Phase Between Forcing and Response - Wiley Online Library

Research Article

Journal of Geophysical Research: Oceans DOI 10.1002/2017JC013138

Breakpoint forcing revisited; phase between forcing and response S. Contardoa

, G. Symondsb

and F. Dufoisb

a

CSIRO Oceans and Atmosphere Flagship, Crawley, Western Australia, Australia

b

UWA Oceans Institute, University of Western Australia, Crawley, Western Australia,

Australia

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.

Abstract Using the breakpoint forcing model [Symonds et al., 1982], 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 π 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 π/2. The phase lag decreases as the beach slope decreases and the group frequency increases, approaching approximately π/16 within reasonable limits of the parameter space. The phase between the breakpoint-forced This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/2017JC013138 © 2018 American Geophysical Union Received: May 26, 2017; Revised: Jan 14, 2018; Accepted: Jan 26, 2018 This article is protected by copyright. All rights reserved.

outgoing long wave and the wave group is between π/2 and π and it increases as the beach slope decreases and the group frequency increases, approaching 15π/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 modelling.

Index terms 4546 Nearshore processes 4560 Surface waves and tides Keywords Infragravity waves Long waves Breakpoint forcing

Text 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

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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 π out of phase with the group. They hypothesized that the incoming bound long wave reflects from non-uniformities 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 and Huntley, 2002; Baldock, 2012; Contardo and 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 et al., 2000; Baldock and Huntley, 2002; Baldock et al., 2004] and field observations [Masselink, 1995; Pomeroy et al., 2012; Contardo and Symonds, 2013; Inch et al., 2017; Moura and Baldock, 2017]. These analyses (except for Baldock et al. [2000]) were based on the phase differences between the wave groups and the long waves. Bound waves and wave groups are π out of phase in intermediate water and shift toward π/2 in shallow water [Bowers, 1992; List, 1992]. Due to this π/2 shift, a positive correlation is

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expected to precede the 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 breakpointforced long waves. Several authors [i.e. Masselink, 1995; Baldock et al., 2000; Pomeroy et al., 2012; Contardo and Symonds, 2013]) 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 π 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. 4 This article is protected by copyright. All rights reserved.

SHB82 express the solutions in term of the non-dimensional parameter χ: 𝜎 2𝑋 𝜒= 𝑔 tan 𝛽

Equation 1

where σ is the group radial frequency, X is the breakpoint mean position, tan β is the beach slope and g is the gravitational acceleration. The elevation amplitude of the forced long waves depends on χ and on the non-dimensional amplitude difference, Δa, 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 non-dimensional limits of the breakpoint excursion are x1 and x2 which are equal to 1 – Δa and 1 + Δa respectively. For high values of Δa, the model is valid only for small values of χ (eq. 14 SHB82). Thus we use a small value of Δa (0.1) for all calculations, as Δa does not affect the phase relationships. Using the shallow water approximation, SHB82 express the forcing function as a Fourier series (eq. 8 in SHB82), so at the group frequency it becomes: 1 𝜕(𝑎2 ) = 2𝑎1 (𝑥) cos 𝑡 2𝑥 𝜕𝑥

Equation 2

where a is the incident wave amplitude and

𝑎1 (𝑥) =

sin 𝜏 𝜋

Equation 3

and 𝜏 = cos−1 (

𝑥−1 ) ∆𝑎

Equation 4

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

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𝜁𝑆𝐿𝑊 = 𝐴1 𝐽0 (2√𝜒𝑥) sin 𝑡 + 𝐴2 𝐽0 (2√𝜒𝑥) cos 𝑡,

Equation 5

within the breakpoint excursion: 𝜁𝐵𝑃𝐸 = 𝐴1 𝐽0 (2√𝜒𝑥) sin 𝑡 + [𝐴3 𝐽0 (2√𝜒𝑥) + 𝐴4 𝑌0 (2√𝜒𝑥) + 𝜂𝑝 ] cos 𝑡

Equation 6

and seaward of the surf zone: 𝜁𝑂𝐿𝑊 = 𝐴1 [𝐽0 (2√𝜒𝑥) sin 𝑡 − 𝑌0 (2√𝜒𝑥) cos 𝑡],

Equation 7

where x represents the offshore distance non-dimensionalized by the mean breakpoint position, and t is time non-dimensionalized by the radial group frequency. J0 and Y0 are zeroorder Bessel functions, of the first and second kind respectively, and ηp is a particular solution [SHB82]. A1, A2, A3, A4 are constants (expressions given in Appendix A) and vary with the non-dimensional parameters Δa and χ as shown in Figure 1. Note in Equation 7, the sign of the second term is different from eq. 22 in SHB82. The minus sign is needed to obtain a seaward propagating wave. As noted in SHB82, for χ ≈ 1.2 (χ = χmax), the elevation amplitude offshore of the breakpoint is maximum (Figure 8 in SHB82). For χ ≈ 3.7 (χ = χ0), 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 = 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.

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Re-arranging Equation 5 (Appendix B), we write the solution for the elevation of the SLW in the surf zone as: 𝜁𝑆𝐿𝑊 = √(𝐴1 2 + 𝐴2 2 )|𝐽0 (2√𝜒𝑥)| cos(𝑡 + 𝛷𝑆𝐿𝑊 ),

Equation 8

and the phase lag between the standing long wave (SLW) in the surf zone and the forcing is given by: 𝐴1 ) 𝐴2 = 𝐴1 −tan−1 ( ) + 𝜋 { 𝐴2 −tan−1 (

𝛷𝑆𝐿𝑊

if A2 𝐽0 (2√𝜒𝑥) ≥ 0

Equation 9

if A2 𝐽0 (2√𝜒𝑥) < 0

Similarly the phase between the OLW in the offshore zone and the forcing term can be retrieved from Equation 7:

𝜁𝑂𝐿𝑊 = |𝐴1 |√𝐽0 2 (2√𝜒𝑥) + 𝑌0 2 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑂𝐿𝑊 ),

Equation 10

and the phase lag between the outgoing long wave (OLW) and the forcing in the offshore zone is given by: 𝐽0 (2√𝜒𝑥) )+𝜋 𝑌0 (2√𝜒𝑥) = 𝐽0 (2√𝜒𝑥) tan−1 ( ) 𝑌0 (2√𝜒𝑥) { tan−1 (

𝛷𝑂𝐿𝑊

𝑖𝑓 𝐴1 𝑌0 (2√𝜒𝑥) ≥ 0

Equation 11

𝑖𝑓 𝐴1 𝑌0 (2√𝜒𝑥) < 0

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 breakpoint-forced 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.

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The breakpoint forcing signal is usually identified, from observations and numerical modelling, from the phase relationship between the IBFLW and the wave group offshore, and the phase relationship between the OBFLW and the wave group offshore [Masselink, 1995; Pomeroy et al., 2012; Contardo and Symonds, 2013]. 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), 𝜁𝑆𝐿𝑊 = 𝜁𝐼𝐵𝐹𝐿𝑊 + 𝜁𝑅𝐿𝑊

Equation 12

so Equation 5 can be written as: 𝜁𝑆𝐿𝑊 = 𝐴𝑠𝑧 [𝐽0 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑆𝐿𝑊 ) − 𝑌0 (2√𝜒𝑥) sin(𝑡 + 𝛷𝑆𝐿𝑊 ) +

Equation 13

𝐽0 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑆𝐿𝑊 ) + 𝑌0 (2√𝜒𝑥) sin(𝑡 + 𝛷𝑆𝐿𝑊 )] with 𝐴𝑠𝑧 =

√𝐴1 2 +𝐴2 2 2

.

The expression for the elevation amplitude of the IBFLW is given by the first two terms in Equation 13: 𝜁𝐼𝐵𝐹𝐿𝑊 = 𝐴𝑠𝑧 [𝐽0 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑆𝐿𝑊 ) − 𝑌0 (2√𝜒𝑥) sin(𝑡 + 𝛷𝑆𝐿𝑊 )]

Equation 14

which can be written, following Appendix B, as:

𝜁𝐼𝐵𝐹𝐿𝑊 = 𝐴𝑠𝑧 √𝐽0 2 (2√𝜒𝑥) + 𝑌0 2 (2√𝜒𝑥)cos(𝑡 + 𝛷𝐼𝐵𝐹𝐿𝑊 ) with the phase between the IBFLW and the forcing term given by:

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Equation 15

𝑌0 (2√𝜒𝑥) ) 𝐽0 (2√𝜒𝑥) = 𝑌0 (2√𝜒𝑥) 𝛷𝑆𝐿𝑊 + tan−1 ( ) +𝜋 𝐽0 (2√𝜒𝑥) { 𝛷𝑆𝐿𝑊 + tan−1 (

𝛷𝐼𝐵𝐹𝐿𝑊

𝑖𝑓 𝐽0 (2√𝜒𝑥) ≥ 0 Equation 16 𝑖𝑓 𝐽0 (2√𝜒𝑥) < 0.

Similarly, the expression for the elevation amplitude of the RLW is given by the last two terms in Equation 13: 𝜁𝑅𝐿𝑊 = 𝐴𝑠𝑧 [𝐽0 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑆𝐿𝑊 ) + 𝑌0 (2√𝜒𝑥) sin(𝑡 + 𝛷𝑆𝐿𝑊 )]

Equation 17

and rearranged as:

𝜁𝑅𝐿𝑊 = 𝐴𝑠𝑧 √𝐽0 2 (2√𝜒𝑥) + 𝑌0 2 (2√𝜒𝑥)cos(𝑡 + 𝛷𝑅𝐿𝑊 )

Equation 18

with the phase between the RLW and the forcing given by: 𝑌0 (2√𝜒𝑥) ) 𝑖𝑓 𝐽0 (2√𝜒𝑥) ≥ 0 𝐽0 (2√𝜒𝑥) 𝑌0 (2√𝜒𝑥) ( ) + 𝜋 𝑖𝑓 𝐽0 (2√𝜒𝑥) < 0 . 𝐽0 (2√𝜒𝑥)

𝛷𝑆𝐿𝑊 − tan−1 ( 𝛷𝑅𝐿𝑊 = −1

𝛷𝑆𝐿𝑊 − tan {

Equation 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 the shoreline but at a cross-shore position such that 𝑌0 (2√𝜒𝑥) = 0, that is when 2√𝜒𝑥 = 0.87 i.e. x = 0.19 / χ. 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: 𝜁𝑂𝐿𝑊 = 𝐴𝑜𝑧 [𝐽0 (2√𝜒𝑥) sin(𝑡 + 𝛷𝑂 ⁄2) − 𝑌0 (2√𝜒𝑥) cos(𝑡 + 𝛷𝑂 ⁄2) + 𝐽0 (2√𝜒𝑥) sin(𝑡 − 𝛷𝑂 ⁄2) − 𝑌0 (2√𝜒𝑥) cos(𝑡 − 𝛷𝑂 ⁄2)]

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Equation 20

where ΦO 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: 𝛷𝑂 = 2𝛷𝑆𝐿𝑊 +π

Equation 21

and 𝐴𝑜𝑧 = 𝐴𝑠𝑧 .

Equation 22

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: 𝜁𝑂𝐵𝐹𝐿𝑊 = 𝐴𝑠𝑧 [−𝐽0 (2√𝜒𝑥) cos(𝑡 − 𝛷𝑆𝐿𝑊 ) − 𝑌0 (2√𝜒𝑥) sin(𝑡 − 𝛷𝑆𝐿𝑊 )]

Equation 23

Equation 23 is re-arranged (Appendix B) as follows:

𝜁𝑂𝐵𝐹𝐿𝑊 = 𝐴𝑠𝑧 √𝐽0 2 (2√𝜒𝑥) + 𝑌0 2 (2√𝜒𝑥)cos(𝑡 + 𝛷𝑂𝐵𝐹𝐿𝑊 )

Equation 24

with the phase between the OBFLW and the forcing given by: 𝑌0 (2√𝜒𝑥) ) +𝜋 𝐽0 (2√𝜒𝑥) = 𝑌0 (2√𝜒𝑥) −𝛷𝑆𝐿𝑊 − tan−1 ( ) 𝐽0 (2√𝜒𝑥) { −𝛷𝑆𝐿𝑊 − tan−1 (

𝛷𝑂𝐵𝐹𝐿𝑊

𝑖𝑓 𝐽0 (2√𝜒𝑥) ≥ 0 Equation 25 𝑖𝑓 𝐽0 (2√𝜒𝑥) < 0 .

The phase lags account for the travelling time between the breakpoint and the cross-shore positions. 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 = 0 and χ = 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 ±∞ 10 This article is protected by copyright. All rights reserved.

at the shoreline, as the expressions for the incoming and outgoing components include zeroorder Bessel functions of second kind, and they are in phase at x = 0.095 (x = 0.19 / χ). The phase between the SLW and the forcing (ΦSLW), given by Equation 9, and the phase between the OLW and the forcing (ΦOLW), given by Equation 11 are shown in Figure 3a. ΦSLW is independent of the cross-shore position (x < x1), except for phase shifts by π at node locations, consistent with a standing wave. For x > x2, ΦOLW increases with x, consistent with an outgoing progressive wave (see Equation 10). For x < x1, the standing wave can be decomposed into the incoming and reflected components with corresponding phases ΦIBFLW and ΦRLW respectively as shown in Figure 3b and c. For x > x2, the OLW is composed of two outgoing waves with phases ΦOBFLW (Equation 25) and ΦRLW (Equation 19), also shown in Figure 3b and c. The addition of the component phases shown in Figure 3b and c returns the phases for the total solution shown in Figure 3a, i.e. for x < x1, ΦSLW = ΦIBFLW + ΦRLW and for x > x2, ΦOLW = ΦOBFLW + ΦRLW. It is also important to note the phases shown in Figure 3 all depend on χ. Figure 4a and b show the phases ΦSLW and ΦOLW, respectively, as functions of χ. Figure 4a shows that ΦSLW increases with χ independently of the cross-shore position except for shifts by π when nodes are crossed, e.g. at χ = 2.892 and χ = 1.606, for x = 0.5 and x = 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, ζn, to zero (i.e. equating 𝐽0 (2√𝜒𝑥) to zero) and solving for the roots of χ. For instance, 𝐽0 (2√𝜒𝑥) = 0 at χ = 2.892 and 1.606 (indicated with dashed gray lines in Figure 4a) for x = 0.5 and x = 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). ΦOLW also increases with χ but varies with x as shown in Figure 3.

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We note that the OBFLW and the RLW are in phase for χ = χmax and π out of phase for χ = χ0 as shown in Figure 5 which represents the difference between ΦOBFLW and ΦRLW versus χ. This is in agreement with SHB82 who noted that the elevation of the OLW was maximum for χ = χmax and zero for χ = χ0. 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 = X as functions of χ. Figure 6 shows ΦIBFLW(x = X) is less than π/2 (for χ > 0.001) and decreases as χ increases. According to Equation 16, ΦIBFLW > 0 means the IBFLW lags behind the forcing. Figure 6 also shows ΦOBFLW (x = X) is greater than π/2 (for χ > 0.001) and increases as χ increases. 5. Discussion and conclusions 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 (Masselink, 1995; Baldock et al., 2000; Pomeroy et al., 2012; Contardo and Symonds, 2013) assumed the incoming breakpoint forced long waves were in phase with the incident wave groups while the outgoing breakpoint forced long waves were π 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 χ. Using typical values for surf zone width X = 100 m, group frequency = 0.06, and beach slope tan= 0.02 gives χ ~ 2 and, according to Figure 6, ΦIBFLW ~ 8 and decreases as χ increases. Similarly, at χ ~ 2 Figure 6 shows ΦOBFLW ~ 7and decreases as χ increasesIn both cases much larger values of χ 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 a. We do note that the phase of the standing long wave in the 12 This article is protected by copyright. All rights reserved.

surf zone, SLW → 0 as χ → 0 as shown in Figure 4a. . This case is consistent with Figure 6 in SHB82 where, as χ → 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

SLW -> 0 occurs in the presence of very steep beaches, or very low group frequencies. For larger values of χ, -< SLW < . 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 π out of phase, they conclude that the breakpoint is approximately eight meters shoreward of that location. Now considering, the phase might be a bit lower than π, the breakpoint would be approximately two meters seaward of where it was estimated, i.e. six meters 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 χ (and on the group frequency), the cross-spectra method (used in [Contardo and 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. Firstly 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 13 This article is protected by copyright. All rights reserved.

Fourier decomposition. By relaxing the shallow water approximation the forcing would include the depth dependent part of the radiation stress [Longuet-Higgins and 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  to /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 and 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 modelling may provide insight to the long wave response on complex bathymetry with irregular wave forcing.

Appendix A – Expression of constants 14 This article is protected by copyright. All rights reserved.

Using the boundary conditions at x = x1 and x = x2, the constants in the elevation solutions are expressed as follow: 𝜂𝑝 (𝑥2 )𝐽1 (2√𝜒𝑥2 ) + 2𝑥2 0.5 𝜂𝑝′ (𝑥2 )𝐽0 (2√𝜒𝑥2 )⁄4𝜒 𝐴1 = 𝐴4 + 𝐽1 (2√𝜒𝑥2 )𝑌0 (2√𝜒𝑥2 ) − 𝐽0 (2√𝜒𝑥2 )𝑌1 (2√𝜒𝑥2 ) 𝐴2 = 𝐴3 +

𝜂𝑝 (𝑥1 )𝑌1 (2√𝜒𝑥1 ) + 2𝑥1 0.5 𝜂𝑝′ (𝑥1 )𝐽0 (2√𝜒𝑥1 )⁄4𝜒 𝐽0 (2√𝜒𝑥1 )𝑌1 (𝑥1 ) − 𝐽1 (2√𝜒𝑥1 )𝑌0 (𝑥1 ) Equation a1

𝜂𝑝 (𝑥2 )𝑌1 (2√𝜒𝑥2 ) + 2𝑥2 0.5 𝜂𝑝′ (𝑥2 )𝑌0 (2√𝜒𝑥2 )⁄4𝜒 𝐴3 = 𝐽1 (2√𝜒𝑥2 )𝑌0 (2√𝜒𝑥2 ) − 𝐽0 (2√𝜒𝑥2 )𝑌1 (2√𝜒𝑥2 ) 𝐴4 =

𝜂𝑝 (𝑥1 )𝐽1 (2√𝜒𝑥1 ) + 2𝑥1 0.5 𝜂𝑝′ (𝑥1 )𝐽0 (2√𝜒𝑥1 )⁄4𝜒 𝐽0 (2√𝜒𝑥1 )𝑌1 (2√𝜒𝑥1 ) − 𝐽1 (2√𝜒𝑥1 )𝑌0 (2√𝜒𝑥1 )

Appendix B – Trigonometry formulas We use the trigonometric addition formulas to calculate phase relationships: cos a cos 𝑏 + sin 𝑎 sin 𝑏 = cos(𝑎 − 𝑏)

Equation b1

We pose: 𝑎1 cos 𝑡 + 𝑎2 sin 𝑡 = 𝑅 cos(𝑡 − 𝛼)

Equation b2

with 𝑎1 = R cos α and 𝑎2 = 𝑅 sin 𝛼. Then 𝑎2 ) 𝑎1 α={ 𝑎2 tan−1 ( ) + 𝜋 𝑎1 tan−1 (

𝑖𝑓 𝑎1 ≥ 0 Equation b3 𝑖𝑓 𝑎1 < 0

And Equation b4

𝑅 = √𝑎1 2 + 𝑎2 2

15 This article is protected by copyright. All rights reserved.

Acknowledgments and data 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.

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nearshore, J. Geophys. Res. Ocean., 118(12), 7095–7106, doi:10.1002/2013JC009430. Inch, K., M. Davidson, G. Masselink, and P. Russell (2017), Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions, Cont. Shelf Res., 138(March), 19–31, doi:10.1016/j.csr.2017.02.010. List, J. H. (1992), A model for the generation of two-dimensional surf beat, J. Geophys. Res., 97(C4), 5623, doi:10.1029/91JC03147. Longuet-Higgins, M. S., and R. W. Stewart (1964), Radiation stresses in water waves; a physical discussion, with applications, Deep Sea Res. Oceanogr. Abstr., 11(4), 529–562, doi:10.1016/0011-7471(64)90001-4. Longuet-Higgins, M. S., and R. W. Stewart (1962), Radiation stress and mass transport in gravity waves, with application to “surf beats,” J. Fluid Mech., 13(4), 481, doi:10.1017/S0022112062000877. Masselink, G. (1995), Group bound long waves as a source of infragravity energy in the surf zone, Cont. Shelf Res., 15(13), 1525–1547, doi:10.1016/0278-4343(95)00037-2. Moura, T., and T. E. Baldock (2017), Remote sensing of the correlation between breakpoint oscillations and infragravity waves in the surf and swash zone, J. Geophys. Res. Ocean., 122(4), 3106–3122, doi:10.1002/2016JC012233. Munk W. H. (1949), Surf Beats, Eos Trans, (30), 849–854. Pomeroy, A., R. Lowe, G. Symonds, A. Van Dongeren, and C. Moore (2012), The dynamics of infragravity wave transformation over a fringing reef, J. Geophys. Res. Ocean., 117(11), 1–17, doi:10.1029/2012JC008310. Symonds, G., D. A. Huntley, and A. J. Bowen (1982), Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res., 87(C1), 492, doi:10.1029/JC087iC01p00492. 17 This article is protected by copyright. All rights reserved.

Tucker, M. J. (1950), Surf Beats: Sea Waves of 1 to 5 Min. Period, Proc. R. Soc. Lond. A. Math. Phys. Sci., 202(1071), 565–573, doi:10.1098/rspa.1950.0120.

18 This article is protected by copyright. All rights reserved.

Figure 1 Constants, A1, A2, A3 and A4 for different values of Δa and χ.

Figure 2 Elevation, versus cross-shore position, at t = 0, of the total breakpoint-forced long wave and its incoming and outgoing components. Gray dashed lines represent the limits of the breakpoint excursion. χ = 2, Δa = 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 χ. Dashed vertical gray lines represent x1 and x2. Δa = 0.1.

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

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Figure 5 Difference between ΦOBFLW and ΦRLW versus χ. Vertical gray dashed lines represent χ = χmax and χ=χ0.

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Figure 6 ΦIBFLW (blue line) and ΦOBFLW (orange line) and their difference (yellow line). x = X.

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