ESTIMATION OF BOUND AND RELEASED INFRAGRAVITY WAVES BASED ON WAVE OBSERVATION AND NUMERICAL SIMULATION IN SHALLOW WATER

OBSERVATION AND NUMERICAL SIMULATION IN SHALLOW WATER Katsuya Hirayama, Port and Airport Research Institute, hirayama@pari.go.jp Hiroaki Kashima, Ministry of Land, Infrastructure, Transport and Tourism, kashima-h2w7@mlit.go.jp Yoshiyuki Uno, ECOH Corporation, yoshiyuki_uno@ecoh.co.jp INTRODUCTION It is mentioned that observed infragravity waves consist of bound waves propagating with short-wave groups, released waves due to reduction of short crest waves and free waves existing in a field. Though it is difficult to distinguish among them, a standard spectrum for infragravity waves is defined by using a relation to a wind wave spectrum. In this study, a comprehensive definition of standard spectrum is newly proposed to estimate infragravity wave heights with the relation between the ratio of wave height and the ursell number of observed wave property, represented by selected data of wave observation in shallow water. Moreover, the release process of bound waves at a harbor entrance is reproduced in numerical simulation using a Boussinesq model for short-wave transformation. These results are verified by comparison to infragravity waves observed at outside/inside of a harbor for a month. OUTLINE OF METHODS Hiraishi et al. (1997) has proposed an infragravity wave spectrum related to the Bretschneider-Mitsuyasu (BM in short) spectrum for wind wave by a boundary frequency: fba which is decided by an original parameter;l=fp/fba (Here, fp: peak frequency). l is related to the square root of ratio of infragravity wave energy to total wave energy: RL=(m0L/m0)0.5.  df f S m    0 0  df f S m ba f


INTRODUCTION
It is mentioned that observed infragravity waves consist of bound waves propagating with short-wave groups, released waves due to reduction of short crest waves and free waves existing in a field.Though it is difficult to distinguish among them, a standard spectrum for infragravity waves is defined by using a relation to a wind wave spectrum.In this study, a comprehensive definition of standard spectrum is newly proposed to estimate infragravity wave heights with the relation between the ratio of wave height and the ursell number of observed wave property, represented by selected data of wave observation in shallow water.Moreover, the release process of bound waves at a harbor entrance is reproduced in numerical simulation using a Boussinesq model for short-wave transformation.These results are verified by comparison to infragravity waves observed at outside/inside of a harbor for a month.Hiraishi et al. (1997) has proposed an infragravity wave spectrum related to the Bretschneider-Mitsuyasu (BM in short) spectrum for wind wave by a boundary frequency: fba which is decided by an original parameter;l=fp/fba (Here, fp: peak frequency).l is related to the square root of ratio of infragravity wave energy to total wave energy: RL=(m0L/m0) 0.5 .

 df
The relational function between them for not only BM spectrum but also JONSWAP (JS in short) spectrum is newly proposed as Eq. 2.
(2) For BM spectrum: A=0.1227, B=6.3206For JS spectrum (; peak parameter): A=-0.0009 2 +0.0292+0.0967,B= 0.0154 2 -0.3248+6.5601  1 shows that distribution of RL for BM or JS calculated with wave spectra observed at outside of harbor where the water depth is h=16m for a month in every 2 hours.Ursell number: Ur=HsLs 2 /h 3 is also calculated per every 2 hours by averaging the statistical values of wave train observed in each 20 minutes.Here, Hs is significant wave height and Ls is wave length for significant wave period: Ts at h.Each relational function between RL and Ur is estimated as Eq. 3 with the selected data whose peak of wind wave spectrum is single in storm periods.For BM spectrum: RL=0.0058Ur+0.0475(3a) For JS spectrum: RL=0.0061Ur+0.0440(3b) Also, each relational function between l and Ur is estimated as Eq. 4, and it is drawn in Fig. 1 with Eq. 2, respectively.
Fig. 2 shows that each relation between RL=HL/Hs and Ur expressed by Eq. 3a or the combination of Eqs. 2 and 4 for BM, comparing the relation between H2nd/Hs and Ur.
Here, HL is infragravity wave height and H2nd is semitheoretical second-order wave height for JS spectrum with =1 (Kato & Nobuoka, 2005), which is equal to the modified BM spectrum.It indicates that the observed bound wave heights: HLb may be estimated with the wave properties.That is, while Ur is greater than 10, the observed HL can be explained as H2nd (=HLb).However, it is supposed that the free wave heights: HLf are dominant even in the selected data while Ur is smaller.On the other hand, the released wave heights at inside of harbor are estimated with an interpolation matrix obtained by the results of numerical simulation for representative several incident wave conditions.For an example, Fig. 3 shows the distribution of wind wave heights for the offshore wave given at outside of the harbor, whose properties: Hs=2.6m,Ts=12s and principal direction: NNE with directional spreading parameter: Smax=75.The matrix

RL
Ur can be also applied to an estimation of occurrence frequency for harbor oscillation (Hirayama et al., 2015) while another approach is proposed (Lopez et al., 2015).Using a Boussinesq model, the released waves can be calculated because the short-wave groups, which induce the second-order wave-wave interaction (Schaffer, 1993;etc.),are reduced due to reproduction of partial reflection on wave absorbing works and breaking and runup on complex bathymetries.4 shows the comparison between observed and estimated heights of infragravity wave whose frequency is less than fba at outside of harbor.In both cases based on BM and JS, it is recognized with referring to Fig. 2 that the observed HLb can be estimated well because the observed HL are rarely underestimated.Moreover referring to Fig. 1, it can be understood that their overestimation is caused by existing HLf those are dominant while Ur is small.Therefore, HLf observed at outside of harbor can be estimated as: HLf =(HL 2 -HLb 2 ) 0.5 Figure 5 shows the comparison between observed and estimated heights of infragravity wave whose frequency is from 1/300 to 1/30 [Hz] at inside of the harbor.It is recognized that the observed HL can be explained by the released wave heights: HLr, those are obtained in the calculations using the Boussinesq model.Therefore it is surmised that the free waves rarely exist in the harbor.

CONCLUSIONS
This paper mentions the following items: -By using the newly proposed spectrum for infragravity waves, the height of offshore bound waves whose frequency is less than the boundary frequency can be estimated by the ratio of infragravity wave height to wind wave height while ursell number is greater.-By using a Boussinesq model to calculate the reduction of short-wave groups at a harbor entrance, infragravity wave heights in a harbor can be estimated as released wave heights in case that free waves rarely exist there.In a future work, the wave train which consists of both wind and infragravity waves will be generated from the standard spectrum with considering distribution of their direction, in order to estimate infragravity waves those may include free waves in a harbor.
JS spectrum Figure 1 -Distribution of RL on Ur and relation function Considering fba=fp/l estimated with Eqs.1-2 by a convergence calculation, Fig.

Figure 2 -
Figure 2 -Comparison of the ratio of between semitheoretical and observed bound wave height

Figure 3 -
Figure 3 -Distribution of wind wave height in a harbor calculated by using a Boussinesq model to estimate released wave heights model (inside of the harbor)