Qualitative Description of Swashing Motion States on Mild Beach Slope

The swashing motion on mild beach slope is dominated by the motion of low frequency waves (LFWs). Companying such a motion, there are two types of swashing motion states, occurrence or no occurrence of LFW’s collision. The present study distinguishes the two states qualitatively by relating it to the number of generated LFWs for the case of two incident wave groups. A simplified swashing index is established theoretically for this purpose. A series of related experiments were performed to observe the generated out-going LFWs on different mild slope from 1:20 to 1:160 and to determine the critical value of the swashing index. Numerical simulations based on higher order Boussinesq equations are also performed to help the recognition of the LFWs generated in the experiment.


Introduction
Swash zone, the region of beach profile covered intermittently by fluid due to water wave motion, is important for the researches of coastal hydrodynamics and morphology. The waves remaining on foreshore slope after wave breaking rush up and down the beach with significant momentum, and exert strong action on sandy bottom, which leads to large local morphological response related to foreshore beach deformation. The relevant studies have revealed that the presence or lack of swash collisions may be the reason for foreshore accretion or erosion and the related influence factor may be the ratio of the time duration of swash excursion to the incident wave period (being larger than one leads to a steepening foreshore; otherwise, a flattening foreshore) (e.g. Holland and Puleo, 2001). But up to now, the research results on qualitative description of swashing motions on different beach slopes are rare in the literatures, especially for mild slope. This paper presents such a research work by examining a series of wave swashing experiments on a large range of beach slopes and related numerical results. The qualitative description is established theoretically and experimentally by introducing a simplified index for swashing states (the ratio of swashing duration to wave group period).
The previous researches have noticed the difference between swashing motions on steep and mild beaches. The swa-f ⩽ shing motion on steep slope is characterized by the swash flow which is mainly resulted from the collapse of high-frequency bores (with the frequency f >0.05 Hz), whereas the swashing motion on mild slope is by standing low frequency motions (with 0.05 Hz) (see Baldock et al., 1997). Mase and Iwagaki (1985) ran a series of tests in the laboratory with random waves and found that the ratio of the number of individual run-up waves to that of incident waves decreased as the beach slope decreased. Baldock et al. (1997) presented experimental data of swash motions for regular waves, wave groups, and random waves on an impermeable beach with slope 1:10. For the case of bichromatic wave groups, they found considerable interaction in the swash zone between subsequent bores, which often caused the smallest bores at the beginning of the group to run up further than the subsequent larger bores. Erikson et al. (2005) applied the ballistic model to account for the interaction between up-rush and back-wash at still water shoreline and within swash zone. They considered a wave group (a wave packet) propagating onto a permeable beach formed by sand and showed that for the relative gentle foreshore slope 7:100, the theoretical results of the maximum run-up length can be markedly improved with the consideration of swash interaction. For the steep foreshore slope 1:5, the improvement is little, which is due to the swashing time duration shorter than the incident wave period for this steep slope case (not long enough to allow the interaction to happen). Different from the researches mentioned above, the present study aims at the effects of different beach slopes on swashing motion; the dependence of the ratio of swashing duration to wave group period on beach slope is investigated. The main focus is on the case of mild slope, as it is found in the research process that for mild slope the swashing motion is dominant by that produced by low frequency waves (LFWs), and the corresponding qualitative description should be treated differently from that for steep slope. To this end, the large range of slopes, 1:20 to 1:160, are considered. In order to study the swashing motion by LFW, the incident wave is taken as two wave groups (two wave packets). This simple type of wave group makes the resulting swashing motion be simple too, that is, there are only the two types of LFW's swashing motions, occurrence or no occurrence of LFW's collision and the corresponding number of generated out-going LFWs during swashingis easily identified, which is one or two. (Another consideration is that this wave type can also avoid the standing wave structure on beach formed by the reflection of out-going LFW at wave paddle). A corresponding theoretical analysis is also performed, which leads to the establishment of a simplified formula for judging the number of generated out-going LFWs.

Experimental set-up
The experiment was conducted in the wave flume at the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology, which is 60 m long, 0.7 m wide and 1.0 m deep. At one end of the flume is the piston-type wave generator for wave generation, and at the other end is a beach model made of concrete. Different beaches with slopes from 1:20 to 1:160 were adopted and they were divided into two test groups, Test A and Test B. Test A used five beaches with slopes of 1:20, 1:40, 1:60, 1:80 and 1:100, which all begin at 10.0 m from the wave generator and have the water depth of 32.5 cm in the horizontal bottom, shown in Fig. 1. Test B used six beaches with slopes of 1:80, 1:90, 1:100, 1:120, 1:140 and 1:160, which also began at 10.0 m from the wave generator but were built on a concrete horizontal platform of 12.5 cm in height beginning at 3.0 m from the wave generator in order to increase the water depth in front of the wave maker to help the wave generation. The water depth before the platform was still kept to be 32.5 cm as in Test A, and the water depth over the platform is 20.0 cm. So the platform raises the water depth before the wave generator from 20 cm to 32.5 cm, which makes the wave generation produce a higher wave height. Test A and Test B have two common beaches, slopes 1:80 and 1:100, which can be used to make intercomparison between the two test results.
Standard capacitance-type wave gauges were used to measure wave surface elevations. The measuring accuracy of the gauges is 0.5%. Their arrangements of gauges are different for different slopes, and will be shown in the figures for measured results presented later. The LFW produced by the wave groups was obtained in the experiment by low-past filtering of the recorded time history of the wave elevations at each wave gauge by fast Fourier transformation with cutoff frequency lower than that of short waves.
To obtain around the problem of the second wave reflection on the wave paddle, only two wave groups were considered. So the disturbance of wave field by reflected waves from the paddle was absent in the experiment. To generate the required incident wave groups, the computer signal designed for controlling wave maker is determined as follows. Consider the produced wave group consisting of modulated sine waves plus bound long waves (set-down), and has the surface elevation: where is the short wave free surface elevation, is the bound long wave free surface elevation. is expressed in the form YIN Jing et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 412-423 413 , ( ), where is the time, x is the spatial coordinate with the x-axis having the origin at the mean position of wave-maker paddle and pointing shoreward, and is taken as the paddle's mean position, is the maximum wave height in the wave groups, and are the period and wave length of the short wave, and are the period and wave length of wave group with n being the numbers of short waves contained in a wave group, is the number of wave groups (wave packets), and is adopted, corresponding to two wave groups, is the modulation parameter of short waves, and is taken to produce the strongly modulated wave group for the present study. η L The expression for is given by which is the solution of the following governing equations for the horizontal bottom: and , where is the still water depth, is the velocity of long wave, is the water density, is the oscillating part of the radiation stress, expressed by where is the group velocity, is the celerity of short wave, is the short wave energy, and is the gravitational acceleration.
The corresponding velocity of fluid particle is the sum of short and long wave velocity components, , with and , the latter being the solution to Eq. (4a), where k and z are the wave number and vertical coordinate with the origin at still water level. With the above results, the displacement of wave maker paddle can be determined by the solution of the equation .
For both Test A and Test B, the wave groups with different maximum wave height and period were produced, each having the short wave period T=1.0 s or T= 1.3 s. Four short wave heights (H max =4.0 cm, 6.0 cm, 8.0 cm, 10.0 cm) and three group periods (n=8, 10, 14) were chosen for T=1.0 s, and three short wave heights (H max =4.0 cm, 6.0 cm, 8.0 cm) and two group periods (n=6, 10) were chosen for T=1.3 s. Table 1 lists these experimental cases, in which each test case is labeled in terms of the short wave period T, maximum wave height H max and the number of short waves in one group, n. For example, AT1H2n10 represents the case of Test A with the first period (T=1.0 s), the second wave height (H max =6.0 cm) and n=10; BT2H3n06 represents the case of Test B with the second period (T=1.3 s), the third wave height (H max =8.0 cm) and n=6.

Numerical simulations
The experiment shows that the generated out-going LFW is small in magnitude and not easily detected (see the results in Figs. 2 and 3). This may become more serious for the cases of very small slope, as the milder the slope, the smaller the magnitude (see Figs. 3 and 4). To help the identification of the out-going LFW, the numerical simulations corresponding to the experimental cases were performed by applying higher-order Boussinesq equations (BouN2D4) of Zou and Fang (2008) in terms of the surface elevation and depth-averaged velocity with the revision accounting for the shoreline boundary condition and wave breaking. The   numerical discretization method is as that used in Zou and Fang (2008), and the space and time steps in numerical solution were set as , . The adopted model has the nonlinear property of being fully nonlinear up to and the Padé[4, 4] dispersion, so it can describe the fully nonlinear wave motion near shoreline. The incident wave boundary is taken at the horizontal bottom with the internal wave making technique applied, which can be set to produce the same incident wave as that of the experiment.
As examples of numerical results, Figs. 2 and 3 show the surface elevations at different locations for Test AT2H2-n10 with slope 1:60 and Test BT2H2n10 with slope 1:100, respectively. The corresponding low frequency waves are also presented by filtering the time series of the surface elevation. It is seen that the agreement between the numerical results and data is good. The quantitative descriptions of the simulation accuracy are described by the relative curve errors defined as:   where and are the wave height of short wave and its mean value, the superscripts p and m stand for the computation and measurement, respectively; the subscripts s and l for short and long waves, respectively. So, is the relative errors between the two curves of the computed and measured short wave surface elevations, is that for the long waves; and are the numbers of sample data. In Eq. (5a), different from Eq. (5b), the error due to the phase difference of computed and measured short wave free surface elevations is not accounted for, as this difference can be seen directly from the shifting of the computed from the measured in the figure and its inclusion will cause extraordinary large values of the relative error which are not useful here. Owing to the long wave length, this problem is not serious for LFWs (the phase difference is small relative to wave length), so it is considered in Eq. (5b). Here only the representative values of and are given for the analysis: in Fig 52.48% for surf zone position x=22.0 m. From these results and the surface elevations at other gauges in the two figures, it is seen that the simulation accuracy for the short wave is better than that for the long wave: the computed surface elevation of short wave is very close to the measureme-nt, even after wave breaking, but that of long wave has a larger relative error, which gets larger when approaching shoreline. As shown by the numerical results of only LFW presented in the next section (Figs. 5 and 6), the error with long wave does not affect significantly the accuracy of computed out-going LFW concerned by the present study, especially for that appearing in the offshore region with horizontal bottom.

Swashing motion states on mild slope
Here we discuss the features of swashing motion on different slopes. For relative steep slope, it is known that there are two types of wave interactions in swashing motions (e.g. Erikson et al., 2005): one is the catch-up and absorption where the front of a wave moving landward is passed by a subsequent long wave bore moving in the same direction; the other is the collision where two separate fronts collide as the back-wash of a preceding swash lens meets the front of a subsequent swash wave during the up-rush phase. Different from such a swashing motion state, the swashing motions on mild slope are dominated by LFW motions. These swashing motions involve the following three effects of mild slope.
The first effect is that the wave breaking on mild slope can destroy a large part of the short waves before the waves get to still water shoreline and leads to the waves taking part in swashing motion being mainly the LFW. This can be seen in Figs. 2 and 3, in which the short wave near still water line become small rags superimposed on the relative large LFW (revealed in the figures both by the experimental data and by the numerical results).
The second effect is that the mild slope can lead to a relative large magnitude of LFW. This is due to the larger phase lag between the LFW and short wave envelope during shoaling on a mild slope, as the phase lag has effects on the energy transfer from the grouped short waves to the accompanying long waves. The relevant explanation can be referred to Janssen et al. (2003) and Battjes et al. (2004). They showed that the phase lag of the group-induced long wave beneath the envelope of the incident short-wave groups leads to the incident bound waves having significant amplitude growth during their shoaling. For the present case, the difference in phase lag with different slopes can be seen in Figs. 2 and 3 for the locations of wave shoaling and before wave breaking (after wave breaking, this phase lag cannot be identified, as the group-shaped wave envelope is destroyed). The quantitative description of the phase lag can be done by calculating the coherence coefficient between the envelope of short wave crest and long wave beneath it: and are the crests of short wave and its mean value, the other quantities are defined as in Eq. (5). The fol-  0.54 for the last shoaling zone position x=16.0 m in Fig. 3 with slope 1:100. From these values of r and the comparison of locations of LFW's troughs in these figures, it can be seen that the phase lag gets larger for the milder slope: the value of is smaller and the lowest point of LFW's trough has a larger distance from the peak of wave envelope over it for slope 1:100 in Fig. 3 than that for slope 1:60 in Fig. 2.
The third effect is that a mild slope causes small reflection of set-down, so the effect of the reflection on the LFW's growth is also small.
All the above three effects mean that there are more LFWs which can propagate up to the shoreline and take part in the swashing motion for mild slope cases, so the swashing motion state on mild slope can be illustrated mainly by the motions of LFWs. One of the remarkable features of such a motion is that the catch-up motion is little and the collision or no collision between incident and reflected LFWs are the two types of motion states. The collision state is the case that the back-wash of a preceding swash of an LFW has the time to meet with the subsequent rushing-up LFW. So due to the collision, only one out-going wave will be formed for the case of two incident wave groups considered in the present study. The no collision state is the case that the time interval between the two successive uprushing LFWs is long enough so that there is no chance for them to meet during swashing motion. For this case, the each up-rushing LFW can move back freely after it reaches its summit and propagated back to offshore region. So the number of out-going wave generated in this case will be the same as that of incident wave groups, two out-going waves.
The above correspondence between the swashing motion state and the number of generated LFWs means that we can use the number of generated LFWs as an indicator to distinguish the different swashing motion states: one out-going LFW corresponds to the collision state and two to the no collision state. So in the following discussion, the number and evolution process of the generated out-going LFWs are examined. ⩽ Fig. 4 presents the LFWs obtained from time series of long waves measured by Gauge 2 located at the horizontal bottom (see Fig. 1) for Test AT1H2n10 and Test AT2H1n06, with the slopes varying from 1:20 to 1:100. As shown by the inverse triangles in the figure, the incident wave of two wave groups generates two types of outgoing long waves: one with one crest and the other with two crests. This difference in the number of long wave crest relies on beach slope: for relatively steeper beach slopes (slope 1:80), the number of out-going long wave crest is the same as the number of wave groups, but for relatively gentler beach slope 1:100 it is smaller than the number of wave groups; only one crest coming back from the shoreline. > ⩽ To show the temporal and spatial evolution process of the LFWs for a wide range of mild beach slopes, Figs. 5 and 6 present the time series of the LFW recorded along the wave propagation course for the tests of AT2H3n10 with slopes 1:80, 1:100 and BT2H3n10 with slopes 1:90, 1:160. The generated out-going LFW is indicated by the dash line with an arrow pointing offshore; one line for one out-going crest and two lines for two out-going crests. The results of numerical simulations by the higher order Boussinesq equations are also included in the figures to help the identification of the LFW in the data. The numerical simulations give the similar results to the experiment (with a little bit larger LFW near shoreline region, as discussed for Figs. 2 and 3), which give us the confidence on the recognition of out-going LFW which has a small magnitude in offshore region (for small slopes 1/160-1/120). It can be seen that the crest of the outgoing LFW is two for slopes 1:90, but is one for slopes 1:100.
Next we need to discuss the critical condition for the transition between the collision and no collision state of swashing motions, or equally the critical slope for the transition from one out-going wave to two out-going waves, which is necessary for better understanding of the problem. This can be directly seen from the results of Figs. 5 and 6, which is 1:100. Obviously, this result is not very conclusive, as this critical condition should also depend on incident wave parameters. So in the next section we pursue a theoretical model for this condition, which accounts for the possible major influence factors related to this problem.

Index for the number of out-going LFW
The introduction has mentioned that the possibility of the collision of swashing waves should be dependent on the ratio of the swashing period ( ) to wave period. For mild slope cases (the main cases considered in the present experiment), this wave period should be taken as the wave group period , as it is shown in the previous section that the LFW becomes dominant over the short wave during swashing motion on mild slope with the latter having the similar amplitude as the former due to stronger breaking of short waves on mild slope. So here we introduce the following ratio, as the swashing index, the index parameter for determining the collision or no collision swashing motion state (indicated by the number of out-going LFWs). The following determines the critical value of SI from both the theoretical analysis and the experimental result. T r We determine the swashing period theoretically through applying the ballistic model for description of the collision process, as shown in Fig. 7 (see Ho et al. (1963), Shen and Meyer (1963), Erikson et al. (2005) collapsed bore or a long wave as a unit mass moving up and down the foreshore under the action of gravity. When the up-rush reaches its maximum, the velocity U is zero. This gives the time for the maximum up-rush , which is also the time required for the fluid particle to return to its starting location (x=0), so the swashing period is double the time : . Here is the initial velocity and is usually determined by applying the classical dam-break equation as suggested by Svendsen and Madsen (1984): , with H being the wave height taken at the still water shoreline and C an empirical coefficient ranging from 1 to the maximum of 2. This result is especially applicable for the swashing driven by short wave bores which happens on steep slope (Erikson et al., 2005), as the short wave is breaking wave and is controlled by local water depth. On the contrary, the swashing driven by LFWs will depend on the overall propagation process of LFW, as the LFW remains unbreaking up to shoreline, whose situation up to shoreline will depend on the history of its propagation. So for mild slope cases, the above formula for is not acceptable; the short wave has been destroyed largely by wave breaking before reaching the still water shoreline, and the left is dominated by the LFW. This can be seen clearly from the time series of the surface elevation near the still shore line, the records at x=25.0 m in Fig. 2 (which is 4.5 m away from still water shore line), and x=24.0 m in Fig. 3 (which is 6.0 m away from still water shore line). Therefore, for the mild slope cases, a new formula for need to be established, and we present a result shown in Appendix. Different from mentioned above, which relates to the local wave height H, the present formula relates to the wave parameters at breaking point, which reads with for the present wave group where the subscript b means the value at the breaking point. The corresponding swashing period becomes from , in which the subscript b stands for the value at the breaking point. The approximation has been made in Eq. (9). As we can see in Appendix, the above result is obtained by considering the effect of the radiation stress as two parts: one is the mean set-up , produced by the steady part of the radiation stress , and the other is the oscillating velocity of LFW at the breaking point, , produced by the oscillating part of the radiation stress . Although the result Eq. (9) is obtained by applying some approximations (such as , ), these approximations are acceptable, as they are widely used in investigation of surf zone hydrodynamics (e.g. Bowen et al., 1968;Bowen, 1969).
Substituting Eq. (9) into Eq. (7) gives the expression for the swash index, In application of this formula, the breaking wave height can be determined by the empirical equation given by Le Méhauté and Koh (1967), , where and is the incident wave height and length in deep water. The water depth at the breaking point, , is calculated by . For the present experiment cases, is calculated from the wave parameters on the horizontal bottom by applying the wave energy conservation law with taken as the maximum wave height .
It is seen in Eq. (10) that the value of the depends on beach slope , period of wave group and height of short wave . As listed in Table 1, to account for these dependences, the present experiment was designed to let these parameters vary, especially for the slope , which varies in a large range from 1:160 to 1:20 in order to consider the possible mild slopes. For the simple wave groups of the present experiment, the different values of SI calculated with these parameters will correspond to the two different numbers of generated LFWs, one or two. As indicated by the definition Eq. (7), the critical value of SI for predicting the number of generated out-going LFWs is ought to be around 1.0, that is, if 1, we have , the number of the generated out-going LFWs will be two, the same as the number of the incoming wave groups, because for this case there is no time left for the reflected long wave to meet the incoming long waves over foreshore; On the contrary, if , we have , the number of the reflected long wave will be one, because for this case the reflected long wave will have the time to meet the incoming long waves over foreshore and interacts with it. To describe the determination of this critical SI from the experimental results, Fig. 8 presents the number of out-going crests, , against SI. Because SI increases with the increasing wave height and decreases with the increasing wave group period and beach slope , the data in the figure are given for three cases with fixed (Fig. 8a), fixed (Fig. 8b) and YIN Jing et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 412-423 T g N LFW SI < 0.24 SI > N LFW fixed n (equivalent to fixed ) (Fig. 8c), respectively, each case containing two panels with the upper for Test A and the lower for Test B. The figure shows that for all the panels, the common results are: is two for the range of smaller SI, , and is one for the range of larger SI, 0.69. Between the two critical values of SI, can be one or two, depending on the test condition. So the results can be summarized as: SI ⩽ 0.24 1 or 2, 0.24 <SI< 0.69 (11) This gives the judgment condition for determining the number of generated out-going LFWs, which is based on the present experimental results and theoretical Eq. (10). As this theoretical formula is a simplified one, which only includes the contribution of bound long wave, the corresponding result Eq. (11) is approximate. This is also the reason why the critical values in Eq. (11) is not around 1.0. Actually, if we can make some neglected factors be accounted for, the resulting critical values will be increased and approach 1.0. To show this, the following section presents the effect of breaking point oscillation.

Contribution of breaking point oscillation
It is known that the LFWs are usually classified as two types: the bound long wave (set-down) and free long wave (both being collectively called surf beat). The former is produced due to the nonlinearity of wave motion and is bound to its carrier wave (wave group or irregular wave), and the latter is the released bound long wave by short wave break-δ = 1.0 ing or the forced wave by the shoreward and seaward movements of time-varying breaking point (Schäffer, 1993). The above result for SI does not account for the latter type of LFWs. Actually, if this is done, the result of SI will be improved, as the wave groups considered here have the larger mudulation and this may make the breaking point oscillation have a significant influence on the result. Schäffer (1993) has presented the analytical solutions of LFW produced by breaking point oscillation, but the solutions are not easily applied here for the establishment of a simple expression of the swash index due to the complex form of the solutions. Therefore, here we peruse a simple approach. To this end, we utilize the decomposition of long wave elevation introduced in Appendix: , and we only consider the influence of the breaking point oscillation on , and wave-induced set-up; an analytical expression can be obtained for this case. From the expression of in surf zone (see Appendix), we can obtain the vertical velocity of the mean free surface caused by the oscillation of the breaking point, . Then, we have Here the material derivative means following the moving breaking point, since moves with the breaking point, and the material derivative means following the fluid particle moving with the long wave velocity caused by the movement of the breaking point. The corresponding swashing velocity (run-up velocity at shoreline) can be determined as: in which where has been neglected during performing the material derivative with respect to t due to . The contribution of the above velocity to the swash index is with for the present study. As in Eq. (9), has been approximated by the bottom slope in Eq. (16). The critical value of corresponding to the transit from one to two generated out-going LFWs can be obtained from the critical case with in Figs. 5 and 6, which shows that for fixed and ( , ) a little bit decrease of slope, the slope 1:90, will lead to the transit from one out-going LFW to two out-going LFW. Substituting these values of , and into Eq. (14) yields =0.568. So if we include this result in the critical values in Eq. (10), the contribution of the breaking point oscillation can be accounted for. Notice that is perpendicular to given by Eq. (A7) in Appendix (the former being proportional to and the latter to ), so the incorporation of the critical value of can be carried out by the vector sum of the two velocities, that is, replacing the values of SI in Eq. (10) by the values of , which leads to the changes of 0.69 to 0.89, 0.24 to 0.62. Then, the judgment condition Eq. (11) can be changed into . This result demonstrates the argument mentioned before that the critical values of SI approaches to 1.0 when more contributions are considered. δ Both Eq. (11) and Eq. (17) can be adopted for the judgment of the number of the generated out-going long wave, but the former is simpler: it only needs the information of bound long wave considered, whereas the latter is a little bit complicated; it also involves the breaking point oscillation information, such as the modulation parameter (determining the oscillation amplitude). Eq. (8) or Eq. (9) we can see one of the differences between the swashing motions driven by LFW and by short wave bores: the iniatial velocity or the period of swashing produced by LFW is proportional to the wave group's maximum breaking wave height to the power of 3/2, instead to (as shown by for the swashing motion driven by short wave bores). Furthermore, if the contribution of breaking point oscillation, the velocity given by Eq. (14), is added, the additional proportionality, being proportional to the modulated maximum wave height , will appear in the expression for (or ).

Conclusions
The qualitative description of swashing motion states on mild beach slope, occurrence or no occurrence of LFW's collision, is obtained by relating the states to the number of the generated out-going LFW, one or two, for the case of two incident wave groups. This treatment is based on the feature of swashing motion on mild slope, which is dominated by LFW. The judgment condition for the number of out-going LFWs is established in terms of the swash index , the ratio of the swashing duration to wave group period. A simplified formula of the index is introduced by the theoretical analysis based on the equations governing the motion of LFWs. The related laboratory experiments and corresponding numerical simulations using higher-order Boussinesq equations for different beaches from 1:160 to 1:20 and different incident wave conditions were performed for the determination of the critical value of the index.
The obtained results show that for the swashing motions on mild slope, which are dominated by LFWs, the swashing period is different from that for the swashing motions on steep slope, which are driven by short wave bores: the former is proportional to the wave group's maximum breaking wave height to the power of 3/2, but the latter to (as shown by ). Furthermore, if the contribution of breaking point oscillation is accounted for, the swashing period will be also proportional to the modulated maximum wave height . The present result is further used to explain the generation of LFWs by irregular waves propagating towards mild slope beach, as a time series of irregular wave with narrow-banded spectrum can be seen as a sum of multiple pairs of double wave groups with different frequencies and amplitudes.

Appendix: Initial velocity for swashing motion produced by the LFW
Consider the short wave averaged nonlinear shallow water equations with and U being the surface elevation and velocity of LFW (U is the depth averaged velocity including the Stokes drift), , the mean total water depth, , the radiation stress. For the non-breaking wave, we have the expression for given by Eq. (4) in the main text; for breaking waves, we have , as the wave height of breaking wave is determined by the relation ( is the wave breaking index), and the corresponding. Then, the total expression for is of the form: where is the instantaneous breaking point determined by with h and H being the local water depth and wave height of short wave. Since in Eq. (A3) is oscillatory, the solution to Eqs. (A1) and (A2) is very complicated. As a simple form of the swash index is desired in the present study, here we simplify the problem by fixing the breaking point at its time-mean position: ( is determined by , with being the averaged wave height). This model has been proposed by Schäffer (1993), and to avoid the wave height discontinuity at the fixed breaking point caused by this approach the short wave modulation is allowed to be transformed into the surf zone, i.e. in terms of the variable definitions of the present study, letting the short wave envelope vary as xx cos 2θ ( ) (with the amplitude determined by the local wave height of breaking wave). Here we further simplify this problem by temporally only considering the stationary part of the above , , with the varying part treated, together with the breaking point oscillation outside the mean breaking point, as the effect of the breaking point oscillation and considered separately in the main text. This decomposition is acceptable as least at the regime of linear theory. Thus, the corresponding radiation stress in Eq. (A3) can be expanded into a steady part ( ) plus an oscillating part ( ): with xx cos 2θη (0) S (1) xx cos θ +S (2) xx cos 2θ Corresponding to the above decomposition, the surface elevation can be divided into a stationary part produced by and an oscillating part produced by : . This means that satisfies the equation (the velocity corresponding to is zero) and satisfies the same form of governing equations as Eqs. (A1)-(A3) but with replaced by . So is the wave-induced set-up of the classical definition (Bowen et al., 1968). The merit of the above treatment is that is now zero in surf zone ( ), and the governing equations for can be expressed in characteristic form 422 YIN Jing et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 412-423 [ ∂ ∂t where . Here we have used a constant beach slope, , the case considered in the present study. Eqs. (A6) and (A7) also incorporate the contribution of the set-up , but in an implicit way, embodied in the total water depth , which is equivalent to the change of the mean water depth from h to . It is well known that Eqs. (A6) and (A7) possess the two conservative quantities: , and . The conservation of along can be utilized to establish the relation between quantities at still water line (expressed by the subscript 0) and breaking point (expressed by subscript b), i.e.
The calculation of the second term, , needs the expression for , and we approximate it by , neglecting the contribution of due to its zero mean value, with determined by the solution to Eq. (A8) in surf zone (see Bowen et al., 1968;Bowen, 1969), , where . So we have . can be determined by , but it is negligible compared with and ( , with ). With neglecting , we have .
The time interval in Eq. (A8) for the wave group to propagate from the breaking point to still water shore line can be determined as: where the relation has been used. Then, we have .
Substituting Eq. (A9) into Eq. (A8) with applying obtained above yields U b can be determined by the continuity of the velocity U cross breaking point, and calculated according to the solution outside surf zone. We obtain this solution from the linear version of Eqs. (A1)-(A3) with only considering the component oscillating at wave group period, and have Here we have made the radiation stress be calculated by the shallow water assumption . The bottom slope effect has been assumed to be negligible in Eq. (A11), as the mild slope is considered here (for the mild slope case, a geometrical optics solution for or can be found, which also has the form of Eq. (A11) but with ). Another approximation is neglecting of the homogenous solutions (free LFW), which is determined by the matching condition cross breaking point and is small compared with the bound long wave, the larger magnitude of the latter being seen from the small denominator of Eq. (A11) for shallow water (the near resonant bound long wave near breaking point). Owing to the above approximation, the solution adopted here is only the contribution of the bound long wave. Since in the definition of the swashing period in the main text should be taken as the maximum up-rush velocity of swashing motion, in Eq. (A10) needs to be taken as its maximum, the amplitude in Eq. (A11). Taking this amplitude as in Eq. (A10) yields where we have approximated by in the numerator, but for the denominator is still expressed in the accurate form, , in order to account for the difference in the denominator with enough accuracy. Applying the wave breaking condition gives the initial velocity for swashing motion produced by LFW: As mentioned above, the above solution includes the approximation of omitting the effect of breaking point oscillation. A remedy to this is presented in Section 6 of the main text.