PUBLICATIONS Journal of Geophysical Research: Oceans RESEARCH ARTICLE 10.1002/2017JC013138 Key Points:  The phase lag between breakpoint-forced long wave and wave group depends on beach geometry and wave group parameters  Incoming breakpoint-forced long waves and incident wave groups are not in phase

Correspondence to: S. Contardo, [email protected]

Citation: Contardo, S., Symonds, G., & Dufois, F. (2018). Breakpoint forcing revisited: Phase between forcing and response. Journal of Geophysical Research: Oceans, 123. https://doi.org/10.1002/ 2017JC013138 Received 26 MAY 2017 Accepted 26 JAN 2018 Accepted article online 5 FEB 2018

Breakpoint Forcing Revisited: Phase Between Forcing and Response S. Contardo1

, G. Symonds2

, and F. Dufois3

1

CSIRO Oceans and Atmosphere Flagship, Crawley, WA, Australia, 2UWA Oceans Institute, University of Western Australia, Crawley, WA, Australia, 3ARC Centre of Excellence for Coral Reef Studies, The UWA Oceans Institute, University of Western Australia, Crawley, WA, Australia

Abstract Using the breakpoint forcing model, for long wave generation in the surf zone, expressions for the phase difference between the breakpoint-forced long waves and the incident short wave groups are obtained. Contrary to assumptions made in previous studies, the breakpoint-forced long waves and incident wave groups are not in phase and outgoing breakpoint-forced long waves and incident wave groups are not p out of phase. The phase between the breakpoint-forced long wave and the incident wave group is shown to depend on beach geometry and wave group parameters. The breakpoint-forced incoming long wave lags behind the wave group, by a phase smaller than p/2. The phase lag decreases as the beach slope decreases and the group frequency increases, approaching approximately p/16 within reasonable limits of the parameter space. The phase between the breakpoint-forced outgoing long wave and the wave group is between p/2 and p and it increases as the beach slope decreases and the group frequency increases, approaching 15p/16 within reasonable limits of the parameter space. The phase between the standing long wave (composed of the incoming long wave and its reflection) and the incident wave group tends to zero when the wave group is long compared to the surf zone width. These results clarify the phase relationships in the breakpoint forcing model and provide a new base for the identification of breakpoint forcing signal from observations, laboratory experiments and numerical modeling.

1. Introduction After Munk (1949) and Tucker (1950) identified ‘‘surf beat’’ as seaward propagating long waves correlated to and lagging the incident short wave group, two mechanisms, bound wave release and breakpoint forcing, were proposed to explain the correlation between short waves (wind-sea and swell) and outgoing long waves, near the surf zone. Longuet-Higgins and Stewart (1962) described long waves, bound to the wave group and propagating p out of phase with the group. They hypothesized that the incoming bound long wave reflects from nonuniformities in the transmitting medium before the shoreline, resulting in the observed outgoing long wave. Some authors suggested the release could happen in shallow water if the bound wave frequency and wavenumber satisfy the free wave dispersion relationship (Baldock, 2012; Baldock & Huntley, 2002; Contardo & Symonds, 2013). Symonds et al. (1982) proposed the time-varying breakpoint forcing as a mechanism generating long waves as a result of the time varying setup associated with incident wave groups. This mechanism generates long waves in the outer surf zone which propagate shoreward and seaward. The waves propagating shoreward are reflected at the shoreline producing a standing wave in the surf zone. Since the breakpoint forcing model was developed by Symonds et al. (1982), several studies have attempted to identify the mechanism preferentially forcing long waves (also called infragravity waves) in numerical models (List, 1992), laboratory experiments (Baldock & Huntley, 2002; Baldock et al., 2000, 2004), and field observations (Contardo & Symonds, 2013; Inch et al., 2017; Masselink, 1995; Moura & Baldock, 2017; Pomeroy et al., 2012). These analyses (except for Baldock et al., 2000) were based on the phase differences between the wave groups and the long waves. C 2018. American Geophysical Union. V

All Rights Reserved.

Bound waves and wave groups are p out of phase in intermediate water and shift toward p/2 in shallow water (Bowers, 1992; List, 1992). Due to this p/2 shift, a positive correlation is expected to precede the

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negative correlation, in the lagged cross-correlations between the wave envelope (representing the group) and the incoming infragravity wave signal. List (1992) proposed there should be a positive correlation between waves groups offshore and breakpoint forced long waves at the shoreline (including a phase correction associated with the propagation time across the inner surf zone). List (1992) also suggested that if bound waves were present in the surf zone, the phase lag between the incoming long wave in the surf zone and the short wave envelope was hardly affected by the presence of breakpoint-forced long waves. Several authors (i.e., Baldock et al., 2000; Contardo & Symonds, 2013; Masselink, 1995; Pomeroy et al., 2012) have assumed incoming breakpoint-forced long waves and wave groups to be in phase and outgoing breakpoint forced long wave and wave groups to be p out of phase (accounting for propagation time). However this is not stated in Symonds et al. (1982) as the phase relationships are not examined. In this paper, we explore the phase relationship between the incoming wave group and long wave response using the breakpoint forcing model (Symonds et al., 1982). We demonstrate that the phase between the long waves and the short wave envelope depends on the group frequency, the incident wave height and the beach slope.

2. Background The time-varying breakpoint model (Symonds et al., 1982, hereafter: SHB82) parametrizes the radiation stress, in two dimensions (one spatial, cross-shore, and time), in a surf zone where the breakpoint position varies with time, due to normally incident bichromatic wave groups on a plane beach. Standing wave solutions are found in the surf zone and, seaward of the breakpoint, the solutions are in the form of outgoing waves. Solutions are found at the group frequency and its higher harmonics. SHB82 express the solutions in term of the nondimensional parameter v: v5

r2 X g tan b

(1)

where r is the group radial frequency, X is the breakpoint mean position, tan b is the beach slope, and g is the gravitational acceleration. The elevation amplitude of the forced long waves depends on v and on the nondimensional amplitude difference, Da, between the smallest and the largest waves in the group. The model assumes a small breakpoint excursion compared to the incoming wave group wavelength. Following SHB82, the nondimensional limits of the breakpoint excursion are x1 and x2 which are equal to 1 – Da and 1 1 Da, respectively. For high values of Da, the model is valid only for small values of v (equation (14) SHB82). Thus, we use a small value of Da (0.1) for all calculations, as Da does not affect the phase relationships. Using the shallow water approximation, SHB82 express the forcing function as a Fourier series (equation (8) in SHB82), so at the group frequency it becomes: 1 @ ða 2 Þ 52a1 ðx Þcos t 2x @x

(2)

where a is the incident wave amplitude and a1 ðx Þ5

sin s p

(3)

and s5cos

21

  x21 Da

(4)

The solutions to the elevation equation proposed in SHB82, for the fundamental mode (n 5 1), are as follows. In the surf zone:

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Figure 1. Constants, A1, A2, A3, and A4 for different values of Da and v.

pffiffiffiffiffi pffiffiffiffiffi fSLW 5A1 J0 ð2 vx Þsin t1A2 J0 ð2 vx Þcos t;

(5)

  pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi fBPE 5A1 J0 ð2 vx Þsin t1 A3 J0 ð2 vx Þ1A4 Y0 ð2 vx Þ1gp cos t

(6)

within the breakpoint excursion:

and seaward of the surf zone: pffiffiffiffiffi pffiffiffiffiffi fOLW 5A1 ½J0 ð2 vx Þsin t2Y0 ð2 vx Þcos t ;

(7)

where x represents the offshore distance nondimensionalized by the mean breakpoint position, and t is time nondimensionalized by the radial group frequency. J0 and Y0 are zero-order Bessel functions, of the first and second kind, respectively, and gp is a particular solution [SHB82]. A1, A2, A3, A4 are constants (expressions given in Appendix A) and vary with the nondimensional parameters Da and v as shown in Figure 1. Note in equation (7), the sign of the second term is different from equation (22) in SHB82. The minus sign is needed to obtain a seaward propagating wave. As noted in SHB82, for v  1.2 (v 5 vmax), the elevation amplitude offshore of the breakpoint is maximum (Figure 8 in SHB82). For v  3.7 (v 5 v0), the elevation amplitude seaward of the breakpoint is zero.

3. Method 3.1. Total Solution We express the theoretical formulation for the phase lags in the surf zone and the offshore zone, relative to the forcing, which is maximum at t 5 0 (see equation (2)). When the biggest (smallest) wave in the group breaks, the forcing function is maximum (minimum), i.e., the forcing is in phase with the wave group envelope. Rearranging equation (5) (Appendix B), we write the solution for the elevation of the SLW in the surf zone as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi fSLW 5 A1 2 1A2 2 jJ0 ð2 vx Þjcos ðt1USLW Þ; (8) and the phase lag between the standing long wave (SLW) in the surf zone and the forcing is given by:

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Journal of Geophysical Research: Oceans   8 > 21 A1 > 2tan > < A2 USLW 5   > A1 > > : 2tan21 1p A2

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 pffiffiffiffiffi if A2 J0 2 vx  0  pffiffiffiffiffi if A2 J0 2 vx < 0

(9)

Similarly the phase between the OLW in the offshore zone and the forcing term can be retrieved from equation (7): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi fOLW 5jA1 j J0 2 ð2 vx Þ1Y0 2 ð2 vx Þcos ðt1UOLW Þ; (10) and the phase lag between the outgoing long wave (OLW) and the forcing in the offshore zone is given by: 8  pffiffiffiffiffi !  pffiffiffiffiffi > J0 2 vx > 21 > tan  pffiffiffiffiffi 1p if A1 Y0 2 vx  0 > > < Y0 2 vx (11) UOLW 5  pffiffiffiffiffi ! > >   J 2 vx p ffiffiffiffiffi > 0 21 > tan  pffiffiffiffiffi > if A1 Y0 2 vx < 0 : Y0 2 vx 3.2. Decomposition of the Total Solution These phase relationships are for the total long waves resulting from breakpoint forcing. In the surf zone, the long wave generated is a standing wave (SLW), i.e., the linear superposition of the incoming breakpointforced long wave (IBFLW) and its reflection at the shoreline (RLW). In the offshore zone, the total outgoing long wave (OLW) is composed of an outgoing breakpoint-forced long wave (OBFLW), generated at the breakpoint, and the RLW. The breakpoint forcing signal is usually identified, from observations and numerical modeling, from the phase relationship between the IBFLW and the wave group offshore, and the phase relationship between the OBFLW and the wave group offshore (Contardo & Symonds, 2013; Masselink, 1995; Pomeroy et al., 2012). Here we decompose the total long waves (SLW and OLW) into their incoming and outgoing components (IBFLW, OBFLW, and RLW). This way, we calculate the phase between the IBFLW and the wave group, and the phase between the OBFLW and the wave group. We decompose the SLW in the surf zone into its incoming and outgoing (reflected) components. A standing wave is equivalent to two equal waves propagating in opposite directions (i.e. of opposite phase), fSLW 5fIBFLW 1fRLW

(12)

pffiffiffiffiffi pffiffiffiffiffi fSLW 5Asz ½J0 ð2 vx Þ cos ðt1USLW Þ2Y0 ð2 vx Þsin ðt1USLW Þ pffiffiffiffiffi pffiffiffiffiffi 1 J0 ð2 vx Þcos ðt1USLW Þ1Y0 ð2 vx Þsin ðt1USLW Þ

(13)

so equation (5) can be written as:

with Asz 5

pffiffiffiffiffiffiffiffiffiffiffiffiffi A1 2 1A2 2 . 2

The expression for the elevation amplitude of the IBFLW is given by the first two terms in equation (13): pffiffiffiffiffi pffiffiffiffiffi fIBFLW 5Asz ½J0 ð2 vx Þcos ðt1USLW Þ2Y0 ð2 vx Þsin ðt1USLW Þ (14) which can be written, following Appendix B, as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi fIBFLW 5Asz J0 2 ð2 vx Þ1Y0 2 ð2 vx Þcos ðt1UIBFLW Þ with the phase between the IBFLW and the forcing term given by: 8  pffiffiffiffiffi!  pffiffiffiffiffi > Y0 2 vx > 21 >  U 1 tan if J0 2 vx  0 pffiffiffiffiffi > SLW > < J0 2 vx UIBFLW 5  pffiffiffiffiffi! > >  pffiffiffiffiffi Y0 2 vx > 21 >  > USLW 1 tan pffiffiffiffiffi 1p if J0 2 vx < 0: : J0 2 vx

(15)

(16)

Similarly, the expression for the elevation amplitude of the RLW is given by the last two terms in equation (13):

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pffiffiffiffiffi pffiffiffiffiffi fRLW 5Asz ½J0 ð2 vx Þ cos ðt1USLW Þ1Y0 ð2 vx Þsin ðt1USLW Þ

(17)

and rearranged as: fRLW 5Asz

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi J0 2 ð2 vx Þ1Y0 2 ð2 vx Þcos ðt1URLW Þ

with the phase between the RLW and the forcing given by: 8  pffiffiffiffiffi!  pffiffiffiffiffi > Y0 2 vx > 21 >  USLW 2tan pffiffiffiffiffi if J0 2 vx  0 > > < J0 2 vx URLW 5  pffiffiffiffiffi! > >  pffiffiffiffiffi Y0 2 vx > 21 >  pffiffiffiffiffi 1p if J0 2 vx < 0: > USLW 2tan : J0 2 vx

(18)

(19)

Note the expressions for the elevation amplitude of the IBFLW and OBFLW are constructed with Bessel functions. With Bessel functions, although the reflection occurs at the shoreline (where the horizontal current velocity of the standing wave is zero), the incoming and reflected waves are not in phase at  pffiffiffiffiffi pffiffiffiffiffi the shoreline but at a cross-shore position such that Y0 2 vx 50, that is when 2 vx 50:87 i.e., x 5 0.19/v. We decompose the OLW, in the offshore zone, into its two components: the OBFLW and the RLW. Equation (7) may be written as the sum of two outgoing progressive waves as follows: pffiffiffiffiffi pffiffiffiffiffi fOLW 5Aoz ½J0 ð2 vx Þ sin ðt1UO =2Þ2Y0 ð2 vx Þcos ðt1UO =2Þ pffiffiffiffiffi pffiffiffiffiffi (20) 1 J0 ð2 vx Þsin ðt2UO =2Þ2Y0 ð2 vx Þcos ðt2UO =2Þ where UO is the phase between the two outgoing waves. Assuming the first two terms in equation (20) represent the reflected long wave, and equating with the last two terms in equation (18), it can be shown that:

UO 52USLW 1p

(21)

Aoz 5Asz :

(22)

and

The elevation amplitude of the OBFLW is given by the last two terms in equation (20) and, using equation (21) and equation (22), is given by: pffiffiffiffiffi pffiffiffiffiffi fOBFLW 5Asz ½2J0 ð2 vx Þ cos ðt2USLW Þ2Y0 ð2 vx Þsin ðt2USLW Þ (23) Equation (23) is rearranged (Appendix B) as follows:

Figure 2. Elevation, versus cross-shore position, at t 5 0, of the total breakpoint-forced long wave and its incoming and outgoing components. Gray dashed lines represent the limits of the breakpoint excursion. v 5 2, Da 5 0.1.

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Figure 3. Phase lags (a) between the total breakpoint-forced long waves and the forcing, (b) between the IBFLW and the forcing in the surf zone and phase lag between OBFLW and the forcing in the offshore zone, (c) between the RLW and the forcing, versus distance from shoreline for different values of v. Dashed vertical gray lines represent x1 and x2. Da 5 0.1.

fOBFLW 5Asz

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi J0 2 ð2 vx Þ1Y0 2 ð2 vx Þcos ðt1UOBFLW Þ

(24)

with the phase between the OBFLW and the forcing given by:

UOBFLW 5

8 > > > 2USLW 2tan21 > > < > > > 21 > > : 2USLW 2tan

 pffiffiffiffiffi! Y0 2 vx  pffiffiffiffiffi 1p J0 2 vx  pffiffiffiffiffi! Y0 2 vx  pffiffiffiffiffi J0 2 vx

 pffiffiffiffiffi if J0 2 vx  0 (25)  pffiffiffiffiffi if J0 2 vx < 0:

The phase lags account for the traveling time between the breakpoint and the cross-shore positions.

Figure 4. Phase lag between the total breakpoint-forced long wave, at three cross-shore positions in (a) the surf zone (USLW) and (b) offshore (UOLW), and the forcing at xb. Gray dashed lines point at nodes in elevation. Da 5 0.1.

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4. Results From the Model Figure 2 shows an example of the total elevation (equation (5) to equation (7)) and its components, IBFLW (equation (16)), OBFLW (equation (24)), and RLW (equation (18)), at t 5 0 and v 5 2. The total solution (black line) is the sum of the IBFLW (blue line) and the RLW (red line) in the surf zone (x < x1) and the sum of the OBFLW (green line) and the RLW (red line) in the offshore zone (x > x2). The elevations of the IBFLW and the OBFLW tend to 61 at the shoreline, as the expressions for the incoming and outgoing components include zero-order Bessel functions of second kind, and they are in phase at x 5 0.095 (x 5 0.19/v). The phase between the SLW and the forcing (USLW), given by equation (9), and the phase between the OLW and the forcing (UOLW), given by equation (11) are shown in Figure 3a. USLW is independent of the cross-shore position (x < x1), except for phase shifts by p at node locations, consistent with a standing wave. For x > x2, UOLW increases with Figure 5. Difference between UOBFLW and URLW versus v. Vertical gray dashed x, consistent with an outgoing progressive wave (see equation (10)). lines represent v 5 vmax and v 5 v0. For x < x1, the standing wave can be decomposed into the incoming and reflected components with corresponding phases UIBFLW and URLW, respectively, as shown in Figures 3b and 3c. For x > x2, the OLW is composed of two outgoing waves with phases UOBFLW (equation (25)) and URLW (equation (19)), also shown in Figures 3b and 3c. The addition of the component phases shown in Figures 3b and 3c returns the phases for the total solution shown in Figure 3a, i.e., for x < x1, USLW 5 UIBFLW 1 URLW and for x > x2, UOLW 5 UOBFLW 1 URLW. It is also important to note the phases shown in Figure 3 all depend on v. Figures 4a and 4b show the phases USLW and UOLW, respectively, as functions of v. Figure 4a shows that USLW increases with v independently of the cross-shore position except for shifts by p when nodes are crossed, e.g., at v 5 2.892 and v 5 1.606, for x 5 0.5 and x 5 x1, respectively, (red and green lines in Figure 4a). The positions of the standing wave nodes in the surf zone are found from equation (5), by equating the elevation amplitude, fn, to  pffiffiffiffiffi  pffiffiffiffiffi zero (i.e., equating J0 2 vx to zero) and solving for the roots of v. For instance, J0 2 vx 50 at v 5 2.892 and 1.606 (indicated with dashed gray lines in Figure 4a) for x 5 0.5 and x 5 x1, respectively, (blue and red lines, respectively, in Figure 4). Note the SLW is not in phase with the forcing, unless A1 is zero (Figure 3a, x < x1 and Figure 4a). UOLW also increases with v but varies with x as shown in Figure 3. We note that the OBFLW and the RLW are in phase for v 5 vmax and p out of phase for v 5 v0 as shown in Figure 5 which represents the difference between UOBFLW and URLW versus v. This is in agreement with SHB82 who noted that the elevation of the OLW was maximum for v 5 vmax and zero for v 5 v0. In Figure 3 the phases include the propagation time from the forcing region to any given x. Since we are interested in the phase between the breakpoint-forced long wave response and the forcing we consider the phase of the IBFLW and OBFLW at x 5 X as functions of v. Figure 6 shows UIBFLW(x 5 X) is less than p/2 (for v > 0.001) and decreases as v increases. According to equation (16), UIBFLW > 0 means the IBFLW lags behind the forcing. Figure 6 also shows UOBFLW (x 5 X) is greater than p/2 (for v > 0.001) and increases as v increases.

5. Discussion and Conclusions

Figure 6. UIBFLW (blue line) and UOBFLW (orange line) and their difference (yellow line). x 5 X.

The aim of this study was to examine the phase relationships between breakpoint-forced long waves and the incident wave groups using the breakpoint forcing model of SHB82. Previous studies (Baldock et al., 2000; Contardo & Symonds, 2013; Masselink, 1995; Pomeroy et al., 2012) assumed the incoming breakpoint forced long waves

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were in phase with the incident wave groups while the outgoing breakpoint forced long waves were p out of phase with the wave groups. However, we have shown the long wave response to breakpoint forcing is dependent on the non-dimensional parameter v. Using typical values for surf zone width X 5 100 m, group frequency r 5 0.06, and beach slope tanb 5 0.02 gives v  2 and, according to Figure 6, UIBFLW  p/8 and decreases as v increases. Similarly, at v  2 Figure 6 shows UOBFLW  7p/8 and decreases as v increases. In both cases much larger values of v require unrealistic values of the parameters given above and, in order to satisfy the criterion expressed by equation (14) in SHB82, would require vanishingly small values of Da. We do note that the phase of the standing long wave in the surf zone, USLW ! 0 as v ! 0 as shown in Figure 4a. This case is consistent with Figure 6 in SHB82 where, as v ! 0, the shoreline amplitude at the group frequency approaches the difference in steady state setup for the smallest and largest waves in the group. Therefore USLW -> 0 occurs in the presence of very steep beaches, or very low group frequencies. For larger values of v, -p < USLW < p. Contardo and Symonds (2013) identified the signal from breakpoint forcing on a barred beach from the cross-correlation between the short wave envelope and the outgoing component of the long wave, and find a six second time lag between the two, at a location just offshore of the bar. Assuming the short wave group and the OBFLW are p out of phase, they conclude that the breakpoint is approximately 8 m shoreward of that location. Now considering, the phase might be a bit lower than p, the breakpoint would be approximately 2 m seaward of where it was estimated, i.e., 6 m shoreward of the measurement location. However it cannot be precisely estimated since the short wave group is not bichromatic nor regular, and the bathymetry is complex. It is important to note as well that, since the phase is dependent of v (and on the group frequency), the cross-spectra method (used in Contardo & Symonds, 2013) should be preferred to the cross-correlation method in which the frequency information is lost. In deriving the phase relationships between the long wave response and the incident wave group forcing we have used the breakpoint forcing model described by SHB82 who make a number of significant assumptions. First the model assumes the shallow water approximations for both long and short waves. In particular by assuming the shallow water approximation for the radiation stress tensor, and linear depth dependent wave dissipation through the surf zone, SHB82 were able to represent the time dependent forcing as a square wave in time, given by a constant shoreward of the breakpoint, and zero seawards of the breakpoint. These assumptions allowed an analytic expression for the forcing term through a Fourier decomposition. By relaxing the shallow water approximation the forcing would include the depth dependent part of the radiation stress (Longuet-Higgins & Stewart, 1964). Some numerical studies have included the full expression for the radiation stress terms (Pomeroy et al., 2012). SHB82 also ignore the contribution to the long wave response associated with the incoming bound wave. In intermediate depths the bound wave is phase locked to the incoming wave groups, and the phase difference shifts from p to p/2 as the group propagates into shallow water. In shallow water the wave group frequency and wavenumber may satisfy the free wave dispersion relationship and the bound wave may be released and propagate shoreward as a free wave. Our results show the breakpoint forced response is not phase locked to the incoming wave groups. In the event that the bound wave is released in shallow water the phase of the incoming long wave in the surf zone will result from the combination of the IBFLW and the released bound wave. The released bound wave and the IBFLW both reflect at the shoreline and contribute to the outgoing signal though the phase of the released bound wave is not known. However, breakpoint forcing also results in an outgoing free wave seaward of the surf zone. Numerical models which include radiation stresses do include both bound and free wave responses and may shed some light on the combined response. It is difficult to identify the signal from breakpoint forcing from observations, and in situ observations are lacking (i.e., Moura & Baldock, 2017). Our results are a step in the process of being able to identify breakpoint forcing signal from observations. Once the breakpoint forcing model is well understood, numerical modeling may provide insight to the long wave response on complex bathymetry with irregular wave forcing.

Appendix A: Expression of Constants Using the boundary conditions at x 5 x1 and x 5 x2, the constants in the elevation solutions are expressed as follow:

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 pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi gp ðx2 ÞJ1 2 vx2 12x2 0:5 g0p ðx2 ÞJ0 2 vx2 =4v  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi A1 5A4 1  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi J1 2 vx2 Y0 2 vx2 2J0 2 vx2 Y1 2 vx2 A2 5A3 1

 pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi gp ðx1 ÞY1 2 vx1 12x1 0:5 g0p ðx1 ÞJ0 2 vx1 =4v  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi J0 2 vx1 Y1 ðx1 Þ2J1 2 vx1 Y0 ðx1 Þ

 pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi gp ðx2 ÞY1 2 vx2 12x2 0:5 g0p ðx2 ÞY0 2 vx2 =4v  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi A3 5  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi J1 2 vx2 Y0 2 vx2 2J0 2 vx2 Y1 2 vx2  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi gp ðx1 ÞJ1 2 vx1 12x1 0:5 g0p ðx1 ÞJ0 2 vx1 =4v  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi A4 5  pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi J0 2 vx1 Y1 2 vx1 2J1 2 vx1 Y0 2 vx1

(A1)

Appendix B: Trigonometry Formulas We use the trigonometric addition formulas to calculate phase relationships: cos a cos b1sin a sin b5cos ða2bÞ

(B1)

a1 cos t1a2 sin t5R cos ðt2aÞ

(B2)

We pose:

with a1 5 R cos a and a2 5Rsin a: Then   8 > 21 a2 > tan > < a1 a5   > a2 > > : tan21 1p a1

if a1  0 (B3) if a1 < 0

And pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R5 a1 2 1a2 2

Acknowledgments Financial support for this research is provided by the Royal Australian Navy as part of the Bluelink project. We thank the reviewers for their valuable comments which contributed to the improvement of the manuscript. No data was used in producing this manuscript.

(B4)

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