Journal of Coastal Research 1213-1220 Fort Lauderdale, Florida Fall 1997

Predicting Shoaling Wave HeightsJohn R.C. Doering

Department of Civil and Geological EngineeringUniversity of ManitobaWinnipeg, MBCanada R3T 5V6

ABSTRACT _

.tllllllll:.~susss~~,.",a+ S--

DOERING, J.R.e ., 1997. Predicting Shoaling Wave Heights. Journal of Coastal Research, 13(4), 1213-1220. FortLauderdale (Florida), ISSN 0749-0208.

Data obtained from laboratory experiments conducted on two (1:40 and 1:20) planar beach slopes in a random waveflume were used to examine the accuracy and robustness of the method of wave height prediction proposed by Hughesand Miller (1987). In part icular, the data were used to examine the sensitivity of the method to bottom slope, wavesteepness ii.e., peak enhancement), and relative depth (dIL ). The results indicate that the accuracy of the method isrelated to both spectra l satura tion and bottom slope. Overall , the accuracy of the predictions is good (i.e., relative errorless than 10%) provided d/L > - 0.2; accuracy deteriorates furth er in shallower water.

ADDITIONAL INDEX WORDS: Shoaling wave height s, wave flume, waves, shallow water waves, beach.

with

BACKGROUND

(3)

(4)

(1)

(2)

4>Pr..lf/ ~) = e - S/4 I f /fp) '\

f./Jjf, (p, ')', o) = y e 1'1 fI' J ~:<T t l;' ~ ',

where ex is th e equilibrium range parameter, g is the accel­era t ion du e to gr avity, f is frequency, cfJPM(f/fp) is the shapefactor propo sed by Pierson and Moskow itz (1964 ), fp is th e

Th e TMA spectrum, from which HUGHES and MILLER(1987 ) developed their method of wave height pr ediction, isnamed afte r the three data se ts used to develop it, viz., Texel ,MARSEN, and ARSLOE. It is obta ine d by repl acing PHIL­LIPS' (1958 ) equilibrium range scaling (denoted here by E/ f))for wind-generate d deep water waves, which is used in theJONSWAP equation (HASSELMANN et aZ. 1973), by the ex­pr ession propo sed by KITAIGORODSKII et aZ. (1975 ) for th eequilibrium range of wind-gen erated finite depth waves. Th eJONSWAP equat ion can be expressed in th e form

The purpose of this paper is to examine th e robu stness ofth e HM method of wave height prediction. Previous work haslargely adhered to satisfying the assumptions set forth byBo uws et al. (1985) and HUGHES and MILLER (1987). How­ever, it is useful to know the exte nt to wh ich th ese assump­t ions can be relaxed. In th e followin g sect ion the backgroundleading up to the HUGHES and MILLER (1987) method ofwa ve height prediction is bri efly reviewed. Th e experimentsthat provided th e data used to test this method are th en de­scribed. The observations, discussion , a nd conclus ions follow.

The prediction of wave height is fund am ental to many as­pect s of coastal enginee ri ng . Often th e task in volves th e es ­timation of an inshore water wave height from a mea suredor hindcast offshore, i.e., deep water , wave hei gh t. Histori­cally, th e most common method of t ran sforming a wave fieldhas been to use some form of lin ear th eory. Th e height atbreaking has been pred icted by invokin g a criteri on that lim­its wave steepness or th e ratio of wave height to water depth.However, unlike monochromatic waves, a random wave tr ainhas no well-defin ed loca tion of break ing a nd no specific waveheigh t associated with breaking. As a result, th e shoa ling andbr eaking of random waves is typ icall y modeled in one of twoways: in terms of th e va riability of statist ical param eters ofth e wave field , such as a cha ra cte rist ic wave height, e.g., Hmo;or in terms of th e evolut ion of th e spect ra l den sity function.

Two gen eral approac hes to describe the shoa ling andbreaking of random waves have bee n advanced. Th e first de­scribes wave shoa ling as a local dep th depend ent process. Th esecond more recent meth od, integra tes th e energy flux bal ­anc e equat ion along th e path of pr opagation , th ereby yieldinga wave height depend ent on th e integr al of shoa ling pr ocesses( B ATT.l E S and J ANSSEN, 1978; THORNTON and GUZA, 1983;BATTJES and STIVE, 1985; and many others).

While various numerical models ca n pr ovide an accuratestatistic al description of wave heights through th e shoalingregion and in the surf zone , a "back-of-t he-envelope" methodof estimating shoaling wave heights is quite useful , particu­larl y in the preliminary stages of a design. Th e method pro­posed by HUGHES and MILLER (198 7), herein after refe re d toas simply HM, is one such means of computi ng potentiall yaccurate estimates of wave height in a very simple fashi on.

95180 received 2 December 1995: accepted in revisions 8 Augu st 1996.

INTRODUCTION

1214 Doering

Substituting the empirical expression for u given by Bouwset al. (1985), viz.,

(14)

The data used for this study were obtained from experi­ments conducted in the National Water Research Institute'swind-wave flume at th e Canada Centre for Inland Waters.The flume is 103 m long, 4.5 m wide , and it has a maximumwater depth of 1.5 m. "Random" waves were generated usingthe GEDAP software package developed by the National Re­search Council of Canada (FUNKE and MANSARD, 1984). Cor­rections were made to the piston-type waveboard drive signalto suppress spurious, second-order waves that arise throughthe mechanical generation of waves. All of the "random" wav­etrains were created from DHH target spectra, after DONE­LAN et al. (985). The parameters of a DHH spectrum aresimilar to those of a JONSWAP spectrum. However, a DHHspectrum has an r: high frequency tail rather than the f-5

tail that is characteristic of a JONSWAP spectrum. In addi­tion, a DHH spectrum exhibits stronger peak enhancementat short fetches than a JONSWAP spectrum. Six peak fre­quencies (fp = 0.9,0.8,0.7,0.6,0.5, and 0.4 Hz) and two waveage values (Uj cp ) 0.83 and 5.0, equivalent to peak enhance­ment ('Y) factors of 1.7 and 5.9, respectively, were used. Thepeak enhancement factors of 1.7 and 5.9 correspond to "fully­developed" and "strongly-forced" spectra, respectively. As thewaves where mechanically generated, the wave age param­eter or peak enhancement factor was used only to influenceHmo. Each run contained approximately 500 waves ; four dif­ferent realizations of each target spectral shape were run. Asummary of the runs is given in Tabl e 1. A water depth of1.00 m was used for all runs.

The experiments were condu cted on two impervious (ply­wood) planar beach slopes, 1:40 and 1:20. The toe of bothbeaches was located 27.78 m from th e (mean position of the )waveboard; see Figure 1. Wave heights were measured usingsurface-piercing capacitance-type wave probes. The electron­ics packages for the wave prob es were designed and built at

THE EXPERIMENTS

H VU~ = 0 0445­L;0.75 • gO.25·

Hence, for a constant wind speed, U (one of the as sumptionsin the derivation of the TMA spectrum) the implication ofHUGHES and MILLER'S (987) result is the ratio Hm"lLp 0 75 isconstant through the shoaling region provided the waves arein equilibrium with the wind ; i.e., the spectru m is "fully-de­veloped". Thus, given Hmo and the peak period 'I'; at somedepth d, the wave height Hmo can be predicted at any otherlocation, either inshore or offshore , by simply specifying thedepth, i.e., L ; = {(Tp , d ) where Tp is assumed to be invariantwith depth. It should be noted that Resio (987) also derivedthe result Hm"lLp 0 .75 is constant for equilibrium wind seas infinite water depth.

Of course, the HUGHES and MILLER(987) method of waveheight prediction does not account for many of the other pro­cesses that can modify wave height during shoaling; for ex­ample, refraction, diffraction, reflection, or directionality.However, it does provide a potentially accurate yet simplenumerical means to compute wave heights for fully-devel­oped, wind-generated, single-peaked spectra sh oaling overflat or gently sloping bottoms.

(9)

(5)

(6)

(7)

(13)

(11)

(0)

(12)H _ ex"2Lmo

- __ Po

tr

E(k ) = lhu k - 3 1\J(k ,fp,d ),

1EJ!f, d) = (2 7r)4ag2f-5<f>J!woJ,

J!i:7rex = 0.0078U L

g p

into equation (12) and rearranging, HUGHES and MILLER(987) obtained

1= -uk - 24 p •

Noting that Hmo = 4VE;, and k p = 27r1L p, then equation(11) can be written as

ET M A(fJ = EJfJ4>PM(flfp)4>if,fp,'Y,u ). (8)

Although the TMA spectrum is (perhaps) more useful writ­ten explicitly in terms of frequency space dependent param­eters, Bouws et al. (1985) note that it can be compactly ex­pressed in wavenumber space by the one-dimensional spec­trum

where I\J is a spectral shape function. The results of Bouwset at. (985) suggest that the TMA spectrum provides a goodfit to the equilibrium range of finite depth spectra. VINCENT(984) has shown that integrating equation (9) over thisrange (i.e., kp ~ 00) provides a close approximation to the totalspectral energy, Eo where

[

k-3( w, d)ak( w, d)]<f>J!w) = aw ,

k -3( w 001 ak( W , 00), 1 aw

Wd = 27T{~,k is wavenumber, and d is the local water depth, in the placeof Elf) in equation 0), gives the expression from Bouws etal. (985),

where

Lx 1

Eo = "2uk-3 If{k, fp, d) dk .k p

It is assumed that the contribution of the shape factor I\J canbe safely neglected over this range, hence,

Lx 1

E = -uk-3 dko 2

k p

peak spectral frequency, 4>Jf,fp,'Y,u) is the JONSWAP shapefunction, 'Y is the peak enhancement factor, and tr is the peakwidth parameter. Substituting the KrTAIGORODSKII et al.(1975) expression for the equilibrium range of a finite depthspectrum, i.e.,

J ourn al of Coastal Resear ch, Vol. 13. No.4, 1997

Sho al Wave Height Pr edi ction 1215

Table 1. Summary of Spectral Realizations

Run IDs Type y MH z] H mo[m]

GR30-33 DHH 1.7 0.9 0.06GR34-37 DHH 5.9 0.9 0.11GR05-08 DHH 1.7 0.8 0.08GR06-09 DHH 5.9 0.8 0.14GRIO-13 DHH 1.7 0.7 0.10GR14-17 DHH 5.9 0.7 0.19GR18-21 DHH 1.7 0.6 0.14GR22-25 DHH 5.9 0.6 0.25GR26-29 DHH 1.7 0.5 0.20GROI-04 DHH 1.7 0.4 0.32

the National Water Research Institute. Wave probe calibra­tion data show that these inst ruments are very linear (r2 >0.999) and have excellent long term gain st ability. Ten waveprobes (WP's) were installed on the 1:40 beach slope. The firstwave probe was located at the toe of the beach, which was27.78 m from the waveboard. The remaining nine waveprobes were installed on 4 m centers; this yields a decreasein depth of 0.1 m between adjacent wave probes. Ten wave

LIST OF SYMBOLS

Symbol

aCp

d

ft:

.gu.;kLt.,Tr,U,

y

J.Lo(T

w

Designation

Wave amplitudePh ase speed of the spectral peakStill wa ter depthFr equ encyPeak spectral freq uencyAccele ration du e to gravityWave height ( = 4 \f'iL:,)Radi an wavenumber ( = 27r/L)

Wave lengthWave length of peak per iodWave per iodPeak wa ve peri odWind speed

Equilibrium range param eterPe ak enhanc ement factorRelat ive error of predictionsIntegrat ed spectral varianceSpect ral peak width param et erSpectral sha pe functionSpectral sha pe funct ionRadi an frequ en cy

wave board wave staff wood beach

WP ISWL

1.5m1.0m flume floor

27 .78mIiE-4.0m

1:40 Beach Selup

\ wood beac hr wave staff

WP l WP 10.r.>:1- ./~

l.0m ........ vflume floor~................

.--- ........~

i..,

1:20 Beach Selup

Figure 1. Cross-secti on of 1:10 sca le experimenta l arra ngements used for th e 1:40 and 1:20 beach slopes .

J ourn al of Coasta l Research, Vol. 13, No. 4, 1997

1216 Doering

10-3

10-4

10-5 l- i! \\~'",

~ 10-6a

\ \\<I)u; 10-7'C

~~I "':-\10-8

10 9

0.410-10

10-1 100 10110-20.3

/7'' i¢<

IX

0.20.1

0.2

0.1

f" ~

1lI / "

J! ( ///,"*,/ 'Il-

.Ill ~

0.05 ~ ~;.r:)/ x

oiL/o

0.15

0.4 " - - - ----,-- - - ---,---- - - -----,--- - - ----:1

0.35

0.3

'8 0.25

'0~uii~e,

Observed [rn] Frequency [Hz]

Figure 2. Observed wave heights (Hmo) versus thos e predicted using theHM method for "y = 1.7 spectra shoaling on the 1:40 beach slope. Datafrom WP 1 was used as the initial condition.

Figure 3. Measured surface elevation spectrum ifp = 0.6 Hz, 't = 1.7)versus the target spectrum. Data are from WP 1 at the toe of the 1:40beach slope. Of = 0.061 Hz and there ar e 80 d.o.f.

probes were also installed on the 1:20 beach slope. WP 1 waslocated at the toe of the 1:20 beach; that is, in the same lo­cation and depth as WP 1 was on the 1:40 beach. The re­maining nine probes were placed on 2 m centers. This yieldsa decrease of 0.1 m between adjacent probes, as for the 1:40beach slope. The objective was to position the ten wave probes(WP 1 to 10) on the 1:20 beach in the same depth of wateras the respective wave probe was on the 1:40 beach.

Wave reflection from each beach was measured using awave-wire array. The 1:40 beach yielded a reflection coeffi­cient of approximately 4% for the largest (peak) period of 2.5s waves, whereas a 7% reflection coefficient was obtained forthe same period on the 1:20 beach. Shorter waves are, ofcourse, reflected less, while longer period waves arising fromradiation stress effects associated with wave groupiness aremore strongly reflected.

Analog outputs (i .e., wave probes) were lowpass filtered at10 Hz then sampled digitally (with 12 bit resolution) at 20Hz.

DATA ANALYSIS AND DISCUSSION

The laboratory data were used to assess the accuracy androbustness of the method of wave height prediction proposedby HUGHES and MILLER (1987) . Wave heights were charac­terized using H mo' which is defined here (in the usual man­ner) as 4~, where J..lo is the integrated spectral variance.Figure 2 shows the observed wave heights versus those pre­dicted using the HM method for all the 'Y = 1.7 spectral re­alizations shoaled on the 1:40 beach slope; refer to Table 1.The data from the deepest wave staff, WP 1, located at th e

toe of the beach, were used to predict H mo at each of the nineshallower stations for each run. The 45° line indicates that,on average, the HM method underpredicts the wave heightson this beach. Least-squares regression, constrained to passthrough the origin (i .e., with an intercept of zero), yields H pred

= 0.95Hobs (r2 = 0.95), which indicates a tendency to under­predict by approximately 5%. This observed tendency of theHM method to underpredict wave heights in shallower waterdepths is similar to the laboratory observations obtained byKAMPHUIS (1991) on a 1:10 initially planar sand beach. How­ever, it is not consistent with the field observations ofHUGHES and MILLER(1987) that show no particular tenden­cy.

One of the assumptions leading to the HM method of waveheight prediction is spectral saturation. Since these waveswere created by a piston-type wave generator this assump­tion might not be met. Figure 3 shows a wave spectrum mea­sured at WP 1 versus the target spectrum. Although theagreement around the spectral peak is excellent, the mea ­sured spectrum lies below the target spectrum for frequenciesgreater than approximately 1 Hz. One way to increase (orperhaps achieve complete) energy saturation at high frequen­cies would be to use data from a shallower water depth. It isknown from visual observations that active breaking did notoccur before WP 4 for any of the runs considered. The resultof using WP 4 to predict wave heights for both shallower anddeeper water depths is shown in Figure 4. Compared to Fig­ure 2, the data are now relatively symmetric with respect tothe 45 ° line . Least-squares regression (constrained to passthrough the origin) of this data yields Hp,ed = 1.00Hobs (r2 =0.96).

Journal of Coastal Research, Vol. 13, No.4, 1997

Shoal Wave Height Prediction 1217

~

j.~

JJl(X

:r;f'

J"xxilc X

">II'

1I

j0.3 0.4

(a)0.4 ,----~---~----~--

0.3

0.35

0.25

Observed [m]

0.4

0.35

0.3 {0.25

/~a jfX

¥ ' a"0 I,"s 0.2 III.S:! , 'j< "0"0 V~ '# ... llC .~.... -.--( .;'.0.. 0.15 "0

v....0..

0.1 ,7 Xi

0.05•.rs'*fX.,,o x

,p"

I0 /0 0.1 0.2 0.3 0.4

Figure 4. Observed wave heights (H m . ) ver su s those pr edicted using theHM method for "y = 1.7 spect ra shoaling on th e 1:40 beach slope. Dat afrom WP 4 wer e used as th e initial condition.

Ob served [m]

Another assumption leading to the HM method of waveheight prediction is the bottom is relatively flat and smoothlyvarying. Figure 5a shows the observed wave heights versusthose predicted, using WP 1 as the initial condition, by theHM method for all the 'Y = 1.7 spect ra shoaled on the 1:20beach slope; i.e., these are the same initial wavetrains shoaledon a beach of twice the previous slope. Compared to Figure2, the quality of the shallow water predictions on the 1:20 isclearly not as good as that for th e more gentle 1:40 beachslope . Least-squares regression performed in a manner sim­ilar to above , yields H pred = 0.87Hobs (r 2 = 0.95), which indi­cates a tendency to underpredict by 13%. Stive (1980) hasnoted, "the water motion at each depth seems to be stronglylocally controlled". However, there is clearly a limit to theslope for which this can be true. On a steep slope the incidentwave field is unable to respond quickl y enough to th e rapidchange in water depth associated with the short (horizontal)distance that is available for spectral evolution. Thi s mightalso explain the observations of KAMPHUIS (1991), who useda 1:10 initial sand beach slope to shoal the waves. The sug­gestion from Figures 2 and 5a is the quality of the HM meth­od of wave height prediction is linked to bottom slope; in par­ticular, the method appears to yield more accurate predic­tions on relatively flat slopes than on steep slope s, which isin keeping with theoretical expectations. Figure 5b shows theresults of using WP 4 as the initial condition to predict waveheights in both shallower and deep er water depths. Least­squares regression yields H pred = 0.97Hobs (r 2 = 0.95). Onceagain, the tendency to underpredict ha s been be reduc ed byusing WP 4 as the initial condition.

It should be noted that the use of other WP's were inve s-

(b)0.4

1

0.35

0.3 ~//

/ f-a 0.25 .xX

*, x

"0 f>'¥~ 0.2oKX/~:.a

~ / jj....0.. 0.15 t '- ll<

~.\

0.1 t{ x x

0.05/t - x

/ 'zz

00 0.1 0.2 0.3 0.4

Observed [m]

Figure 5. (a) Obser ved wave heights (H m') ver su s tho se predicted usingthe HM method for "y = 1.7 spect ra shoaling on th e 1:20 beach slope.Data from WP 1 wer e used as th e ini tial condit ion. (b) Sam e as for part(a ) except dat a from WP 4 were use d as th e initial condition.

Journal of Coastal Research, Vol. 13, No, 4, 1997

1218 Doering

I I i ii i '110-2 ii i i II i

1O-3 l ,: '.

1O-4 l /r-<:.,

.€ 10-5 r " ."c::

~

Q)

-.....--'\C 10 ' t

'~~Q)uc:::e

~ 10-7

~.'

~

10-8

, I10-9 I

100 10110-10

10-110-2

Frequency [Hz]

Figure 6. Su rfa ce elevation spect ra for -y = 1.7 (- -) and for -y = 5.9(- - - ), Data ar e from WP 1 at the toe of th e 1:40 beach slope. Sf = 0.061Hz and there ar e 80 d.o.f

tigated as ini tial conditio ns to predict wave heights. WhenWP's 3 or 2 are used, there is a tendency to und erpredictwave heights. Using shallower WP's (i.e., 6 to 10) lead s toboth over- and und erprediction as the resul ts are linke d towhich runs have und ergone active bre aking. For example, themajority of th e 2.5 sec, 32 em waves for runs GROI-04 wouldhave broken at WP 5, while many of the 1.6 sec, 14 ern wavesfor runs GR18-21 would not; conse quently, both over- andunderprediction occurs .

Perhaps, an oth er way to achieve (or increase) energy sa t­uration at high frequencies would be to increase the stee p­ness of the waves; this could be achieved by increasing thepeak enhancement param eter "y. A compari son of a "y = 1.7(runs GRI 8-21) spectrum vers us "y = 5.9 (runs GR22-25) isshown in Figure 6. Hmo for "y = 5.9 is nearly twice th at of v= 1.7, i.e., 0.25 m versus 0.14 m, respectively. Figure 7 showsthe HM predictions of wave height for all th e "y = 5.9 spectrashoaling on th e 1:40 and 1:20 beach slopes; WP 1 was usedas the ini ti al condition for th ese plot s, th e resul ts should becompared to Figures 2 and 5a, respectively. Clea rly, the qual­ity of the predictions are better th an those for which "y = 1.7.Least-squares regression of the data in Figures 7a and 7byields H pr ed = 0.99Hobs (r 2 = 0.96) and H pred = 0.95Hobs (r 2 =0.95), respectively. Relative to Figures 2 and 5a, the tendencyto underpredict has has reduced, as th e data lie much closerto the 45° line. Figure 6 indicates that saturation of high fre­quencies is more closely met for the "y = 5.9 spect ral reali­zatio ns since the high frequency tail for the "y = 5.9 spectrumlies above that for v = 1.7. Figure 8 shows the outcome whenWP 4 is used to predict the wave heigh ts. The use of WP 4yields a small change in the wave heights predicted on the 1:

(a)

0.4 1

0.35

0.3

~ 025 [ .// >/1s 0.2

-Ifx

~ 0.15 .,,:.Y J

0.1 ~ :r>Ii

Z X , /

O.05t l /0.3 0.4

O'0.1 0.20

Observed [m]

(b)0.4

0.35

0.3

E 0.25

..-1l 02[ •... cP

. ·/~[;

0.. 0.15/I~ :

0.1 ~ ~,po '"

<lO'

0.05 ' "Ok:

0.1 0.2 0.3 0.40

Observed [rn]

Figure 7. Observed wave heights (H m6 ) ver sus those predicted using theHM method for -y very strongly-forced spectra shoaling on (a) the 1:40bea ch slope and (b) th e 1:20 beach slope. Data from WP 1 were used asth e initia l condition .

Journal of Coasta l Research, Vol. 13, No.4, 1997

Shoal Wave Height Pr ediction 1219

(a)

.f'o...

...

0.5

0.6 ,-------.------,--------r---,------------,

(a)0.4 ,------.-----.--------~

0.35

0.3

0.50.40.30.20.1

..." 00'''' B

g~o

0.4

0.3

0.40.30.20.1

0.2

0.1

0.15

E 0.25

Observed [m] dlL

+-0.4 f ~

-0.5~~~

t-0.6

0 0.1 0.2 0.3 0.4 0.5

(b)0.4

0.35

0.3

E 0.25 "....0........

"0 /,f u..ls 0.2 ,~.::l

,!~(J

"0 ~Q) Q)....~ 0.15

"/~':~

0.1 lP)~lllil '

00,' ~

0.05~

a 'a 0.1 0.2 0.3 0.4

-0.1

-0.2

-0.3

o

Observed [ill] dlLFigure 8. Observed wave heights (H mo) versus those predicte d usi ng theHM met hod for v very st rongly-forced spectra shoa ling on (a) th e 1:40beach slope and (b) the 1:20 beach slope. Data from WP 4 were use d asthe initia l condition.

Figu re 9. Relat ive error of the HM predictions for peak enha ncement ...,= 1.7 ( + ) and v = 5.9 (0) ru ns shoaling on (a) the 1:40 beach slope and(b) the 1:20 beach slope, res pect ively, as functio n of dlLp •

J ournal of Coasta l Research, Vol. 13, No. 4, 1997

1220 Doering

40 slope (Figure Sa), but leads to a noticable improvement inthe wave heights predicted on the 1:20 slope (Figure Sb).

To assess the overal quality of HM predictions of waveheight, the relative error, defined here as

predicted - observede=

observed

was examined. Figures 9a and 9b show the relative error forboth the 'Y = 1.7 (+) and 'Y = 5.9 (0) spectra shoaling on the1:40 and 1:20 beaches, respectively, as a function of d/Lp , therelative depth; WP 4 was used as the initial condition. Thequality of the predictions on both beaches is relatively good(i.e., e < 10%) for d/Lp > 0.2. However, as the relative depthdecreases further, the quality of the predictions deterioratessignificantly. In shallow water, ti.e., dll.; < 0.05), the relativeerror increases to = 0.5. It is interesting to note that the HMmethod overpredicts the relatively shallow water waveheights on the 1:40 beach slope yet underpredicts those onthe 1:20 beach . An attempt was made to correlate the relativeerror with the Ursell parameter, which incorporates the ef­fects of both relative wave steepness (ak term) and relativedepth (kd term). However, there was no discernable improve­ment in the observed trend.

CONCLUSIONS

Data obtained from laboratory experiments conducted ontwo planar beach slopes (1:40 and 1:20) were used to examinethe accuracy and robustness of the method of wave heightprediction proposed by HUGHES and MILLER (1987 ). Forspectra with a peak enhancement factor 'Y = 1.7 shoaled ona 1:40 beach slope, the results of the analysis indicate thatthe HUGHES and MILLER(1987) method can provide accurateest ima tes of wave heights provided the (implicit) requirementof energy saturation is satisfied. There is a tendency to un­derpredict wave heights when spect ra l saturation is incom­plete. When the same initial spectra are shoaled on a 1:20beach slope, the inshore wave heights are further underpre­dieted . The suggestion is the HM method is relatively sensi­tive to bottom slope , and will underpredict on slopes ste eperthan 1:40. Increasing the peak enhancement factor, 'Y, to 5.9leads to better predictions on both the 1:40 and 1:20 beachslopes, relative to those for v = 1.7. The improvement is like­ly related to spectral saturation. For both of the peak en­hancement factors examined, th e HM method yields relative­ly accurate wave height predictions (i .e., within 10%) provideddlL; > 0.2. In shallow water (d/Lp -s 0.05) relative errors ofup to 60% where observed. Consequently, the HM methoddoes not appear to be well-suited to predicting wave heightsnear or in the surf zone as noted by VINCENT (1984) andHUGHES (1984). In summary, th e method proposed byHUGHES and MILLER (1987 ) appears well-suited to providingreasonable estimates (rela tive error < 10%) of wave height

for a wide range of single-peaked, saturated spectra shoalingon a relatively gentle slope (0(1:40)) provided d/L > 0.2.

ACKNO~DGEMENTS

This research was conducted while th e author was a Vis­iting Fellow with Dr. M.A. Donelan at the National WaterResearch Institute, Canada Centre for Inland Waters, Bur­lington, Ontario. The laboratory experiments were funded byP.E.R.D . under project number 62124 , and were lead by Dr.M.G. Skafe!. The laboratory experiments were conducted bythe author with the assistance of T. Nudds, K. Davis , G. Vo­ros , and B. Taylor. The analysis and synthesis of th ese resultswas supported by the author's NSERC research grant.

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Journal of Coastal Research, Vol. 13, No.4, 1997