SEPARATION OF LOW-FREQUENCY WAVES BY AN ANALYTICAL METHOD

INTRODUCTION Waves in the infragravity frequency band are an important wave motion in the coastal zone. Many researches have shown that low-frequency waves or infragravity waves can lead to resonant responses of harbors (Bowens, 1977) and moored ships (Nagai et al., 1994), have important role in the coastal morphology process (Yu and Mei, 2000), and have significant influence on the design of coastal structures (Kamphuis, 2000). Among the generation mechanisms of these low-frequency waves, release of bound subharmonic waves (Longuet-Higgins and Stewart, 1962) and moving breaking points (Symonds et al., 1982) are widely recognized. Separating low-frequency waves into specific components (i.e. incident free long waves, reflected free long waves and incident bound long waves) is vital to analyze the low-frequency waves and realize the generation mechanism of these low-frequency waves (Baldock et al., 2000). Several methods (e.g., Kostence, 1984; Bakkenes, 2002) are available for separating low-frequency waves. However, based on Fourier transform, the amplitude and phase information in time can not be directly obtained by these methods. In this paper, a new method is proposed for separating low-frequency waves over sloping bathymetries in the real-time domain by constructing the analytical signals of the low-frequency waves measured simultaneously at three spaced locations. Hence, the amplitude and phase information of low-frequency waves in time can be obtained.


INTRODUCTION
Waves in the infragravity frequency band are an important wave motion in the coastal zone.Many researches have shown that low-frequency waves or infragravity waves can lead to resonant responses of harbors (Bowens, 1977) and moored ships (Nagai et al., 1994), have important role in the coastal morphology process (Yu and Mei, 2000), and have significant influence on the design of coastal structures (Kamphuis, 2000).
Among the generation mechanisms of these low-frequency waves, release of bound subharmonic waves (Longuet-Higgins and Stewart, 1962) and moving breaking points (Symonds et al., 1982) are widely recognized.Separating low-frequency waves into specific components (i.e.incident free long waves, reflected free long waves and incident bound long waves) is vital to analyze the low-frequency waves and realize the generation mechanism of these low-frequency waves (Baldock et al., 2000).
Several methods (e.g., Kostence, 1984;Bakkenes, 2002) are available for separating low-frequency waves.However, based on Fourier transform, the amplitude and phase information in time can not be directly obtained by these methods.
In this paper, a new method is proposed for separating low-frequency waves over sloping bathymetries in the real-time domain by constructing the analytical signals of the low-frequency waves measured simultaneously at three spaced locations.Hence, the amplitude and phase information of low-frequency waves in time can be obtained.

SEPARATION PRINCIPLE
The schematic diagram for the separation method is shown in Figure 1.The mean position of the wave maker is defined as x = 0 m.Three wave gauges, which are located at x 1 , x 2 and x 3 , are used to record the surface elevations simultaneously.The space between x 1 and x 2 is ∆x, and the space between x 2 and x 3 1 State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116023, P.R.China is δ∆x, where δ is a nondimensional value larger than 0. In a study of low-frequency waves generated by bichromatic wave groups, which are composed by waves with frequency ω 1 and ω 2 , the low-frequency wave series with frequency ∆ω (∆ω = |ω 1 -ω 2 |) can be obtained by a band-pass filter.Generally, the measured low-frequency waves are composed by the incident free long waves, η I , the reflected free long waves, η R , and the incident bound long waves, η B .The amplitude changes between the measured locations are determined using the linear shoaling theory.Then, the low-frequency waves at the three wave gauges can be expressed as: ) ) , cos cos cos x t ks a t x ks a t x ) , cos cos cos x t ks a t x where κ is the local wave number of free long waves and determined by the linear dispersion relation: ∆k is the wave number of incident bound waves and equal to the difference of the local wave numbers of the two primary waves; a I , a R and a B are the amplitudes of the incident, reflected and bound long waves, respectively, and φ I , φ R and φ B are the corresponding phases.ks 1 is the amplitude ratio between waves at x 1 and x 2 , and ks 2 is the amplitude ratio between waves at x 1 and x 3 , and both of them are determined by shoaling coefficient: It is difficult to separate long waves via Eqs.(1), ( 2) and (3) directly.However, through constructing the analytic signal, the phase changes of waves between the measured locations can be extracted, then, the composed long waves can be separated easily.The analytic signals of the measured long waves can be constructed by the Hilbert transform (Cohen, 1995) or the Morlet wavelet transform (Mallat, 1999) and can be express as: ( ) x t a e a e a e x t ks a e e ks a e e ks a e e x t ks a e e ks a e e ks a e e where i is the imaginary unit, κ c1 , ∆k c1 ∆x, κ c2 and ∆k c2 ∆x are the wave number changes between the measured locations and can be determined by numerical integrals.By further deriving the Eqs.( 6), ( 7) and ( 8), the analytical forms of the incident bound long waves, reflected and incident free long waves at x 1 can be obtained as follows: The real parts of Eqs. ( 10), ( 11), ( 12) are the time series of the separated waves.

TEST OF THE METHOD BY NUMERICAL RESULTS
To examine the efficiency and validity of the present separation method, numerical example with known incident bound, incident and reflected free waves, is numerically generated.In this test, a water shows calculated results using the present method.The results show that the errors are very small and can be neglected.Figure 3 shows the separation results of another numerical case, using the same wave parameters but at a steeper slope, β = 1/10.It is found that the errors for the case at the beach with slope β = 1/10 is more obvious than that at the beach with slope β = 1/40, indicating that the accuracy of the present method is closely related to the slope of beaches.  is the calculated amplitude.The tests show that the errors of the present method are very small over constant depth, even in very shallow water depths.
shows the separation errors over a mild slope beach with β = 1/40.The errors are very small at places where water depth kh is larger than 0.2.But, at places where kh is smaller than 0.2, the errors exceed 10%, thus this method is invalid.The separation errors for steeper slope beaches over variable water depths are shown in Figure 6 and Figure 7.The results illustrate that the separation errors at steeper beaches are larger than that of milder slope beaches.At shallow water depth, closer gauge locations can improve the separation results.However, at very shallow water depth where kh is smaller than 0.2, the present method is invalid, even with a closer spacing gauge location.
The above numerical tests have shown that the present method can give correct results at places where kh is larger than 0.2 over both mild and steep slope beaches.In the next section, this new method will be used to study an experiment for the generation of low-frequency waves over a mild slope beach.The surface elevations on the bathymetry are recorded with 22 capacitance wave gauges; their positions are delineated in Figure 8.The gauges are positioned with a 1.0m or 2.0mspacing across the offshore horizontal profile and across the offshore part of the slope.To increase spatial resolution of the measurements in the vicinity of wave breaking, gauge density is increased to 0.5m near the discontinuity in slope.The control signal to the wave paddle is computed using a second-order wave-maker theory (Schäffer, 1996), which significantly suppresses the spurious free waves.The additional lag ∆φ = φπ can be obtained by a correlation calculation, where φ is the phase lag of the bound long wave behind the short wave envelop.In this study, the additional lags ∆φ = 2τf g , where τ is the lag time of the maximum negative correlation between the bound long waves and the short-wave group envelops.The cross-shore variations of the additional lags for the experimental cases are shown in Figure 13.This shows that before initial breaking, the phase lags between bound long waves and envelops is close to zero.After initial breaking, the phase lags change sharply, where the bound long waves are behind the short waves about π/2, while in the inner surf zone, the phase lags are about π, which is similar to the conclusion of Battjes et al. (2004).

CONCLUSIONS
A new 2D method for separating low-frequency waves in real time domain over sloping bathymetries is proposed by constructing the analytical signals of the measured waves.The efficiency and accuracy of this method are demonstrated using numerically simulated data.The results show that the present method can give accurate results at relative water depth (kh) larger than 0.2.At shallow water depth (kh < 0.2), the errors is larger than 10%, so this method is invalid.
Using the present method, experiments of low-frequency waves generated by bichromatic wave groups over at mild slope beach are studied.The separated results show that the release of bound subharmonic waves is the dominant mechanics of generation of surf beat at mild slope beaches.The study of the local correlations between the incident bound low-frequency waves and the short wave envelopes along the tank shows that the phase lags between bound long waves and envelops is close to 0 before initial breaking, while the phase lags change sharply after initial breaking.The bound long waves are behind the short waves about π/2 at the initial breaking point, while the phase lags are about π in inner surf zone,, which is consistent with the previous experimental studies.

Figure 1 .
Figure 1.Schematic diagram for separating low-frequency waves over a sloping bathymetry.

depth h 1
Figure 2

Figure 2 .
Figure 2. Comparisons between the target and calculated surface elevations of incident bound long waves (a), incident (b) and reflected (c) free long waves at x1 over a sloping beach with β = 1/40.

Figure
Figure 3． Comparisons between the target and calculated surface elevations of incident bound long waves (a), incident (b) and reflected (c) free long waves at x1 over a sloping beach with β = 1/10.

Figure 8 .
Figure 8. Schematic drawing of the experimental setup

Figure 9 .
Figure 9.The time series of the bound long waves at x = 7m for the experimental cases; the red horizontal lines are the theory results of Longuet-Higgins and Stewart (1962); (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 11 and
Figure 11 and Figure 12 show the calculated results at several locations along the beach for Case 1 and 3, respectively.The amplitudes of the bound wave increase as water depth decrease and the time series are almost uniform before wave breaking.After breaking, the bound wave amplitudes decrease and the time series are not uniform anymore.The incident free waves are almost not change before initial breaking, and increase sharply in the surf zone.Meanwhile, the reflected free waves are almost not change along the shore.The results indicate that the release of bound long waves is the main mechanics of generation of low-frequency waves in the mild slope beach, which is consistent with the results of Dong et al. (2009).

Figure 11 .Figure 12 .Figure 13 .
Figure 11.The separation results at several locations along the flume for Case 1；initial breaking occurs at x = 19m.