# Low Frequency Waves Detected in a Large Wave Flume under Irregular Waves with Different Grouping Factor and Combination of Regular Waves

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

_{50}) of 0.25 mm, with a narrow grain size distribution (d

_{10}= 0.154 mm and d

_{90}= 0.372 mm) and a measured settling velocity (w

_{s}) of 0.034 m/s. The experimental profile and equipment distribution is presented in Figure 1. The x-coordinate origin is at the wave paddle at rest condition before starting the waves and positive toward the shoreline. The movable bed profile started after 31 m of concrete with a section 1:20 slope from x = 31 to 37 m prior to a plane bed, from x = 37 to 42 m, followed by a 1:15 slope plane beach (Figure 1). Prior to running each wave condition, the water depth was performed by manual reshaping and then compacted by running 10 minutes of ‘smoothing’ wave conditions, in order to return almost the same initial profile. The water depth at the toe of the wedge-type wave paddle was 2.5 m [44].

- random waves with different Grouping Factors (GF);
- combination of free partial standing long waves plus monochromatic short waves (hereinafter, combined waves);
- regular monochromatic;
- bichromatic waves (including bound long waves).

^{2}≈ 0.13 for 500 waves. [48,49,50,51]. Greater values of GF suggest that small waves tend to be succeeded by small waves, and large waves by other large waves.

## 3. Methods

_{SZ}, bed elevation data have been corrected. In fact, it is worth mentioning that the total beach volume of each profile was not the same. Due to profiler measurement errors, in particular the inability to accurately measure ripple volumes and some non-uniformity of the profile, calculation of ΔV along the whole profile never returns identically zero. Therefore, errors in the calculated ΔV were corrected by distributing the mismatch in sediment volume along the whole profile, leading to a zero value of ΔV. Generally, the error distributed is of a few millimeters, hence the correction does not significantly affect the volume computation. This approach derives from the method proposed by and Baldock et al. [18] for calculation of the net time-averaged sediment transport. The analysis assumed a depth of closure for the sediment transport calculations at x = 60 m (at a water depth of approximately 1 m), and applied the sediment continuity correction over the active profile, that is, landward of x = 60 m or shallower than 1 m. This enables greater resolution and improved accuracy in the ΔV calculations. Closure errors corresponded to a mean error in vertical elevation across the profile that ranged from 3 mm to 15 mm, with an average of 9 mm over all tests, which is consistent with the estimated accuracy of the bed profiler. However, other methodologies could be applied to reconstruct transverse profiles and to study coastal evolution, in particular performing a comparison of aerial photographs according to Muñoz-Pérez et al. [52].

#### 3.1. Spectral Analysis

#### 3.2. Eigen Analysis

^{2}, and h is the average water depth.

^{2}/g represents the Eigenvalue for the problem and ω the angular frequency. Equation (3) is finite differenced using centered second-order derivatives. The corresponding expansion, orthogonality condition, and dispersion relation are given by:

_{n}and F

_{m}represent the family of eigenmodes of order n and m respectively.

_{1}, F

_{2}, F

_{3}and F

_{4}, have been determined by solving numerically Equations (3)–(6).

## 4. Results and Discussion

^{2}/Hz) and water surface elevation signal plots (related to the wage gauge WG5 at 21.58 m in the wave flume) are presented in Figure 4.

## 5. Additional Considerations

_{1}–F

_{4}) for volume flux in the wave flume, shown in Figure 9 and Figure 10. Mode periods are computed by the eigen analysis based on water depth z. Very interesting results can be identified:

- A strong non-linear pattern of F
_{1}–F_{4}is identified for all the tests in proximity of breaking zone; - Clear opposite behaviours of volume flux eigenmodes are shown for accretive and erosive wave conditions in the case of random and combination waves, except for the test CE_2;
- A different variation of the Eigenmodes for the combination tests in the erosive, clearly, due to the non-linearity effects;
- The monochromatic wave perturbed with the larger long waves for the erosive condition (CE_1) has an opposite variation of the Eigenmodes, than the monochromatic wave perturbed with smaller long waves (CE_2).

- specific eigenmode of the wave flume (generated seiches) induces spreading or downshift of carrier wave frequency, as foreseen;
- grouping of short waves in the inner surf zone could directly induce low-frequency oscillations of the shoreline.

#### 5.1. Influence on Morphodynamic

_{SZ}, in the SZ (here approximated as the emerged beach) between the test start and end. These were obtained using the changes in bed elevation between profiles, above z = 0. Focus on the emerged SZ, during the analyzed first step of experiments, all wave conditions lead to a landward net sediment transport, except cases RE_2 and CE_1. For these wave conditions, the energy density associated with the second harmonic has been found about ten times the power level of the first mode, increasing the disturbance effect of the 2nd harmonic itself. Although erosive random waves have the same wave height and mean period, their morphological effect is quite different. It is noted that the grouping factor could promote the presence of multiple low-frequency motions responsible of nonlinear interactions. For this reason, there is a significant contribution that determines the evolution of surface changes. Unfortunately, the case CE_1 formed part of the tests that developed lateral cross-flume asymmetry; this does not significantly alter the wave height on WG 5, but the profile correction may not be sufficient for SZ sediment volume here computed. Hence, results for that case are not reported in Figure 12.

#### 5.2. Influence on Swash Hydrodynamics

- “accretive” conditions do not necessarily involve smaller runup;
- despite comparable energy levels, random waves give a runup twice higher than combination cases;
- the higher the grouping factor the higher the maximum runup.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Longitudinal cross-section of the Catalonia University of Technology (UPC) flume and detail of acoustic wave gauges and wave gauges location (length in m).

**Figure 4.**Power spectral density and Time series water surface elevation for test: (

**a**) RA_1 (GF = 0.96); (

**b**) RA_2 (GF = 1.08); (

**c**) RE_1 (GF = 1); (

**d**) RE_2 (GF = 1.1); (

**e**) CA_1; (

**f**) CA_2; (

**g**) CE_1 and (

**h**) CE_2.

**Figure 7.**Frequencies plot of measured versus calculated Eigenmodes with formula for rectangular basin for random tests in the case of accretive and erosive condition.

**Figure 8.**Frequencies plot of measured versus calculated Eigenmodes with formula for rectangular basin for combination tests in the case of accretive and erosive condition.

**Figure 9.**Eigenmodes for volume flux q related to the profile measured in random test: (

**a**) RA_1; (

**b**) RA_2; (

**c**) RE_1 and (

**d**) RE_2.

**Figure 10.**Eigenmodes for volume flux q related to the profile measured in random test: (

**a**) CA_1, (

**b**) CA_2, (

**c**) CE_1 and (

**d**) CE_2.

**Figure 11.**Net sand volume variation in the emerged swash zone (water depth > 0) after 24 min of wave generation: positive values represent accretion or landward transport; negative values represent erosion or seaward transport, for random tests.

**Figure 12.**Net sand volume variation in the emerged swash zone (water depth > 0) after 24 min of wave generation: positive values represent accretion or landward transport for combination tests.

Test | H (m) | T (s) | Wave Type |
---|---|---|---|

CA_1 | 0.226 0.038 | 6 30 | Combination |

CA_2 | 0.226 0.038 | 6 15 | Combination |

RA_1 | 0.319 | 6.7 | Random GF = 0.96 |

RA_2 | 0.319 | 6.7 | Random GF = 1.08 |

Test | H (m) | T (s) | Wave Type |
---|---|---|---|

CE_1 | 0.370 0.038 | 3.7 30 | Combination |

CE_2 | 0.370 0.038 | 3.7 15 | Combination |

RE_1 | 0.530 | 4.1 | Random GF = 1 |

RE_2 | 0.530 | 4.1 | Random GF = 1.1 |

Test RA_1 | Harmonics | f (Hz) | T(s) | E (m^{2}/Hz) |

1st harmonic | 0.02319336 | 43.1157895 | 0.00059542 | |

2nd harmonic | 0.04516602 | 22.1405405 | 0.00058067 | |

3rd harmonic | 0.09033203 | 11.0702703 | 0.00155648 | |

Test RA_2 | 1st harmonic | 0.02441406 | 40.9600042 | 0.00052997 |

2nd harmonic | 0.04516602 | 22.1405405 | 0.00089923 | |

3rd harmonic | 0.08544922 | 11.7028571 | 0.00267999 |

Test RE_1 | Harmonics | f (Hz) | T(s) | E (m^{2}/Hz) |

1st harmonic | 0.02197266 | 45.5111111 | 0.00211399 | |

2nd harmonic | 0.04760742 | 21.0051282 | 0.00310199 | |

3rd harmonic | 0.08056641 | 12.4121212 | 0.00424879 | |

Test RE_2 | 1st harmonic | 0.02319336 | 43.1157895 | 0.00041757 |

2nd harmonic | 0.04516602 | 22.1405405 | 0.0038171 | |

3rd harmonic | 0.08911133 | 11.2219178 | 0.0040941 |

E (m^{2}/Hz) | RE_1 | RE_2 | RA_1 | RA_2 |
---|---|---|---|---|

total | 1.22 × 10^{1} | 1.20 × 10^{1} | 3.37 × 10^{0} | 4.00 × 10^{0} |

<0.1 Hz | 1.45 × 10^{−1} | 1.19 × 10^{−1} | 3.97 × 10^{−2} | 6.34 × 10^{−2} |

<0.03 Hz | 2.18 × 10^{−2} | 6.68 × 10^{−3} | 5.26 × 10^{−3} | 5.07 × 10^{−3} |

Test CA_1 | Harmonics | f (Hz) | T(s) | E (m^{2}/Hz) |

1st harmonic | 0.032958984 | 30.34074074 | 0.01233396 | |

2nd harmonic | 0.065917969 | 15.17037037 | 0.00001770 | |

3rd harmonic | 0.108642578 | 9.20449438 | 0.00004718 | |

Test CA_2 | 1st harmonic | 0.025634766 | 39.00952381 | 0.00002758 |

2nd harmonic | 0.041503906 | 24.09411765 | 0.00000737 | |

3rd harmonic | 0.067138672 | 14.89454545 | 0.02348628 |

Test CE_1 | Harmonics | f (Hz) | T(s) | E (m^{2}/Hz) |

1st harmonic | 0.03295898 | 30.34074074 | 0.00314475 | |

2nd harmonic | 0.06713867 | 14.89454545 | 0.00000949 | |

3rd harmonic | 0.10805664 | 12.41212121 | 0.00002413 | |

Test CE_2 | 1st harmonic | 0.023193359 | 43.11578947 | 0.00000319 |

2nd harmonic | 0.040283203 | 24.82424242 | 0.00000234 | |

3rd harmonic | 0.067138672 | 14.89454545 | 0.00393993 |

E (m^{2}/Hz) | CE_1 | CE_2 | CA_1 | CA_2 |
---|---|---|---|---|

total | 1.11 × 10^{1} | 1.32 × 10^{1} | 3.43 × 10^{0} | 3.17 × 10^{0} |

<0.1 Hz | 2.53 × 10^{−2} | 3.14 × 10^{−2} | 9.80 × 10^{−2} | 1.86 × 10^{−1} |

<0.03 Hz | 4.50 × 10^{−3} | 1.35 × 10^{−4} | 1.81 × 10^{−2} | 3.83 × 10^{−4} |

**Table 9.**Eigen values calculated through the formula proposed by Merian [55].

Mode | f (Hz) |
---|---|

1st | 0.02474874 |

2nd | 0.04949747 |

3rd | 0.07424621 |

Test | Measured (m) | Test | Measured (m) |
---|---|---|---|

R_E1 | 0.36 | R_A1 | 0.30 |

R_E2 | 0.41 | R_A2 | 0.37 |

C_E1 | 0.16 | C_A1 | 0.18 |

C_E2 | 0.12 | C_A2 | 0.23 |

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**MDPI and ACS Style**

Riefolo, L.; Contestabile, P.; Dentale, F.; Benassai, G.
Low Frequency Waves Detected in a Large Wave Flume under Irregular Waves with Different Grouping Factor and Combination of Regular Waves. *Water* **2018**, *10*, 228.
https://doi.org/10.3390/w10020228

**AMA Style**

Riefolo L, Contestabile P, Dentale F, Benassai G.
Low Frequency Waves Detected in a Large Wave Flume under Irregular Waves with Different Grouping Factor and Combination of Regular Waves. *Water*. 2018; 10(2):228.
https://doi.org/10.3390/w10020228

**Chicago/Turabian Style**

Riefolo, Luigia, Pasquale Contestabile, Fabio Dentale, and Guido Benassai.
2018. "Low Frequency Waves Detected in a Large Wave Flume under Irregular Waves with Different Grouping Factor and Combination of Regular Waves" *Water* 10, no. 2: 228.
https://doi.org/10.3390/w10020228