REFLECTING CAPABILITY OF SUBMERGED BREAKWATERS

Yong-Sik Cho¹, Bong-Hee Lee² and Tae-Hoon Yoon³

¹Assistant Professor, Department of Civil Engineering, Hanyang University

17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea

(tel) 82-2-2290-0393, (fax) 82-2-2293-9977, (e-mail) ysc59@email.hanyang.ac.kr

²Graduate student, Dept. of Civil Engineering, Hanyang University,

17 Haengdang-dong Seongdong-gu, Seoul 133-791, Korea

³Professor, Department of Civil Engineering, Hanyang University

17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea

Abstract: It is widely known that the submerged breakwater can be used to protect coastal structures and control beach erosion. In this study, a semi-theoretical model based on the eigenfunction expansion method is firstly employed to describe diffraction of waves due to depth changes. Reflecting capability of four different shapes of submerged breakwaters is then estimated and compared by using the model. Both normally and obliquely incident waves are taken into consideration. An efficient and economical shape is recommended for the submerged breakwater.

Keywords: water waves, submerged breakwater, reflection, eigenfunction expansion method

1  INTRODUCTION

The water waves, traveling from offshore to the nearshore zone, may experience diffraction, refraction, reflection, shoaling and breaking resulted primarily from combined effects of irregular bottom topographies, interference with artificial coastal structures and interactions among different harmonic components. Among these physical phenomena, the reflection is believed to play a significant role in formation of multiple offshore ripples frequently observed on the mildly sloping beaches.

Although many studies have recently been carried out on the Bragg resonant reflection of the monochromatic water waves over sinusoidally varying topographies (e.g., O’Hare and Davies, 1992; Liu and Cho, 1993; Cho and Lee, 2000), only few studies have been reported on the application of Bragg resonant reflection to a design of submerged breakwaters minimizing unnecessary beach erosion (Mase et al., 1997; Cho and Lee, 2000). The Bragg resonant reflection, simply Bragg reflection, occurs when the wavelength of incident periodic waves is approximately twice that of a sinusoidally varying bottom topography. Thus, the distance between two adjacent ripples may be in the range of several tens of meters for offshore bars mainly formed due to wind waves.

Recently, Cho and Lee (2000) present a study for the resonant reflection of monochromatic water waves propagating over singly- and doubly-sinusoidally varying topographies. The diffraction of water waves due to a depth change is formulated by using the eigenfunction expansion method and a theoretical model is then developed. The model is first verified by calculating reflection and transmission coefficients over submerged trenches. The computed reflection coefficients of waves over sinusoidally varying topographies are compared with available laboratory measurements and numerical solutions of the mild-slope and the extended mild-slope equations. A very reasonable agreement is observed.

By adopting Cho and Lee’s study, the reflection coefficients of obliquely as well as normally incident monochromatic water waves over a series of submerged breakwaters are calculated in this study. The shapes of submerged breakwaters are triangular, rectangular, trapezoidal and semi-circular. The submerged breakwaters except the rectangular shape are first represented by a finite number of small steps. An efficient and economical shape of submerged breakwaters will then be suggested.

In the following section, the diffraction of monochromatic waves and eigenfunction expansion method are briefly described for completeness. In section 3, reflection coefficients of monochromatic water waves for various shapes of the submerged breakwaters are then calculated and compared. The obliquely as well as normally incident waves are considered. Furthermore, the effects of propagating and evanescent modes are included. Finally, concluding remarks are made in section 4.

2  DIFFRACTION OF MONOCHROMATIC WATER WAVES

The diffraction of water waves due to an abrupt depth change is briefly described for completeness. As plotted in Fig. 1, the depth change is confined to x-direction and thus, the wavenumber in y-direction remains constant due to Snell’s law. Furthermore, the varying topography is connected to constant depths denoted by h at both ends. Fig. 1 also illustrates a representation of an arbitrarily varying topography by a series of small steps as similarly to O’Hare and Davies(1992) and Cho and Lee(2000).

Fig. 1  A schematic representation of a varying topography into a series of small steps

Following Cho and Lee(2000), the velocity potential excluding time and -direction dependent term  at each region is expressed as

                                    (1)

for the right-going wave components and evanescent modes and

                                       (2)

for the left-going wave components and evanescent modes. In equations (1) and (2), , ,  and  are complex amplitude functions to be determined and subscripts  and  represent the number of regions and number of evanescent modes to be considered, respectively. In equations (1) and (2),  is the angular frequency,  and  are wavenumber components of -direction for propagating and evanescent modes, respectively. They are calculated from the relations given as

                                                       (3)

in which is the wavenumber component of -direction. As mentioned previously, remains constant over the whole domain because the variation of the bottom topography is limited only to -direction. The wavenumbers denoted by  and  are estimated by the linear dispersion relations given as

                                                   (4)

To solve equations (1) and (2) two matching conditions are required at each depth discontinuity. The first condition is expressed as

                                                    (5)

which ensures the continuity of horizontal fluxes along a depth discontinuity. The second matching condition is written as

                                                     (6)

which guarantees the continuity of pressures along a depth discontinuity. A detailed description of the eigenfunction expansion method including the matching conditions can be found in Kirby and Dalrymple(1983).

3  REFLECTING CAPABILITY OF SUBMERGED BREAKWATERS

In this section, the theoretical model is applied to investigate the reflection coefficients of monochromatic wave propagating over various submerged breakwaters. As shown in Fig. 2, triangular, rectangular, trapezoidal and semi-circular shapes of breakwaters are employed and tested. The number of submerged breakwaters denoted by  is three, that is .

Reflection coefficients of four different submerged breakwaters consisted of 3 arrays are firstly calculated and compared in Fig. 3. The incident angle of waves is fixed at and dimensions of breakwaters are listed in Table 1. Not only propagating mode but also 4 evanescent modes are taken into consideration throughout the present study. The trapezoidal shape gives the highest peak as shown in Fig. 3. The trapezoidal shape also provides a competitive surface area as shown in Table 2. In other words, the construction cost may be cheaper than the rectangular and semi-circular shapes. Furthermore, the trapezoidal shape produces the second longest wetted perimeter. That is, the trapezoidal shape may dissipate more energy of incident waves than triangular and semi-circular shapes. Thus, the trapezoidal shape is recommended for the submerged breakwater in this study.

Reflection coefficients of trapezoidal submerged breakwaters are then calculated and compared in Fig. 4. The coefficient increases as the number of breakwaters increases as also confirmed by Liu and Cho(1993) and Cho and Lee(2000). The relative water depth providing the peak moves rightward as the number of breakwaters increases. Moreover, the wave energy distribution becomes more compact as the number of breakwater increases. Since the relative water depth,  is varying from  to , the incident waves experience from shallow to deep water depths.

Fig. 2  Four different shapes of submerged breakwaters

Fig. 3  Comparison of reflection coefficients for various submerged breakwaters

              Table 1  Dimensions of submerged breakwaters

            Table 2  Perimeters and areas of submerged breakwaters

Fig. 4  Reflection coefficients of trapezoidal submerged breakwaters ( )

Finally, the effects of incident angles are investigated in Fig. 5. As the incident angle  increases, the maximum reflection moves towards right direction and the magnitude decreases. Dalrymple and Kirby(1986) reported that the wavenumber of obliquely incident waves providing the Bragg resonant reflection is given by

                                                             (7)

in which  represents the wavenumber of a sinusoidally varying topography. Thus, the maximum reflection occurs  for ,  for  and  for . The relative wavenumbers defined as  providing peaks are ,  and  for , and , respectively.

Fig. 5  Reflection coefficients of trapezoidal submerged breakwaters ( )

4  CONCLUDING REMARKS

In this study, a trapezoidal shape is recommended as a shape of submerged breakwaters by calculating and comparing reflection coefficients of four different shapes of breakwaters. Although the trapezoidal shape does not produce the least sectional area and longest wetted perimeter, it reflects more wave energy than any other shape. Thus, the trapezoidal shape is chosen in terms of economical and efficient views.

The recommended trapezoidal submerged breakwater can effectively and economically be used to control beach erosion. Furthermore, the submerged breakwater can protect man-made coastal structures by reflecting a considerable amount of energy of incident waves.

Acknowledgements

This work presented in the paper was supported by funds of National Research Laboratory Program from Ministry of Science and Technologyin Korea. Authors wish to appreciate the financial support.

References

Cho, Y.-S. and Lee, C., 2000. Resonant reflection of waves over sinusoidally varying topographies, Journal of Coastal Research, Vol. 16, pp. 870-876.

Dalrymple, R.A. and Kirby, J.T., 1986. Water waves over ripples, Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 112, pp. 309-319.

Kirby, J.T. and Dalrymple, R.A., 1983. Propagation of obliquely incident water waves over a trench, Journal of Fluid Mechanics, Vol. 133, pp. 47-63.

Liu, P.L.-F. and Cho, Y.-S., 1993. Bragg reflection of infragravity waves by sandbars, Journal of Geophysical Research, Vol. 98, pp. 22733-22741.

Mase, H., Kimura, A and Sakakibara, H., 1997. Resonant reflection and refraction-diffraction of surface waves of due to porous submerged breakwaters, Proceedings of the 25th International Conference on Coastal Engineering, pp. 2366-2376, USA.

Mei, C.C. and Liu, P.L-F., 1993. Surface waves and coastal dynamics, Annual Review of Fluid Mechanics, Vol. 25, pp. 215-240.

O’Hare, T.J. and Davies, A.G., 1992. A new model for surface-wave propagation over undulating topography, Coastal Engineering, Vol. 18, pp. 251-266.