an!7 (Pn(X) COS (ky-- at) () and for a standng edge wave s (l)(, y, t) = an!7 (Pn(X) COS (ky) cos (at) (2) Here s the offshore coordnate measured postve seaward from the shorelne, t s tme, g the acceleraton due to gravty, a s the radal frequency (2r dvded by the perod T), k s the longshore wave number (2r dvded by the wavelength L), and an s the ampltude of the nth mode. The offshore structure s contaned n the term Cbn(X) whch agan s mode dependent. For the case of a plane beach whose depth h()= tan fl, where fl s the beach slope, Ursell [952] showed the dsperson relaton for edge waves to be rt 2 = gk sn (2n + )f (3)

2 , HOLMAN AND BOWEN' INFRAGRAVITY WAVES 6447 Ths places a lower lmt on k for edge waves O.2 _>- (4) and the hghest n for whch (4) s true s called the cutoff mode. For k < O.2/g there ests a contnuum of leaky modes whch are not trapped to the shorelne. A problem for feld studes whch emphasze the cross-shore structure of the nfragravty motons has been the smlarty of edge wave modes and leaky modes for small offshore dstances (see, for nstance, Fgure for the case of a plane beach). In contrast, studes n the longshore should be able to sort out dfferent modes through ther wave number (equaton(3)) or should at least be able to dstngush between the leaky mode and edge wave regmes (equaton(4)). CAMERA CAMERA B2 El3 :.' B? 2 0 FIELD MEASUREMENTS The feld eperment took place on a 7 km long, straght beach to the north of Heceta Head on the md-oregon coast. The subaeral porton of the headland etends about 00 m straght offshore, termnatng the beach on the south end. To the north, the beach s termnated by a second basaltc headland, but wth a much more subtle offshore protruson (Fgure 2). The foreshore slope was typcally Subacqueous bathymetry data s sparse, consstng only of some short surveyed profles from the month prevous to the eperment. The "mean" beach slope was appromately The beach has a sngle, apparently lnear, sand bar. The only survey lne whch reached the bar showed the bar crest to be 75 m offshore (for the tde heght approprate to the data run). The wave data collected conssted of 4 synchronous tme seres of runup dstrbuted unevenly over a 6 km longshore dstance. The data were collected by tme-lapse photography, usng three synchronzed move cameras (one 6 mm, two super 8) lookng alongshore. A 40 m hgh rocky knoll, 5 km north of Heceta Head, provded an deal locaton for two of the cameras, one lookng north, the other south. A number of longshore locatons were dgtzed from each of these flms, the scale n the pcture beng taken ether drectly from prevously surveyed, recognzable landmarks or nterpolated from these ponts. The locatons of the three cameras, labeled, A, B, C from north to south, and the postons of the dgtzed tme seres are shown n Fgure 2. (Camera C vewed the beach oblquely, allowng the dgtzaton of only one range.) The run length was 70 mn wth a frame taken every 2 s. Replcate dgtzatons of the same flm by dfferent operators showed errors on the varance to be about +_ 30 and 40% for ranges Hecto Heod Fg. 2. Plan vew of beach showng locatons of cameras A, B, C and of dgtzed runup tme seres. Heceta Head termnates beach to the south' a smaller headland «km to the north of run A termnates the beach to the north. whch were 250 and 500 m away from the camera, respectvely (5 and 20% error on swash heght). The spectral structure was well reproduced by all operators. A full descrpton of the tme-lapse photography technque and assocated error estmates s contaned n Holman and Guza [984]. The data dscussed below were obtaned on October 9, 980. The vsually observed sgnfcant wave heght, perod, and angle of ncdence were 3.5 m, 5.5 s, and 8øN of normal, respectvely. A longshore current of appromately 0.75 m s- a flowed to the south. From the camera locatons, only one rp current was vsble, 00 m to the north of the knoll. All runup tme seres have been converted from alongslope to the correspondng vertcal component usng the local beach profle. For two of the tme seres, B6 and B7, the closest profle was more than 500 m away. Transformng to the vertcal runup sgnal, Rv, could then alter the varance (and possbly the spectrum) f the foreshore profle changed alongshore. These two runs are not ncluded n the general dscusson whch follows, but are ncluded n the longshore structure analyss..0 EDGE WAVES 0.8 REFLECTED NORMALLY INCIDENT WAVE () o..:o," '.--,/,,,, o I,7 ø ' 3o,' o so - zo -0.2[ ' /..'"X:aa/gtan -'' n=, -- n:5-0.6 Fg.. b(z) versus nondmensonal offshore dstance Z = o'2/gfl for the lowest four edge wave modes and a reflected, normally ncdent wave. RESULTS Fgure 3 shows the tme seres from the 4 ranges. The low frequency nature of the runup s readly apparent. The longshore crest length of ndvdual waves can also be seen. Table ncludes summary statstcs for the 2 data runs (neglectng B6 and B7). The mean varance for the 2 seres m 2 wth a standar devaton of m 2. The equvalent "sgnfcant runup heght" (whch s taken as 4 tmes the square root of the varance and assumes that the process s Gaussan) s m. Ths s 58% of the ncdent sgnfcant wave heght, slghtly less than the 7% rato found by Guza and Thornton [982]. There s some longshore structure to the varance wth a mnmum near A5 or A6. Fgure 4, a typcal spectrum, confrms that the varance s completely domnated by waves wth perods longer than the

3 6448 HOLMAN AND BOWEN: INFRAGRAVITY WAVES TIME (mn.) CI[ ' I I, I ' ' B6,,' v v v --- v E B5 B4 B3 ' ' ' [ ' I ' [ ' [ I E B2 --. A6 ',,,,, A5 [ I [ I t I 4 [ ] I I ' [ ' I a2 [ ' [ ' I [ ' I I I I Fg. 3. Tme seres of sea surfacelevaton, r/, for each of the 4 ranges. Ranges B6 and B7 provde useful phase nformaton but are unrelable n varance due to lack of local beach profle data. ncdent perod. In fact, 99.9% of the total varance s carred by waves longer than 20 s, and % s carred by waves longer than 50 s. Table shows the varaton of ths latter factor over the dfferent seres. Fgure 5 shows the spectra from all 2 longshore locatons. The frequency as now runs from 0.00 to 0.05 Hz reflectng the low frequency domnance. The spectra all look smlar n shape and (wthn a half order of magntude) n energy. Ths s reassurng snce t means that a runup measurement at a sngle locaton such as s often taken n feld eperments s representatve of the beach as a whole. All spectra show energy decay at frequences greater than 0.02 Hz and less than Hz. The very low frequency hll (f < Hz) seems to be largely related to the fallng tde. The fall off n energy on the low frequency sde of the nfragravty band has mplcatons for the nfragravty wave dynamcs, the forcng cannot be red. Most of the fner structure of the spectra are not sgnfcant at the 95% confdence level and vary between spectra. However, between and 0.00 Hz, all spectra show a peak, suggestng the possblty of coherent moton over the entre beach length. Fgure 6 shows the cross-spectral results between ranges B3 and C. The peak between and 0.00 Hz s sgnfcantly coherent at the 95% level despte the fact that the ranges are separated by 5 km. The phase dfference s 80 ø, suggestng a moton whch s standng n the longshore. The longshore structure of ths band was analyzed by usng frequency doman EOF analyss [Wallace and Dckenson, 972; Wang and Mooers, 977]. If the cross spectrum between two ranges and j for a partcular frequency band s gven by U u=c u+qu,j=, N (5) where C and Q are the cospectral and quadspectral estmates TABLE. Summary Statstcs for the 2 Tme Seres Range Percent Sgnfcant Total Varance for Age of Runup Heght, Varance, f < 0.02 Hz, Varance for f < 0.02 Hz, m e m e f < 0.02 Hz m A A2 A3 A4 A5 A6 B B2 B3 B4 B5 C Average ñ ñ ñ ñ 0.30

4 ._ HOLMAN AND BOWEN' INFRAGRAVITY WAVES 6449 IOO - I0- I 95% oo ' I0.0 BI ', ',. ""'", C I 95 % o.] ],,, ",A A,, ß ; > Frequency ( H z ) % OoC).000.oo.c 5.]o.]5.20 Frequency (Hz) Fg. 4. Typcal runup spectrum (B3) wth 75 degrees of freedom. and N s the number of ranges, then the emprcal orthogonal functons are the egenvectors of U u. The egenvectors wll be comple and can be epressed n terms of ampltude and phase. Fgure 7 shows the longshore structure of ampltude and phase for the frst egenfuncton from the frequency band Hz. The orgn of longshore dstance s arbtrarly taken at the farthest northern range. The ampltude structure s hghly varable wth an apparent node at 0.9 km. The ampltude decreases from the apparent mamum at 2.5 km to the sngle pont at 6.0 km, although the lack of data precludes determnng the estance of a second node based on ampltude alone. The phase structure of the egenvector proves the moton to be standng n the longshore. Phase s zero and constan to the north of the ampltude node at 0.9 km, then swtches suddenly to 80 ø and constant to the south. The data pont at 6.0 km ndcates that phase has agan swtched to zero somewhere n the regon of the beach where we have no data, a second longshore node. Note that the prevously neglected data at B6 and B7, whle of questonnable ampltude, show phase relatons consstent wth the rest of the data. The bold trangle, at.0 km on the fgure, shows the locaton of cameras A and B Fg. 6. Cross-spectral results between ranges B and C. Estmates have 24 degrees of freedom. on the knoll. The observaton of the longshore node away from ths ste s reassurng, precludng any camera synchronzaton problems and showng that the knoll dd not strongly nterfere n the hydrodynamcs. The estence of a node n longshore structure s dagnostc of a standng wave moton. Wth only one node actually mapped, we cannot determne an eact wavelength. However, the estence of the second node somewhere n the regon km allows us to place lmts on the wavelength. Assumng the poston of the second node to be at mnmum halfway between the frst node and the 6.0 km pont, and at mamum at the 6.0 km pont, the full edge wave wavelength wll be n the range 5-0 km. From (4) the cutoff wavelength would be 30 km, much longer than ths observed wavelength, provng the moton to be a trapped edge wave. If we take a representatve "mean" beach slope to be 0,02 and the wave perod to be 40 s (the peak n coherence from cross spectra, Fgure 6), the equvalent edge wave mode number wll be between 3 and 7; that s the mode number s small, but not 0 f : Hz, EOF#I.5O I00 - <:[ o X,,/.00 - XXXl] 0 Z'.O X- ' 'X'X'- ' X.. 3.o 4'.0 5'.o 6'.o Longshore Dstance (km) o..-; o,, ' Frequency (Hz) Fg. 5. Superposton of spectra (24 degrees of freedom) from all 2 longshore locatons. Note that the mamum frequency s 0.05 Hz. 80 Fg. 7. Frst egenfuncton for the frequency band Hz. Both ampltude (top) and phase (bottom) are plotted as a functon of longshore dstance. Data ponts are ndcated by crosses. The locaton of cameras A and B s shown by the bold trangle. Queston marks ndcate no data between 2.5 and 6.0 km.

5 , 6450 HOLMAN AND BOWEN' INFRAGRAVITY WAVES and (for the wave to have been a mode, the mean beach f : Hz, EOF 2 slope would have to have been at least 0.055, greater than the.50 observed foreshare slope). Holman and Bowen [982] show that lnear sand bars wll be formed at a nondmensonal offshore dstance of Z = k,,.25 ' 0'2X/gp = 3, where a s the frequency of the generatng edge X,"' wave. These data ndcate Z =.8 for the observed bar (at md to hgh tde) and the 40 s perod. The agreement s appro-.00 mate. That t s not better s not surprsng snce we could hardly suggesthat the observed wave moton generated the 5'.0 Longshore Dstance (km) observed bar; a standng edge wave should not generate a lnear bar. The appromate agreement may ndcate some -X,--? sort of topographc tunng by a pre-estng bar. 0 ø. X-X. ' X-X The smplest mechansm for producng a standng edge ^X'* " ' ' wave s reflecton from longshore barrers. Reflecton from Heceta Head to the south s hard to detect due to the lack of - 8,0ø[ data, but there s lttle to suggest a local antnode. The headland to the north (at -0.5 km) s most defntely not actng as a reflector snce t s too close to the node at 0.9 km. (The Fg. 8. Second egenfuncton for the frequency band Hz. Ths egenfuncton eplans 7ø/,, of the varance n ths band. beach slope would have had to reduce dramatcally for the wavelength reducton. Such was not observed.) There s some queston as to what consttutes a "sgnfcant" reflector for an edge wave. Huntley et al. [98] dscuss edge wave reflecton, although n ther case the reflector s a submarne canyon. For a mode 3 edge wave on a plane beach they show that at least 90% of the edge wave energy wll le seaward of Z = 2, the appromate offshore etent of Heceta Head. Ths suggests that Heceta Head s seen more as roughness by the edge wave and that any effectve reflector should have much larger offshore etent. Ths s equvalent to a sgnfcant edge wave heght, at the antnode, of.34 m. The frst egenvector eplans 57% of the total varance for the 4 tme seres for the frequency band Hz. However, the percentage of the varance eplaned for ndvdual seres vares strongly, rangng from 2.5% for range A5 to over 95% for ranges B5 and B6 (Table 2). The second egenvector for ths band, whch eplans 7% of the total varance for the 4 tme seres (egenvalue '2 = 0.235), s plotted n Fgure 8, and the percentage of varance eplaned for each longshore locaton s lsted n Table 2. Ths egenvector s most mportant n the northern ranges, near the node of the frst mode. The ampltude and phase structure do not correspond to any obvous knd of wave moton. The observaton of a sngle rp current 00 m north of the rocky knoll s of partcular nterest. $asak [977] suggested that standng nfragravty motons may cause rp currents, and $asak and Horkawa [975] showed feld evdence whch suggests that the rp currents may occur at edge wave antnodes (n sea surface elevaton). We, on the other hand, fnd the rp current at the edge wave node. From the frequency doman EOF analyss we can determne the ampltude of the edge wave moton by usng the frst egenvalue,, = If we take range B4 and B5 as representng the antnode of the wave, we fnd the varance at the antnode assocated wth the frst egenvector to be 0.2 m Fgures 9a and 9b show the frst and second egenvectors for the frequency band Hz. These vectors eplan 53 and 28%, respectvely, of the total varance n the band (egenvalues are and 0.87, respectvely). The cross spectrum between B and C (Fgure 6) had shown ths band to be ncoherent, suggestng that coherent process would only operate on shorter length scales. The frst egenvector supports ths TABLE 2. Percentage of the Varance n the Band Hz Eplaned by the Frst Two Egenmodes for Each Longshore Locaton TABLE 3. Percentage of the Varance n the Band Hz Eplaned by the Frst Two Egenmodes for Each Longshore Locaton Percent Varance Eplaned Percent Varance Eplaned Range Frst Mode Second Mode Range Frst Mode Second Mode A 6 28 A A3 56 A A5 2 5 A6 3 4 B B B3 7 4 B B B B7 75 C A A2 A3 A4 A5 A6 B B2 B3 B4 B5 B6 B7 C

6 HOLMAN AND BOWEN: INFRAGRAVITY WAVES 645 contenton, showng a localzed ampltude structure n the regon south of the knoll. The second mode domnates north of the camera ste and shows constant phase north of the knoll, a 80 ø phase shft at the knoll, and a progressve phase shft south of the knoll. These modes suggesthat the knoll eerts a sgnfcant hydrodynamc nfluence for ths band, n. contrast to lower frequences. DISCUSSION Huntley et al. [98] found progressve edge waves on a low-slope beach wth mode 0 domnatng the frequency band Hz, shftng to mode n the band Hz. However, owng, to a lmted array length and the nature of hs analyss technque they could not analyze for hghmode, low frequency motons or for standng edge wave motons. Thus we cannot say whether there s any fundamental dfference between our datasets. In fact, Huntley et al. [98] note suggestons n the data for the presence of standng edge waves whch may be reflectng from the two large submarne canyons whch penetrate close to shore n that area, an dea prevously mentoned by Inntan et al. [976]. The possblty that large-scale topography may place mportant constrants on edge waves s an appealng one from the pont of vew of the data presented here. Frst, the headlands are too small to act as sgnfcant reflectors. Second, the bandwdth of the standng wave moton s somewhat large. In an attempt to ncrease the frequency resoluton, a further analyss was run usng frequency bands of Hz. The phase plots of the egenvectorshowed the strong phase shft ndcatve of the standng wave moton n the band Hz, a 30% varaton n wavelength. However, the Fourer estmates used n the analyss had only 4-6 degrees of freedom, so confdence n the analyss must be low. It s apparent that longer data records are requred to adequately resolve these frequences usng ths technque. If longshore constrants are beng provded by large-scale nner shelf topography, t s unclear at ths tme what that topography s. CONCLUSIONS The sute of runup tme seres presented here show that runup on a relatvely flat beach can become qute large (2.0 m sgnfcant runup heght n an ncdent swell estmated to be u,) o.50 ' X'.oo X''X".0 2_' ø XX -I, 'X, I X, '!! ß f '.0 Longshore Dstance (krn) Hz, EOF#t d.o Fg. 9a. Frst egenfuncton for the frequency band Hz. Ths mode eplans 44% of the varance..5o.:>5.00 '.. k-,, X. ß.X ' / XX 'X-X o _ - -XX,, -80 f Longshore Dstance (km) Hz, EOF#2 Fg. 9b. Second egenfuncton for the band Hz' 23% of the varance s eplaned by ths mode. 3.5 m at the breakpont) and very low frequency (83% of the varance contaned n waves wth perods longer than 50 s). Spectra from 2 dfferent longshore locatons showed smlar general structure decayng for frequences greater than 0.02 Hz and less than Hz. Indvdual peaks between these frequency lmts are not obvously correlated n the longshore wth the ecepton of a peak between and 0.00 Hz whch s always present. Frequency doman EOF analyss showed the longshore structure of the energy to have a node n ampltude assocated wth a clear 80 ø phase shft between two areas of larger ampltudes wth constant phase. Ths structure s dagnostc of a longshore standng wave. Whle the eact wavelength could not be determned, t had to be between 5 and 0 km, much shorter than the 30 km cutoff wavelength for leaky modes, ndcatng that the moton had to be a trapped edge wave. The equvalent edge wave mode number was between 3 and 7, low but not 0 or. The edge wave was of the same appromate perod as would create the observed lnear sand bar although a standng edge wave clearly could not have created a lnear bar. The bar may eert some topographc control on the flud motons. The observed headlands etended seaward onlya fracton of the nferred offshore etent of the edge wave and were probably not sgnfcant reflectors. The longshore reflectors whch dd cause the standngness are unknown. A sngle rp current was noted at the observed node of the edge wave. Acknowledgments. Ths work was supported by the Otfce of Naval Research, Coastal Scences Branch, under contract NR We would also lke to thank Aage Grbskov for techncal support and Pete Howd and Alan Morse for feld assstance. REFERENCES Bowen, A. J., Smple models of nearshore sedmentaton; beach profles and longshore bars, The Coastlne of Canada, edted by S. B. McCann, Pap. 80-0, Geol. Surv. of Canada, Ottawa, 98. Bowen, A. J., and D. L. Inman, Edge waves and crescentc bars, J. Geophys. Res., 76, , 97. Guza, R. T., and E. B. Thornton, Swash oscllatons on a natural beach, J. Geophys. Res., 87, , 982. Holman, R. A., Infragravty energy n the surf zone, J. Geophys. Res., 86, , 98. Holman, R. A., and A. J. Bowen, Bars, bumps, and holes: Models for the generaton of comple beach topography, J. Geophys. Res., 87, , 982.

7 6452 HOLMAN AND BOWEN: INFRAGRAVITY WAVES Holman, R. A., and R. T. Guza, Measurng runup on natural beaches, Coastal Eng., n press, 984. Huntley, D. A., Long perod waves on a natural beach, J. Geophys. Res., 8, , 976. Huntley, D. A., R. T. Guza, and E. B. Thornton, Feld observatons of surf beat,, Progressve edge waves, J. Geophys. Res., 86, , 98. Inman, D. L., C. E. Nordstrom, and R. E. Flck, Currents n sub- marne canyons: An ar-sea-land nteracton, Ann. Rev. Flud Mech., 8, , 976. Katoh, K., Analyss of edge waves by means of emprcal egenfunctons, Rep. Port Harbor Inst., 20, 3-5,98. Sasak, T., Feld nvestgatons of nearshore currents on a gently slopng bottom, Rep. 3, Nearshore Envron. Res. Center, Unv. of Tokyo, Tokyo, 977. Sasak, T., and K. Horkawa, Nearshore current system on a gently slopng beach, Coas. tal Eng. Jpn., 8, 23-42, 975. Ursell, F., Edge waves on a slopng beach, Proc. R. Soc. Set. A, 24, 79-97, 952. Wallace, J. M., R. E. Dckenson, Emprcal orthogonal representaton of tme seres n the frequency doman, I, Theoretcal consderatons, J. Appl. Meteorol.,, , 972. Wang, D. P., and C. N. K. Mooers, Long coastal-trapped waves off the west coast of the Unted States, summer 973, J. Phys. Oceanogr., 7, , 977. Wrght, L. D., J. Chappell, B. G. Thom, M.P. Bradshaw, and P. Cowell, Morphodynamcs of reflectve and dsspatve beach and nshore systems: Southeastern Australa, Mar. Geol., 32, 05-40, 979. A. J. Bowen, School of Oceanography, Oregon State Unversty, Corvalls, OR R. A. Holman, Department of Oceanography, Dalhouse Unversty, Halfa, NS B3H 4J, Canada. (Receved August 0, 983; revsed January 3, 984; accepted February, 984.)