Breaker Depth Analysis Using Critical Wave Steepness

This research developed a breaker depth equation based on the characteristic of the potential velocity solution of Laplace equation. The breaker length equation was obtained using critical wave steepness as boundary condition. Whereas breaker height was obtained from breaker height index equation. The equation is in the form of linear explicit equation with simple calculation. Keywords— breaker depth, critical wave steepness.

The breaker depth equation developed in this research was obtained by performing one of the conservation characteristics of potential solution wave of Laplace equation, i.e. multiplication between wave number and water depth is constant which means that the multiplication of wave number at breaker depth is similar with the multiplication between wave number and deep water depth. Therefore, there is a relation between breaker depth and the condition of wave at the deep water. There is a breaker length variable at the equation. To eliminate breaker length, the critical wave steepness criteria was performed. Using this method, an equation is obtained based on analysis and the law of conservation.

II. COMPUTATION OF BREAKER HEIGHT H b
To perform a computation using breaker index in the form of and ℎ , breaker height should be known. The breaker height is obtained using breaker height index There are a lot of breaker height index equations. This research uses a breaker height that is the average of several breaker height indexes that produces adjoining breaker height. The breaker height indexes that were used are as follows: Komar and Gaughan (1972)  ....... (6) At those equations is breaker height, 0 is deep water wave height, 0 is deep water wavelength and is bottom slope. Table (1) presents the result of breaker height computation using those 6 (six) equations above and their average values. The wave used is the wave with wave period = 8 . , bottom slope = 0.005 and water depth ℎ 0 = 60 , whereas deep water wave height 0 varies between 0.60 -1.8 m.

III. THE COMPUTATION OF BREAKER DEPTH AND BREAKER LENGTH USING THE EXISTING EQUATIONS.
Relation between breaker depth ℎ and breaker length is in the shape of implicit equation where there is a dependency between those two variables. There is an explicit equation to compute breaker depth, i.e. SPM equation (1984) and Van Rijn equation (2011). However, at Van Rijn there is a parameter that has to be tested, therefore this research used SPM equation (1984). The wavelength computation cannot be done using dispersion equation of linear wave theory, considering at breaker depth the value  Then ℎ can be calculated using (8), with i Newton-Rhapson iteration method.

V. COMPARISON OF THE RESULTS OF THE THREE METHODS
The next section will show the result of breaker depth ℎ and breaker length computation using the three methods mentioned above. The bathymetry data is similar to the computation of in section 2. The result of the computation is as follows.  (1) Couple computation between (7) and (8)  (2) Couple computation between (8) and (9)  (3) Computation with (11) and (13) The result of ℎ and computation using the three methods shows that ℎ produced by (3) , i.e. with the proposed method is quite close with methods (1) and (2). Among the three methods, the smallest breaker wave length is produced by (3)

VI. CONCLUSION
The result of breaker wave steepness computation with breaker wave steepness index equation produces breaker wave steepness value that is more or less constant toward wave period as well as deep water height. This shows the presence of critical steepness wave on a wave curve. The proposed equation uses critical wave steepness criteria. The equation uses wave condition in deep water and in the form of explicit equation that is easy to use. In addition, the equation is an analytical product based on the law of conservation. The critical wave steepness criterion is quite important in the development of a simple breaker index; therefore, a research is needed on critical steepness wave, in the laboratory as well analytical.