Breakpoint generated surf beat induced by bichromatic wave groups

Coastal Engineering 39 Ž2000. 213–242 www.elsevier.comrlocatercoastaleng

Breakpoint generated surf beat induced by bichromatic wave groups T.E. Baldock a

a,)

, D.A. Huntley b,1, P.A.D. Bird G.N. Bullock a,4

a,2

, T. O’Hare

b,3

,

School of CiÕil and Structural Engineering, UniÕersity of Plymouth, Palace Street, Plymouth, PL1 2DE, UK b Institute of Marine Studies, UniÕersity of Plymouth, Drake Circus, Plymouth, PL4 8AA, UK Received 26 May 1999; received in revised form 7 October 1999; accepted 1 November 1999

Abstract This paper presents new experimental data on 2-D surf beat generation by a time-varying breakpoint induced by bichromatic wave groups. The experimental investigation covers a broad range of wave amplitudes, short wave frequencies, group frequencies and modulation rates. The data include measurements of incident and outgoing wave amplitudes, breakpoint position, shoreline run-up and the cross-shore structure of both the short and long wave motion. Surf beat generation is shown to be in good agreement with theory wSymonds, G., Huntley, D.A., Bowen, A.J., 1982. Two dimensional surf beat: long wave generation by a time-varying breakpoint. J. Geophys. Res. 87, 492–498x. In particular, surf beat generation is dependent on the normalised surf zone width, which is a measure of the phase relationship between the seaward and shoreward breakpoint forced long waves, and linearly dependent on the short wave amplitude. The cross-shore structure of the long wave motion is also consistent with theory; at maximum and minimum surf beat generation, the mean breakpoint coincides with the nodal and anti-nodal points, respectively, for a free long wave standing at the shoreline. A numerical solution, using measured data as input, additionally shows that the phase relationship between the incident bound long wave and the

)

Corresponding author. Tel.: q44-1752-233664; fax: q44-1752-233658; e-mail: [email protected] E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 4 E-mail: [email protected]. 1

0378-3839r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 9 . 0 0 0 6 1 - 7

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outgoing breakpoint forced wave is consistent with the time-varying breakpoint mechanism. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Surf beat; Wave groups; Long waves; Breakpoint; Bound waves; Coastal

1. Introduction Surf beat is a collective term for low frequency gravity water waves in the coastal zone, first applied by Munk Ž1949. to long period waves observed outside the surf zone which appeared correlated to breaking waves further shoreward. Tucker Ž1950. subsequently showed a correlation between incident short wave groups and the surf beat, with a lag corresponding to the time required for the short waves to propagate to the shore and a low frequency long wave to travel back. Following these observations, a large number of field studies have demonstrated the importance of low frequency motions inside and outside the surf zone and in the swash zone. In these locations surface elevation and velocity spectra are frequently dominated by low frequency energy ŽHuntley et al., 1977; Guza and Thornton, 1982, 1985; Wright et al., 1982; and others.. The low frequency motions are typically highly correlated with short Žwind. wave energy Že.g., Elgar et al., 1992; Herbers et al., 1995b., consistent with generation by short waves, and may be either propagating cross-shore Žleaky waves., refractively trapped Žedge waves., or a mixture of both ŽSuhayda, 1974; Huntley et al., 1981; Oltman-Shay and Guza, 1987.. Surf beat is significant from an engineering perspective since low frequency long waves may modify the incident short waves ŽGoda, 1975., and may play an important role in sediment transport, leading to the formation of bars, cusps and more complex morphology Že.g., Holman and Bowen, 1982; O’Hare and Huntley, 1994.. A number of mechanisms for the generation of surf beat have been proposed ŽLonguet-Higgins and Stewart, 1962, 1964; Gallagher, 1971; Symonds et al., 1982, List, 1992; Schaffer, 1993., which may generate either leaky waves or edge waves. However, the mixture of wave modes on natural beaches complicates the comparison of field data with theoretical models for surf beat generation. Considerable uncertainty therefore still remains as to which mechanisms are important or dominant under different surf zone conditions ŽBattjes, 1988; Hamm et al., 1993.. Furthermore, only one previous laboratory study appears to have been specifically focused on surf beat generation mechanisms Ži.e., Kostense, 1984.. The present paper considers this problem and examines 2-D surf beat generation Žleaky modes only. through carefully controlled laboratory experiments. In particular, bichromatic wave groups are used to study surf beat generation by a time-varying breakpoint ŽSymonds et al., 1982; Schaffer, 1993.. Good quantitative and qualitative agreement is found between the measurements, theory and recent numerical model results presented by Madsen et al. Ž1997.. The laboratory study includes measurements of the inner and outer breakpoint positions and the cross-shore structure of both the short and long wave motion. Section 2 of this paper presents a review of previous work and a simplified discussion of the time-varying breakpoint mechanism of Symonds et al.

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Ž1982. is given in Section 3. The experimental set-up, instrumentation and data analysis methods are outlined in Section 4. Section 5 presents and discusses the experimental data, with final conclusions drawn in Section 6.

2. Previous work Following the observations of Munk Ž1949. and Tucker Ž1950., Longuet-Higgins and Stewart Ž1962; 1964. showed that short wave groups non-linearly force a long wave, bound to the short wave groups. This incident bound long wave Žhereafter abbreviated by IBLW. is not normally considered surf beat, which generally refers to free long waves. However, Longuet-Higgins and Stewart Ž1962; 1964. suggested that surf beat was consistent with the release through wave breaking, and subsequent reflection at the shoreline, of the IBLW, although no release mechanism was Žor has been. proposed. Battjes Ž1988. argues that this is incorrect, since the correlation between the short wave envelope and the IBLW should be greater than that between the short wave envelope and a released and reflected long wave, which is not borne out by the observations of Tucker Ž1950.. Symonds et al. Ž1982. subsequently proposed a mechanism for the generation of surf beat which was directly due to the variability of wave breaking, further elaborated on by Symonds and Bowen Ž1984. for barred beaches. This model proposes that a time-varying breakpoint position Ždue to incident wave groupiness. radiates long waves at the group frequency both shorewards and seawards. If the shoreward propagating long waves reflect at the shoreline, then an interference pattern is set-up and the amplitude of the final seaward propagating wave should vary according to the group frequency and surf zone width. Qualitative support for the Symonds et al. Ž1982. model was provided by the laboratory experiments of Kostense Ž1984., although the quantitative agreement was not so good. Schaffer Ž1993. extended the Symonds et al. Ž1982. model by including short wave forcing within the surf zone Ždue to continued wave grouping. and examined the cross-shore structure of the long wave motion outside the surf zone by including the IBLW. Schaffer Ž1993. found better agreement with the data of Kostense Ž1984., although the surf beat was typically over-estimated by a factor of two, which Schaffer Ž1993. argued was due to the assumption of full shoreline reflection and neglect of long wave dissipation within the surf zone. An alternative analysis of the breakpoint mechanism was also put forward by Mizuguchi Ž1995.. Using a Boussinesq type numerical model which allowed for partial reflection and long wave dissipation, Madsen et al. Ž1997. found good agreement with the data of Kostense Ž1984. and also observed the strong frequency dependence of the seaward propagating free long wave predicted by the Symonds et al. Ž1982. model. The results of Madsen et al. Ž1997. are discussed further in Section 5 in relation to the present experimental data. Other numerical models which have the potential for reproducing surf beat generation include those presented by Hibberd and Peregrine Ž1979., Kobayashi et al. Ž1989., List Ž1992., Watson and Peregrine Ž1992., Roelvink Ž1993., Van Dongeren et al. Ž1994., and Herbers et al. Ž1995a. Žsee Hamm et al., 1993 for a general review..

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A problem with many of these models is that it is frequently not possible to separate the different long wave generation mechanisms, i.e., time-varying break point, release of bound long waves or swash–swash interactions ŽWatson et al., 1994., although the separation is explicit in the model of List Ž1992.. However, Madsen et al. Ž1997. suggest that List’s model will significantly overestimate the amplitude of the incident bound wave. A second potential problem with the model of List Ž1992. is that the long waves are decoupled from the short waves. Consequently, since energy continually flows from the short waves to the bound wave and vice versa ŽLonguet-Higgins and Stewart, 1964; Lo and Dean, 1995., List’s model may not provide a true representation of the bound long wave. Nevertheless, the model of List Ž1992. does show qualitative agreement with field data showing cross-correlations between the offshore short wave envelope and low frequency waves in the surf zone Že.g., Guza et al., 1984; Masselink, 1995.. However, the release of the IBLW Žand subsequent reflection. implies that surf beat should show a quadratic dependence on the short wave amplitude ŽBattjes, 1988., whereas the majority of field data shows a roughly linear dependence on short wave amplitude Že.g., Guza and Thornton, 1982; Elgar et al., 1992; Herbers et al., 1995b; Ruessink, 1998.. The same field data sets tend to show that incident forced Žor bound. wave energy follows the predicted quadratic dependence ŽHerbers et al., 1995b; Ruessink, 1998.. Laboratory and field data therefore to some extent support both the time-varying breakpoint mechanism ŽSymonds et al., 1982; Schaffer, 1993. and the hypothesis of IBLW release of Longuet-Higgins and Stewart Ž1962.. However, in the field, determining the mechanisms by which surf beat is generated is complicated by factors such as spectral width, directionally spread wave fields, refractive trapping and varying topography ŽHerbers et al., 1995a.. Consequently, further laboratory experiments, previously limited to Kostense Ž1984., appear to be beneficial and such experiments are reported in this paper.

3. Surf beat generation by a time-varying breakpoint Prior to discussing the experimental results it is useful to examine features that theoretically may be expected in the data. A simplified version of 2-D surf beat generation by a time-varying breakpoint is therefore outlined below; for details refer to Symonds et al. Ž1982. and Schaffer Ž1993.. Fig. 1 shows a schematic representation of the expected long wave phase structure at two locations either side of, and close to, the mean breakpoint. It is also assumed that the breakpoint excursion is small in the cross-shore direction. The top panel ŽFig. 1a. shows an instant in time when a local maximum in the short wave envelope reaches the mean breakpoint, indicated by the vertical dashed line. Radiation stress gradients induce both an incident breakpoint forced long wave ŽIBFLW. and an outgoing breakpoint forced long wave ŽOBFLW. at the group frequency. An onshore propagating wave crest occurs at this instant in time since larger waves produce a larger dynamic set-up. The IBFLW subsequently reflects at the shoreline and propagates offshore ŽRBFLW.. If the phases of the OBFLW and RBFLW coincide at the outer breakpoint, then

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Fig. 1. Expected long wave phase due to surf beat generation by a time-varying breakpoint. IBFLW, RBFLW, OBFLW — incident, reflected and outgoing breakpoint forced long wave. IBLW — incident bound long wave, RBLW — releasedrreflected ‘‘bound’’ long wave. Ža. Maximum response, mean breakpoint at a nodal point for a free standing long wave. Žb. Minimum response, mean breakpoint at an antinode of a free standing long wave.

constructive interference occurs and the total breakpoint forced free long wave ŽBFLW. attains maximum amplitude. This occurs when the mean breakpoint position approximately corresponds to a nodal point for a free standing long wave at the group frequency. If a measure of the relative phase between the OBFLW and RBFLW is given

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by x s v 2 h brg b 2 , where v is the group frequency Ž2p fG ., h b is the water depth at the mean breakpoint, g is gravity and b is the beach slope, Symonds et al. Ž1982. show that maximum response occurs for a x value of about 1.2. Minimum response occurs when the OBFLW and RBFLW are out of phase at the outer breakpoint, corresponding to an antinode for a free standing long wave and a x value of about 3.7 ŽFig. 1b.. Fig. 1 further shows that, around the mean breakpoint, large gradients are to be expected in the cross-shore variation of the long wave amplitude Žthese appear as discontinuities at the mean breakpoint in Schaffer, 1993.; at maximum response an antinode and node are expected seaward and shoreward of the mean breakpoint, respectively, and vice versa at minimum response. The presence of an additional IBLW is seen to lead to an increase and decrease in these gradients for maximum and minimum response, respectively. For example, with the addition of the IBLW which produces a set-down under a local short wave envelope maximum, the amplitude of the total long wave motion outside the breakpoint will increase for both maximum and minimum response. Consequently, a node will not occur seaward of the breakpoint at minimum response. If the IBLW were released at the breakpoint and subsequently reflected ŽRBLW., then further modifications to the nodal structure would occur, shown by dotted lines. Fig. 1 is also useful for determining the phase relationship between the total outgoing surf beat wave ŽBFLWs OBFLWq RBFLW. and the IBLW, which defines the nodal pattern expected further seaward Žsee Section 5.. For example, at maximum response the outgoing BFLW will be in phase with the IBLW. In contrast, a released and reflected ‘‘bound’’ wave ŽRBLW. would be out of phase with the IBLW ŽFig. 1a.. Finally, at an instant in time when a local short wave envelope minimum reaches the mean breakpoint, all the phases indicated in Fig. 1 simply reverse. 4. Experimental set-up 4.1. WaÕe flume The experiments were carried out in a wave flume 18 m long, 0.9 m wide, with a working water depth of 0.8 m ŽFig. 2.. A plane beach Žgradient b s 0.1. starts 5.65 m from the wave paddle and within the inner surf and swash zones the bed comprises of a single 25 mm thick rigid polyethylene sheet, providing a true smooth uninterrupted surface. The origin of the horizontal co-ordinate, x, is taken as the intersection of the still water line with the beach face, positive onshore. For unbroken monochromatic regular waves with frequencies of 0.1–0.4 Hz, the reflection coefficient for this beach ranges from 0.9 to 0.6. Wave reflection for breaking waves with frequencies of 0.6–1 Hz is negligible Ž3–5%.. Waves were generated by a hydraulically driven wedge type wave paddle using second order generation for long waves ŽBarthel et al., 1983.. A new wave absorption system was developed for this study, which uses digital filtering incorporated within the wave generation software to absorb waves radiated from the far end of the flume. In addition to avoiding the cost of additional electronic hardware, an advantage of this system over conventional methods is that the wave generation signal is not passed

T.E. Baldock et al.r Coastal Engineering 39 (2000) 213–242

Fig. 2. Wave flume and instrumentation.

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through the absorption filter. This allows for easier filter design and results in more accurate incident wave generation. Tests with regular waves indicate that the wave paddle absorbs up to 60% Žin amplitude terms. of reflected waves for frequencies at 0.1 Hz, rising to over 90% above 0.4 Hz. As may be seen from the data presented in Section 5, where both short and long wave amplitudes obtained from different runs are nearly identical, the wave motion generated by the wave paddle is highly repeatable, allowing data to be collected at multiple cross-shore locations. 4.2. Instrumentation Data were collected simultaneously from an array of five surface piercing resistance type wave gauges, mounted on a trolley above the tank, and a run-up wire within the swash zone. Modified wave gauges allowed surface elevation data to be collected in the inner surf and swash zones. Variable wave gauge spacing was used within the array, giving a wide range of gauge separation for combinations of two and three gauges. The absolute accuracy of these wave gauges is of order "1 mm, with a relative accuracy better than "0.2 mm. The run-up wire consists of two 0.8 mm diameter stainless steel wires 12 mm apart, held 3 mm above the bed, and has a resolution better than "0.5 mm in the vertical. Each wire is held at a constant tension of 100 N at the upper end of the beach, passing through a spacer just above the maximum run-up limit. Spacers are therefore avoided in the swash zone, with consequently minimal flow resistance. The inner and outer breakpoints were identified visually against marker rods temporarily fixed in the surf zone. For plunging waves the breakpoint was taken to be the location of the plunger impact; for spilling waves the breakpoint was defined to be when a significant roller had formed on the front face, i.e., not at incipient breaking. Although some subjectivity is involved in this process, tests with regular waves indicated that both these definitions coincided closely with the location of the change from set-down to set-up in the surf zone. However, the accuracy of the breakpoint positions given below is unlikely to be much better than "5 cm. 4.3. Bichromatic waÕe group characteristics In the present study 65 different bichromatic wave groups were investigated, divided into five series ŽA–E., with the group frequency Ž fG s f 1 y f 2 . ranging from 0.1 to 1 Hz ŽTable 1.. Series A and B have a mean primary wave frequency ŽŽ f 1 q f 2 .r2. of approximately 1 Hz, with primary wave amplitudes Ž a1 , a 2 . of 0.025 m ŽA. and 0.0125 m ŽB., representing fully modulated wave groups Ž a2ra1 s d s 1.. Series C has a mean primary wave frequency of 0.6 Hz and amplitudes of 0.025 m Ž d s 1.. Series D and E aim to correspond to numerical model calculations by Madsen et al. Ž1997., and have the same mean primary wave frequency Ž0.6 Hz. and total amplitude Ž a1 q a 2 s 0.08 m. but different modulation rates ŽD: d s 1; E: d s 0.2, weak modulation.. The surf similarity parameter Ž br6Ž HrLo .. for the short waves is in the range 0.35–0.7, suggesting a combination of spilling and plunging breakers. Both the short wave and group frequencies Žand their harmonics. coincide with FFT frequency bins for integer multiples of 512 data points sampled at 25 Hz Ž D f s 0.048828 Hz.. This results in clear spectral signals at each frequency Žsee Section 5. and is essential for accurate separation of incident and reflected waves.

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Table 1 Bichromatic wave group frequencies Amplitudes: ŽA. a1 s a2 s 0.025 m, ŽB. a1 s a 2 s 0.0125 m, ŽC. a1 s a 2 s 0.025 m, ŽD. a1 s a 2 s 0.04 m, ŽE. a1 q a 2 s 0.08 m, a 2 r a1 s 0.2. CaserSeries

f 1 ŽHz.

f 2 ŽHz.

fG ŽHz.

f R ŽHz.

b1010rA,B b1015rA,B b1020rA,B b1025rA,B b1030rA,B b1035rA,B b1040rA,B b1045rA,B b1050rA,B b1055rA,B b1060rA,B b1065rA,B b1070rA,B b1075rB b1080rB b1085rB b1090rB b1095rB b1100rB b6010rC,D,E b6015rC,D,E b6020rC,D,E b6025rC,D,E b6030rC,D,E b6035rC,D,E b6040rC,D,E b6045rC,D,E b6050rC,D,E b6055rC,D,E b6060rC,D,E

1.025 1.074 1.074 1.123 1.123 1.172 1.172 1.221 1.221 1.269 1.269 1.318 1.318 1.367 1.367 1.416 1.416 1.465 1.465 0.634 0.683 0.683 0.732 0.732 0.781 0.830 0.830 0.830 0.879 0.879

0.928 0.928 0.879 0.879 0.830 0.830 0.781 0.781 0.732 0.732 0.683 0.683 0.634 0.634 0.586 0.586 0.537 0.537 0.488 0.537 0.537 0.488 0.488 0.439 0.439 0.439 0.391 0.341 0.341 0.293

0.098 0.147 0.195 0.244 0.293 0.342 0.391 0.439 0.488 0.537 0.586 0.634 0.683 0.732 0.781 0.830 0.879 0.928 0.976 0.098 0.147 0.195 0.244 0.293 0.341 0.391 0.439 0.488 0.537 0.586

0.049 0.049 0.098 0.049 0.049 0.049 0.391 0.049 0.244 0.049 0.098 0.049 0.049 0.049 0.195 0.049 0.049 0.049 0.049 0.049 0.049 0.097 0.244 0.147 0.049 0.049 0.049 0.049 0.049 0.293

The last column in Table 1 gives the repeat frequency Ž f R . for each wave group. This is the frequency at which the short wave phases within the group identically repeat, and is typically a subharmonic of fG . This arises because the short wave phase relationship in an individual group within the same wave train generally differs from that in preceding or following groups Žsee classical sketches of wave groups, e.g., Dean and Dalrymple, 1991, p. 99.. Consequently, slow variations in short wave phase are likely to lead to breakpoint modulations at f R Žand other frequencies., and therefore surf beat generation at frequencies other than fG , which does not appear to have been considered previously. Finally, Table 2 shows the measured inner and outer breakpoint positions Ž x bi , x bo ., h b , and x for each wave group in Table 1. Here h b is defined as the mean still water depth between the inner and outer breakpoints, and is subsequently used to determine x , which ranges from 0.1 to 13.

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Table 2 Bichromatic wave group characteristics, x positive onshore from SWL ŽA. a1 s a 2 s 0.025 m, ŽB. a1 s a 2 s 0.0125 m, ŽC. a1 s a 2 s 0.025 m, ŽD. a1 s a 2 s 0.04 m, ŽE. a1 q a 2 s 0.08 m, a 2 r a1 s 0.2. CaserSeries

b1010rA b1015rA b1020rA b1025rA b1030rA b1035rA b1040rA b1045rA b1050rA b1055rA b1060rA b1065rA b1070rA b1010rB b1015rB b1020rB b1025rB b1030rB b1035rB b1040rB b1045rB b1050rB b1055rB b1060rB b1065rB b1070rB b1075rB b1080rB b1085rB b1090rB b1095rB b1100rB b6010rC b6015rC b6020rC b6025rC b6030rC b6035rC b6040rC b6045rC b6050rC b6055rC b6060rC b6010rD b6015rD b6020rD

fG ŽHz. 0.098 0.147 0.195 0.244 0.293 0.342 0.391 0.439 0.488 0.537 0.586 0.634 0.683 0.098 0.147 0.195 0.244 0.293 0.342 0.391 0.439 0.488 0.537 0.586 0.634 0.683 0.732 0.781 0.830 0.879 0.928 0.976 0.098 0.147 0.195 0.244 0.293 0.341 0.391 0.439 0.488 0.537 0.586 0.098 0.147 0.195

Break point Žm. Outer Ž x bo .

Inner Ž x bi .

y1.10 y1.10 y1.05 y0.90 y0.90 y0.95 y0.85 y0.90 y0.80 y0.90 y0.90 y0.90 y0.90 y0.40 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.45 y0.50 y0.50 y0.50 y0.50 y0.50 y0.50 y0.90 y0.95 y0.95 y0.95 y0.95 y0.95 y0.95 y1.00 y1.05 y1.05 y1.10 y1.35 y1.35 y1.35

y0.15 y0.15 y0.15 y0.15 y0.15 y0.15 y0.20 y0.25 y0.25 y0.25 y0.30 y0.25 y0.25 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.10 y0.15 y0.15 y0.20 y0.20 y0.20 y0.15 y0.10 y0.05 y0.05 y0.05 0.00 y0.25 y0.15 y0.15

h b Žm.

x

0.063 0.063 0.060 0.053 0.053 0.055 0.053 0.058 0.053 0.058 0.060 0.058 0.058 0.025 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.030 0.030 0.030 0.030 0.030 0.030 0.053 0.055 0.058 0.058 0.058 0.055 0.053 0.053 0.055 0.055 0.055 0.080 0.075 0.072

0.24 0.54 0.92 1.26 1.80 2.57 3.22 4.46 5.03 6.67 8.26 9.30 10.79 0.10 0.24 0.42 0.66 0.94 1.29 1.69 2.13 2.64 3.19 3.80 4.45 5.16 6.47 7.36 8.32 9.31 10.37 11.50 0.20 0.47 0.88 1.38 1.97 2.57 3.21 4.07 5.27 6.38 7.57 0.31 0.64 1.11

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Table 2 Žcontinued. CaserSeries

b6025rD b6030rD b6035rD b6040rD b6045rD b6050rD b6055rD b6060rD b6010rE b6015rE b6020rE b6025rE b6030rE b6035rE b6040rE b6045rE b6050rE b6055rE b6060rE

fG ŽHz. 0.244 0.293 0.341 0.391 0.439 0.488 0.537 0.586 0.098 0.147 0.195 0.244 0.293 0.341 0.391 0.439 0.488 0.537 0.586

Break point Žm. Outer Ž x bo .

Inner Ž x bi .

y1.25 y1.25 y1.25 y1.20 y1.15 y1.15 y1.15 y1.15 y1.35 y1.35 y1.35 y1.30 y1.30 y1.25 y1.25 y1.25 y1.30 y1.30 y1.25

y0.15 y0.15 y0.15 y0.10 y0.05 y0.05 y0.05 y0.05 y0.80 y0.75 y0.75 y0.65 y0.65 y0.65 y0.65 y0.75 y0.75 y0.65 y0.65

h b Žm.

x

0.070 0.070 0.070 0.065 0.060 0.060 0.060 0.060 0.108 0.105 0.105 0.098 0.098 0.095 0.095 0.100 0.103 0.098 0.095

1.68 2.40 3.28 3.98 4.65 5.75 6.96 8.29 0.41 0.90 1.61 2.34 3.35 4.45 5.81 7.76 9.82 11.31 13.13

4.4. Analysis techniques In the constant depth region of the flume Ž x - y8 m., the amplitudes of incident bound waves, incident free waves and outgoing free waves at the group frequency were separated by the three wave gauge method presented by Kostense Ž1984., also used by Madsen et al. Ž1997.. The amplitudes of incident and outgoing free waves at subharmonics of fG Žand other frequencies. were determined using the two gauge method of Frigaard and Brorsen Ž1995., again in the frequency domain and for x - y8 m Žthe two methods give consistent results if only free waves are present, but the latter is more convenient.. Amplitude spectra were obtained directly via a FFT using 4096 data points sampled at 25 Hz, with no windowing or frequency averaging applied. Note that each wave group may be considered to be deterministic, rather than a representation of a random process and therefore the spectra shown later do not require the confidence limits associated with stochastic processes ŽBaldock et al., 1996.. 5. Discussion of results 5.1. Incident and radiated waÕe amplitudes 5.1.1. Incident bound long waÕes In order to illustrate the reliability of the wave generation system, and to test the bound wave separation technique discussed above, comparisons between the measured and theoretical IBLW amplitude at x - y8 m are shown in Fig. 3a, with satisfactory

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Fig. 3. Ža. Comparison of measured incident boundwave amplitude with theory. IIII Data points, series A–E. Žb. Measured incident boundwave amplitude vs. x . — —I— — series A, – – = – – series B, — —PeP — — series C, — —PP=PP— — series D, — — D — — series E.

agreement. The theoretical calculations are based on the original finite depth solution for two interacting primary waves Ž a1 , f 1; a2 , f 2 . due to Longuet-Higgins and Stewart Ž1960. Žalso given in Baldock et al., 1996.. This solution is more easily applied to this data than the radiation stress approach that followed ŽLonguet-Higgins and Stewart, 1962, 1964.. Fig. 3b shows the same data for each series A–E, plotted vs. x ŽTable 2.. The IBLW amplitude increases with group frequency and incident short wave height Žand hence x .. The IBLW amplitude is typically small, 0.5–3 mm, at this location Žrelatively deep water., but increases rapidly as the depth reduces Žsee Figs. 9–13.. Note that all the data presented in Figs. 3–7 were obtained in the constant depth region of the flume. 5.1.2. Radiated free long waÕes at the group frequency Although some reflection is possible from the sloping beach seaward of the surf zone, and wave groups propagating up a slope may generate free outgoing waves ŽHamm et al., 1993., numerical calculations ŽA. van Dongeren, A. Reniers, pers. comm.. suggest that these are small compared to the waves generated in the surf zone. In addition, since relatively large outgoing free waves exist, but perfect wave absorption is not possible at

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Fig. 4. Ža. Outgoing free wave amplitude vs. group frequencyrslope, series D. — —I— — Madsen et al. Ž1997. Ž b s1r20., — —PP=PP— — series D. Žb. Outgoing free wave amplitude vs. group frequencyrslope, series E. — —I— — Madsen et al. Ž1997. Ž b s1r40., — —^— — series E. Žc. Outgoing free wave amplitude vs. group frequencyrslope, series A–E. — —I— — series A, – – = – - series B, – – = – – series C, — —PP=PP— — series D, — — D — — series E.

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Fig. 5. Ža. Normalised outgoing free wave amplitude at the group frequency vs. x . — —I— — series A, – – = – - series B, theory, D as 0.2, Symonds et al. Ž1982.. Žb. Normalised outgoing free wave amplitude at the group frequency vs. x . — —PeP— — series C, — —PP=PP— — series D, — —^— — series E, theory, D as 0.2, Symonds et al. Ž1982..

the wave paddle, a small amount of incident free long wave energy must be expected, which may reflect from the far end of the flume. However, in the following, we assume that all outgoing free long waves are generated shoreward of the outer breakpoint by one or more of the mechanisms discussed in Section 2. This largely appears to be confirmed by the cross-shore nodal structure of the long wave motion ŽSection 5.2.. Fig. 4a and b show the amplitude of the outgoing free long wave at the group frequency for series D and E, respectively. These cases were considered by Madsen et al. Ž1997. and the results Žobtained from their figures. are compared to the present data. Note, however, that their group frequencies and beach slope differ from those here. The data are plotted vs. fG rb , based on Schaffer Ž1993., who shows that long Žlow frequency. wave groups over mild slopes behave similarly to short Žhigh. frequency wave groups over steep slopes. For series D ŽFig. 4a. the numerical results are in excellent agreement with the present data, which implies that both the assumptions in their model and the experimental set-up described above are satisfactory. Note that the final data point attributed to Madsen et al. Ž1997., connected by a dashed line, is the

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Fig. 6. Ža. Ratio of outgoing to incident free wave amplitude at the group frequency vs. x . — —I— — Series A, – – = – - series B. Žb. Ratio of outgoing to incident free wave amplitude at the group frequency vs. x . — —PeP— — series C, — —PP=PP— — series D, — — D — — series E.

result for a beach slope of 1r30, rather than 1r20. For series E ŽFig. 4b., both data sets show a maximum in the frequency response at the same value of fG rb , although the amplitude of the outgoing wave found from the present data is considerably larger and there is no secondary maximum at higher values of fG rb . However, the results of Madsen et al. Ž1997. suggest that the amplitude of the outgoing wave increases significantly with beach slope, and therefore both data sets are again consistent. The experimental data show a very clear frequency response, in agreement with surf beat generation by a time-varying breakpoint, further illustrated for all five series A–E in Fig. 4c. Note that for typical laboratory wave heights of 0.1–0.2 m, a x value of 1.2 corresponds to values for fG rb in the range 1.5–2.5, so the results are all consistent with Symonds et al. Ž1982.. In order to compare the measured data directly with the model of Symonds et al. Ž1982., the amplitudes of the outgoing free waves at the outer breakpoint are required. These were estimated from the measurements in the constant depth region of the flume using full linear shoaling Žrather than assuming shallow water.. These amplitudes were then normalised by half the difference in shoreline set-up produced by monochromatic

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Fig. 7. Normalised outgoing free wave amplitude at f / fG vs. x . — —I— — Data, - - - - approximate theory, d s1, after Symonds et al. Ž1982.. Ža. series A, Žb. series C, Žc. series D, Žd. series E.

waves with heights corresponding to the largest and smallest short waves within the group. This was calculated following Longuet-Higgins and Stewart Ž1964. and given by:

h

1 3 g s

2

ž

2 2 2

Ž a1 q a2 . Ž d U .

/

Ž 1.

where g is the ratio of the wave height to water depth at breaking and d U represents the modulation of the wave group. From a series of monochromatic wave measurements, g has been estimated to be 1, typical of steep beaches, and Eq. Ž1. provides good estimates of the shoreline set-up. However, although the theoretical modulation ratio for series

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A–D is d s 1, implying a minimum wave height within the groups of zero, this does not occur in reality. Therefore, based on measurements of the maximum and minimum wave heights within the groups, d U has been taken to be 0.9 and 0.3 for series A–D and series E, respectively. Note, however, that the normalised response is therefore sensitive to both g and d U . The results are shown in Fig. 5a and b, which compare the normalised outgoing free wave amplitude at the group frequency with Symonds et al. Ž1982.. Series A and B Ž d s 1. show a maximum response at a x value of about 1.1, close to that expected from theory. For both series, the normalised amplitude is also very similar, showing that the generation process is linearly dependent on the incident short wave amplitude. Furthermore, although the maxima are significantly smaller than the theoretical value for d s 0.2, this is consistent with Symonds et al. Ž1982., who show that the normalised response reduces quite rapidly as d increases. Therefore, although the assumption of a sinusoidal breakpoint by Symonds et al. Ž1982. restricts the solution to small d Žthe same restriction applies in Schaffer, 1993., in reality surf beat generation also appears to occur in fully modulated groups and exhibits the same dependence on x . Series C and D Ž d s 1, Fig. 5b. show very similar trends; the maximum response again occurs at x f 1.1 and the response is linearly dependent on amplitude. For series E Ž d s 0.2., the measured data is in very good quantitative agreement with the theory, at least in the range 0 - x - 4. However, it is clear that there are no secondary maxima for x f 6, as suggested by Symonds et al. Ž1982., and reasons for this are discussed further in Section 5.2. For series E, Madsen et al. Ž1997. found that the maximum response corresponded to a x value of about 3, in contrast to the present data which gives a corresponding x value of about 1.1. This appears to due to the different beach slopes; Madsen et al. Ž1997. found h b to be about 0.2 m Ž b s 1r40., whereas here h b f 0.1 m Ž b s 1r10.. However, for fully modulated and weakly modulated groups, the results of Madsen et al. Ž1997. show very little difference in the frequency at which the maximum and minimum response occurs Žtheir Fig. 22.. Furthermore, for a fully modulated group the mean breakpoint position will be much closer to the shoreline than for a weakly modulated group, indeed h b will nearly halve, with a similar reduction in x . Somewhat surprisingly, therefore, Madsen et al. Ž1997. found that a weakly modulated group showed significant deviations from Symonds et al. Ž1982., whereas their results show that a fully modulated group behaved more in accordance with theory Žalthough they did not comment on this.. In contrast, the present data show the same trends for both fully and weakly modulated wave groups. Fig. 6a and b show the ratio of the outgoing to incident free wave amplitude Ž R f . at the group frequency. It seems more appropriate to regard this ratio as a ‘‘radiation coefficient’’, rather than a ‘‘reflection coefficient’’, since most of the outgoing free wave energy at the group frequency is generated shorewards of the measurement point Ž x - y8 m.. In addition, according to theory, some of this wave energy is radiated directly offshore and never reflects at the shoreline. The radiation coefficient is very large at maximum response Žof order 5–7., and is typically greater than 2 for all values of x , series A being an exception. Note that, in theory, infinite radiation coefficients should occur, since both Symonds et al. Ž1982. and Schaffer Ž1993. assume incident free

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wave energy is zero. High radiation coefficients at low values of x Žlow group frequencies. also show that the wave absorption system described in Section 4 is highly effective. Indeed, values of R f ) 3 indicate that over 90% of the outgoing free long wave energy measured in the constant depth region of the wave flume originated shorewards of that location. Consequently, the assumption of minimal incident free wave energy appears justified. 5.1.3. Radiated free long waÕes at other frequencies As noted in Section 4.3, the variation in short wave phase between successive groups in an individual wave train should lead to variations in the breakpoint and surf beat generation at subharmonics of fG Žand other frequencies.. In addition, Symonds et al. Ž1982. show that a time-varying breakpoint will also radiate waves at higher harmonics of fG . In order to investigate this further, the amplitude of incident and outgoing free waves at discrete frequencies below the mean primary wave frequency ŽŽ f 1 q f 2 .r2. were calculated as described in Section 4.4. Data at the group frequency, frequencies where the outgoing wave amplitude was less than 0.5 mm and frequencies where R f - 2 were then excluded and the individual outgoing wave amplitudes Žnormalised as above and plotted vs. x from Table 2. are shown in Fig. 7a–d. Note that different cases in the same series frequently show a response at the same, or similar, values of x . Also shown on these plots is an approximate version of the response expected from Symonds et al. Ž1982., where their curve for d s 0.2 has simply been scaled so that the peak normalised value equals 0.5. It should be stressed that this curve is not intended to represent any theory; its main purpose is to show the shape of the expected response function. However, it seems reasonable to suppose that the response at these frequencies will be much weaker than that at the group frequency. Fig. 7 shows similar trends to those in Fig. 5. In each case surf beat generation appears to reach a maximum at x f 1 and minimal outgoing energy was observed in the range 3 - x - 4.5. In the frequency domain, most of the outgoing wave energy corresponded to subharmonics of fG , but outgoing wave energy was also detected at 2 fG and 3 fG . It therefore appears that breakpoint forcing also occurred at higher harmonics of the group frequency, again in accordance with Symonds et al. Ž1982.. 5.2. Cross-shore structure of the long and short waÕe motion Typical amplitude spectra in the constant depth region of the flume, at the outer breakpoint and in the run-up are shown in Fig. 8a and b, which correspond to cases b1025rA and b6020rC. The spectra show the two primary harmonics, a harmonic at the group frequency and a number of sub-and super-harmonics. The harmonics at 1.37 Hz and 0.88 Hz, for cases b1025rA and b6020rC respectively, appear to be near-resonant triads Žsee e.g., Freilich and Guza, 1984., corresponding to the interaction between f 1 and fG Ž f s 2 f 1 y f 2 .. These figures show the resolution of the spectral analysis described in Section 4.4 and the total amplitude Žincident and outgoing. of individual harmonics was subsequently readily obtained from FFTs at multiple cross-shore locations.

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Fig. 8. Ža. Amplitude spectra at x sy11.15 m, x sy1.25 m Ž ; outer breakpoint. and run-up, case b1025rA. x sy11.15 m, - - - - x sy1.25 m, IIII run-up. Žb. Amplitude spectra at x sy11.15 m, x sy1.25 m Ž ; outer breakpoint. and run-up, case b6020rC. x sy11.15 m, - - - - x sy1.25 m, IIII run-up.

5.2.1. Maximum response Fig. 9a shows the cross-shore variation in short wave amplitude for case b1025A Žmaximum outgoing free wave amplitude for series A., where the most shoreward data points are those obtained from the run-up wire. Significant energy transfer is seen to occur from f 1 to the group super-harmonic Ž f 1 q f 2 . Žand presumably fG . prior to breaking. The cross-shore variation in the total amplitude at the group frequency is shown in Fig. 9b, together with the theoretical amplitude of the IBLW and the measured outgoing free long wave, assumed to be the total BFLW Ži.e., the sum of the OBFLW and the RBFLW.. At each cross-shore location the amplitude of the IBLW was calculated following Longuet-Higgins and Stewart Ž1960. using the short wave amplitudes shown in Fig. 9a. Sharp gradients in the measured data occur around the breakpoint, with an anti-node outside the breakpoint and a Žnear total. node inside, in agreement with the discussion in Section 3. A weaker nodal structure exists further offshore. Fig. 9c shows the same data, but together with an analytical solution for a free

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Fig. 9. Ža. Cross-shore variation in short wave amplitude, case b1025rA, fG s 0.244 Hz. — —e— — f 1 , - -I- - f 2 , — —P^P— — 2 f 1 , — —P=P— — 2 f 2 , — —=— — f 1 q f 2 . Žb. Cross-shore variation in wave amplitude at the group frequency, case b1025rA. - -I- - Data Ž fG ., IBLW, — — — — IBLW Žscaled., — —P— — OFLW Žinverse shoaling.. Žc. Cross-shore variation in wave amplitude at the group frequency compared to nodal structure of free standing wave and IBLWqOBFLW, case b1025rA, x bo sy0.9 m, x bi sy0.15 m. IIII Data Ž fG ., J0 , - - - - IBLWqOBFLW.

standing wave and a numerical solution for the sum of the IBLW and total BFLW. The standing wave solution is simply given by a zeroth order Bessel function Ždenoted J0 .,

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with a shoreline amplitude equal to that measured, origin at the mean cross-shore run-up position and valid for x ) y8 m. The numerical solution for the sum of the IBLW and BFLW was obtained by numerically integrating the phase functions for the IBLW and the outgoing free long wave along the length of the flume, each with the amplitudes given on Fig. 9b. At each location the two phase functions were then added, and the maximum amplitude of the combined motion found by time-stepping over the group period. The remaining variable is the phase relationship at the outer breakpoint between the IBLW and the BFLW and, in order to further test the breakpoint generation mechanism outlined in Section 3, this has been estimated from Fig. 1. Finally, it is well known that the solution of Longuet-Higgins and Stewart Ž1960; 1962. over-estimates the amplitude of the IBLW in shallow water, depending on the characteristics of the primary waves. We have therefore scaled the IBLW at the outer breakpoint, so that Žaccounting for phase. the sum of the IBLW and the measured BFLW match the measured amplitude at that point. This typically requires a scaling factor of 0.6–0.3, consistent with the numerical results presented by Lo and Dean Ž1995., who showed that Longuet-Higgins and Stewart’s results overestimated the IBLW amplitude for hrLo - 0.1 Žwhere L o is the deep water wave length for the short waves.. This amplitude is then matched to that at x s y4 m Ž hrLo ) 0.1., calculated as above from Longuet-Higgins and Stewart Ž1960., using a linear splice. This scaled solution is indicated by a dashed line on Fig. 9b Žand equivalent figures below.. The final numerical solution therefore uses the scaled IBLW and for this case Žb1025rA. assumes that the IBLW and BFLW are in phase at the outer breakpoint Žsee Fig. 1.. Shorewards of the mean breakpoint Fig. 9c shows an apparent standing wave, although the large run-up amplitude could be a combination of the IBLW, IBFLW, RBFLW, swash–swash interactions ŽWatson et al., 1994. and direct forcing by short wave grouping remaining at the shoreline ŽBaldock et al., 1997.. Seawards of the breakpoint the free standing wave solution shows no correlation with the data. In contrast, the numerical solution shows fairly good agreement with the data, and although the predicted amplitude of the nodal structure is not perfect, the position of the nodes and anti-nodes match very well. Equivalent results for case b6020rC are shown in Fig. 10a–c Žmaximum outgoing free wave amplitude for series C.. The nodal structure is again similar and the over-estimation of the IBLW amplitude by Longuet-Higgins and Stewart’s solution is greater due to the smaller value of hrLo Žsee Lo and Dean, 1995.. The numerical solution, with the IBLW and BFLW again assumed to be in phase at the outer breakpoint Žsee Fig. 1., is in excellent agreement with the data. Furthermore, for both cases b1025rA and b6020rC, the mean breakpoint is at x f y0.55 m, which corresponds very well with the nodal point for a free standing wave at the group frequency ŽFig. 9c and Fig. 10c.. 5.2.2. Minimal response The data for case b1060rA Žminimal outgoing free wave energy, series A. show virtually no cross-shore nodal structure at the group frequency ŽFig. 11a–c.. This is because the outgoing BFLW is very small and the total wave amplitude at fG is

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Fig. 10. Ža. Cross-shore variation in short wave amplitude, case b6020rC, fG s 0.195 Hz. — —e— — f 1 , - -I- - f 2 , — —P^P— — 2 f 1 , — —P=P— — 2 f 2 , — —=— — f 1 q f 2 . Žb. Cross-shore variation in wave amplitude at the group frequency, case b6020rC. - -I- - Data Ž fG ., IBLW, — — — — IBLW Žscaled., — —P— — OFLW Žinverse shoaling.. Žc. Cross-shore variation in wave amplitude at the group frequency compared to nodal structure of free standing wave and IBLWqOBFLW, case b6020rC, x bo s y0.95 m, x bi sy0.2 m. IIII Data Ž fG ., J0 , - - - - IBLWqOBFLW.

dominated by the IBLW. In this case, the solution of Longuet-Higgins and Stewart Ž1960. provides good estimates of the IBLW amplitude due to the broad banded nature of the wave group Ž f 1 4 f 2 . ŽFig. 11b.. The mean breakpoint for this case is x f y0.6

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Fig. 11. Ža. Cross-shore variation in short wave amplitude, case b1060A, fG s 0.586 Hz. — —e— — f 1 , - -I- - f 2 , — —^P— — 2 f 1 , — —P=P— — 2 f 2 , — —=— — f 1 q f 2 . Žb. Cross-shore variation in wave amplitude at the group frequency, case b1060rA. - -I- - Data Ž fG ., IBLW, — —P— — OFLW Žinverse shoaling.. Žc. Cross-shore variation in wave amplitude at the group frequency compared to nodal structure of free standing wave and IBLWqOBFLW, case b1060rA, x bo sy0.9 m, x bi sy0.3 m. IIII Data Ž fG ., J0 , - - - - IBLWqOBFLW.

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m, which corresponds well with an antinode for a free standing wave at the group frequency ŽFig. 11c. and therefore minimal response is to be expected. Furthermore, using the outer and inner breakpoint positions given in Table 2, Fig. 11c also shows that the breakpoint forcing region is large in comparison with the nodal structure for a free standing wave. This might be expected to reduce the effectiveness of the breakpoint mechanism, further limiting outgoing wave energy. Indeed, if the breakpoint excursion becomes large compared to some factor of the wavelength at the group frequency, then the breakpoint forcing will become ‘‘smeared’’ over the nodal structure for a free standing long wave, and the constructiverdestructive interference mechanism outlined in Section 3 will no longer occur. This can be quantified to some extent by examining the relative magnitudes of the breakpoint excursion and the long wave wavelength. Hence, if D x is the breakpoint excursion, LG is the wavelength of the free long wave and a is a factor to be estimated below, then the breakpoint mechanism requires: D x - a LG

Ž 2.

D x may be approximated by d Hrb , where H is the short wave height seaward of the breakpoint, and LG may be estimated as 6Ž gH .rfG , assuming that the depth at the breakpoint is approximately equal to the short wave height. Eq. Ž2. may then be written as:

d'H fG b'g

-a .

Ž 3.

Since the distance between the nodal points of a free standing long wave is LG r2, and for the breakpoint mechanism to be valid the forcing should be concentrated around an antinode or node, then we would estimate that a should lie in the range 0.2–0.3. Applying Eq. Ž3. to series A and C therefore implies that the interference mechanism will no longer occur for fG greater than about 0.4 Hz, while the corresponding values for series B and D are about 0.5 and 0.3 Hz, respectively. These estimates are consistent with Fig. 4c, where no secondary maxima are observed. However, Eq. Ž3. suggests that series E should show a secondary maximum. A reason for this discrepancy may be that the short wave envelope becomes less sinusoidal at high group frequencies, and therefore the shoreward and seaward breakpoint forcing will differ in magnitude, thereby changing the relative magnitude of the IBFLW and OBFLW. This appears consistent with results presented by List Ž1992. and further data discussed below. Dissipation and partial shoreline reflection may also become significant at the higher group frequencies, particularly since fG approaches the mean primary wave frequency. This is not the case for the equivalent data in Madsen et al. Ž1997., where their group frequencies were much smaller than the mean primary wave frequency. It should be noted that Eq. Ž3. is in some ways equivalent to the assumption of a sinusoidal breakpoint Žand hence small breakpoint excursion. by Symonds et al. Ž1982. and Schaffer Ž1993., although it is perhaps more physically transparent. Eq. Ž3. is readily satisfied under typical field conditions Ž b f 1r30, H f 2 m, d f 0.5 and fG - 0.03 Hz., but may not be for large amplitude waves andror very mild slopes.

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Fig. 12a–c show the data for case b6055rC, again minimal response for series C. For this case, in deep water, the outgoing BFLW is similar in amplitude to the IBLW and a clear nodal pattern exists offshore. However, further inshore, the rapid shoaling of the IBLW, compared to the inverse shoaling of BFLW, tends to reduce the nodal structure ŽFig. 12b.. The numerical solution, with the outgoing BFLW out of phase with the IBLW at the outer breakpoint, again shows very good agreement with the data, with the short length scale of the nodal structure well represented ŽFig. 12c.. The mean breakpoint again coincides with an anti-node for a free standing wave. Referring back to Fig. 1, this phase relationship between the IBLW and the BFLW requires the RBFLW to be dominant over the OBFLW. This may seem unlikely, given that the RBFLW has propagated Žtwice. through the surf zone and reflected at the shoreline. The RBFLW might therefore be expected to be smaller than the OBFLW, due to energy dissipation and partial reflection. However, the numerical model of List Ž1992. for the breakpoint forcing, which is not restricted to a sinusoidal breakpoint variation, shows that the shoreward forcing is greater than the seaward forcing for high groupiness Žlarge d . and a non-sinusoidal short wave envelope. Indeed, the calculations of List Ž1992. for a case with d f 0.5 suggest that the shoreward forcing is about three times greater than the seaward forcing, and this difference will probably increase as d increases. It therefore seems entirely possible that for this case the RBFLW could be larger than the OBFLW, even after accounting for energy dissipation and shoreline reflection, consistent with Fig. 12c. Data for two final cases Žb1010rA and b6010rC. are presented on Fig. 13a and b. These correspond to the lowest group frequency considered in this study and the mean breakpoint is shoreward of the first nodal point for a free standing long wave at this frequency Ž; 0.1 Hz.. Indeed, in comparison to the scale of a free standing wave, the mean breakpoint approaches the shoreline, which is an anti-node. Therefore, from Fig. 1b, an anti-node and node should be expected at the shoreline and seaward of the breakpoint, respectively. However, since the breakpoint is effectively at the shoreline, no standing wave will exist and only a weak outgoing free long wave will occur. This is qualitatively in agreement with the measured data which, although showing a large shoreline amplitude, do not fit the free standing wave solution. The weak nodal structure offshore Ž x - y4 m. is also consistent with the interaction between a weak outgoing free long wave and the IBLW. For these two cases, the long wave structure close to the shoreline Ž x ) y2 m. is similar to that observed by Baldock et al. Ž1997. for bichromatic wave groups and random waves with similar amplitudes, group frequencies and beach slope. Baldock et al. Ž1997. also showed that the low frequency shoreline motion was consistent with direct forcing by short wave grouping remaining in the inner surf zone. Furthermore, Baldock and Holmes Ž1999. showed that a simple model based on short wave bores collapsing at the shoreline could accurately simulate both the low frequency motion and spectral characteristics of the shoreline run-up Žswash.. Therefore, in the limit of the breakpoint approaching the shoreline Ž x ™ 0., the time-varying breakpoint model simply results in direct short wave forcing of the swash due to the dynamic set-up, or run-up, induced by short waves ŽWatson and Peregrine, 1992; Baldock et al., 1997.. In addition, the present data show that as x ™ 0, outgoing wave energy becomes minimal,

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Fig. 12. Ža. Cross-shore variation in short wave amplitude, case b6055rC, fG s 0.537 Hz. — —e— — f 1 , - -I- - f 2 , — —P^P— — 2 f 1 , — —P=P— — 2 f 2 , — —=— — f 1 q f 2 . Žb. Cross-shore variation in wave amplitude at the group frequency, case b6055rC. - -I- - Data Ž fG ., IBLW, — — — — IBLW Žscaled., — —P— — OFLW Žinverse shoaling.. Žc. Cross-shore variation in wave amplitude at the group frequency compared to nodal structure of free standing wave and IBLWqOBFLW, case b6055rC, x bo s y1.05 m, x bi sy0.05 m. IIII Data Ž fG ., J0 , - - - - IBLWqOBFLW.

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in agreement with Symonds et al. Ž1982.. This suggests that, at least for these cases, swash–swash interactions ŽWatson et al., 1994. do not generate significant outgoing wave energy. The data presented above largely appear to verify the time-varying breakpoint model originally developed by Symonds et al. Ž1982. and elaborated on by Schaffer Ž1993.. In addition, although a rigorous examination of the IBLW is not of prime concern in the present paper, the data do suggest that the incident bound energy wave is predominantly dissipated, or transferred to other frequencies, shorewards of the breakpoint. For example, it is clear from Figs. 9–13 that the amplitude of the IBLW at the outer breakpoint is typically much greater than the amplitude of the outgoing long wave. Furthermore, the nodal structure seaward of the breakpoint would be very different from that measured if the IBLW was released and reflected with minimal dissipation Žsee Fig. 1.. However, further work is in progress on this point, in particular experimental studies where the frequency of the breakpoint oscillations and the bound wave frequency differ.

Fig. 13. Ža. Cross-shore variation in wave amplitude at the group frequency, case b1010A, fG s 0.098 Hz, x bo sy1.1 m, x bi sy0.15 m. IIII Data Ž fG ., J0 , - - - - IBLW, — —P— — OFLW Žinverse shoaling.. Žb. Cross-shore variation in wave amplitude at the group frequency, case b6010C, fG s 0.098 Hz, x bo sy0.9 m, x bi sy0.15 m. IIII Data Ž fG ., J0 , - - - - IBLW, — —P— — OFLW Žinverse shoaling..

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Further work will also examine if the time-varying breakpoint mechanism is important in random waves, which will be relevant to more general field conditions.

6. Conclusions New laboratory data on surf beat generation induced by bichromatic wave groups have been presented. Some 65 different wave groups have been investigated, covering a broad range of wave amplitudes, short wave frequencies, group frequencies and modulation rates. Measurements include incident and outgoing wave amplitudes, shoreline run-up Žswash., inner and outer breakpoint positions and the cross-shore structure of both the short and long wave motion. Outgoing free long wave amplitudes show a strong dependence on wave frequency, with maximum response occurring at fG rb f 2, similar to the value found by Madsen et al. Ž1997.. The outgoing long wave amplitude, normalised according to the time-varying breakpoint model proposed by Symonds et al. Ž1982., shows good quantitative and qualitative agreement with that theory. In particular, maximum surf beat generation occurred at x f 1.1, with minimal surf beat generation for x ) 4. The surf beat shows a linear dependence on incident short wave amplitude, in agreement with most field data, and is also dependent on the strength of the short wave modulation. For a weakly modulated wave group, the Symonds et al. Ž1982. model provides excellent estimates of the surf beat amplitude. The cross-shore structure of the long wave motion is again in good agreement with the theoretical model of Symonds et al. Ž1982. if the IBLW seaward of the breakpoint is accounted for ŽSchaffer, 1993.. For example, at maximum response, the mean breakpoint coincides with the nodal point for a free long wave standing at the shoreline. Seaward of the surf zone, the nodal structure of the long wave motion is also in good agreement with a numerical solution describing the interaction of the measured IBLW and the measured total BFLW. These calculations again suggest that the phase relationship at the outer breakpoint between the IBLW and the BFLW is consistent with theory. Finally, as the breakpoint tends to the shoreline, the swash motion becomes directly forced by dynamic set-up, or run-up, due to short waves ŽBaldock et al., 1997., which is the limit of the breakpoint model of Symonds et al. Ž1982. as x ™ 0.

Acknowledgements The authors gratefully acknowledge funding from the European Commission through the MAST III programme, SASME project, Contract no. MAS3-CT97-0081, with additional support from the COAST3D project, Contract no: MAS3-CT97-0086. TB would like to thank the Technical Staff within the SCSE, particularly Steve Edmonds and Ian Morgan. Ian Morgan also produced Fig. 2. The review comments of Prof. R.G. Dean and Prof. D.H. Peregrine are much appreciated.

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