Water Wave Profile at Breaker Point

In this research, a study was done on wave profile and type at breaker point. The study was done using shoaling-breaking model. Output of the shoaling-breaking model becomes the input at water wave surface equation to obtain wave profile. By obtaining the wave profile, the calculation of breaker height becomes more accurate. Keywords— shallow water wave type, cnoidal wave type.


INTRODUCTION
obtained that there are four types of water wave, i.e. Airy's waves, Stoke's waves, Cnoidal waves and Solitary waves. Airy's type, where wave crest is symmetrical with wave trough and wave height is twice of that wave amplitude. This type of wave is found only in a wave with a very small amplitude. Next is Stoke's type, formed in a wave with wave amplitude bigger than wave amplitude of the Airy's type. In this Stoke's type, the wave profile is somewhat asymmetricand the wave height is still close to twice the wave amplitude. In a bigger wave amplitude, cnoidal type of wave is formed. In cnoidal type, the wave profile is asymmetric where the distance between wave trough and neutral line or still water level is very much different with the distance between wave crest and still water level. For a wave with bigger wave amplitude a perfect Cnoidal profile will be formed with wave trough that almost coincides with still water level. This type of wave is also called Solitary wave. Either Cnoidal type or Solitary wave type shows that almost all part of the wave is above the still water level, nevertheless wave height that is twice wave amplitude might occur. In shoaling-breaking model, some use wave amplitude as the variable and some other use wave height as the variable. For this second type, the formulation was done with an assumption that wave height has a value twice that of wave amplitude. With those four types of wave, then the output of shoaling breaking model that uses wave amplitude as the variable or the model that uses half of the wave heightcan not confirm the wave height. To obtain a certainty of the wave height as the result of shoaling-breaking analysis, the analysis of water wave surface profile needs to be done to obtain a better wave height value. This research used water wave surface equation of Hutahaean (2019a) to conduct wave profile analysis at breaker point. The equation used wave constant , wave number and wave amplitude as the parameter. The three wave parameters were obtained from shoaling-breaking analysis of Hutahaean (2019b). At the same time, this research is also a revision on Hutahaean (2019b), where at the research the wave amplitude as the result of shoaling-breaking model was considered has a value of half wave height. Even though it provides a result that is quite close to breaker height from breaker height index equation as the result of laboratory analysis, the result is less than accurate.

II.
SEVERAL TYPES OF WAVE The general form of water wave profile is shown on Fig.1. The form is assymetrical between wave crest and wave trough. If the wave crest elevation against still water level is called ; whereas the wave trough elevation is called , then there is an asymmetry where > | | , whereas wave height is = − .  In the next sections the analysis of wave profile in the deep water is presented, includes the calculation of , , wave height and wave height ratio and the depiction of the wave profile using equation (1). The result of the calculations shows a compatibility with Wilson criteria (1963).

Airy's Waves Type
Airy's waves type have symmetrical profile where = | | and wave height = 2 . This type of wave is found only in a wave with a very small amplitude. Table (2) shows the measurement of Airy's wave profile in the deep water for several wave periods with a very small wave amplitude A, where = 0.503 was obtained, which shows that it is not really symmetrical, but = 2 was obtained. The profile of Airy's wave can be seen in Fig. 2. The value of = 2.099, is still quite close to 2. Therefore, the characteristics of Stoke's wave is that wave profile is asymmetric but ≈ 2 , although it is at its maximum limit. Therefore, in this cnoidal wave type the wave profile is asymmetric and ≠ 2 . If it is approximated with Airy's waves type there will be a quite large error, in this case is 36.2 %. For a wave with bigger wave amplitude, there will be even bigger error

III. WATER WAVE-SURFACE EQUATION
Ƌ Ƌ Substitute and the first term right side was moved to the left and was moved to the right and both equations were divided by , Ƌ Ƌ The integration of the second term right side of the equation can be completed the same way, but with an assumption is a very small number, the integration can be completed by integrating just the element.
Working on an assumption that is a very small number and can be ignored, In accordance with the provision at the velocity potential equation where there is function only, function onlyand function only, then at the water wave surface using variable from velocity potential equation, water surface

IV. THE CALCULATION OF , AND AT BREAKER POINT
The calculation of , and at breaker point was done using shoaling-breaking model (Hutahaean (2019b)). In principle, the model is a transformation analysis of , and from the deep water to breaker point.

The calculation of and in Deep Water
To calculate and two equations were needed. KFSBC and surface momentum equation, with the formulation can be seen in Hutahaean (2019b).
Wave number in the deep water was calculated using the following equation,  Where wave amplitude A is known.

Shoaling-Breaking Model Equations
In the shoaling-breaking analysis, there are 3 (three) variables that change along with the change in the depth, i.e. , and where there is a dependency among the changes of the three variables.
To calculate the three variables, three conservation equations were used, and the formulation of shoaling-

The Calculation of Breaker Height
As has been stated before, the output of shoaling -breaking model is breaker wave amplitude that must be changed into breaker height . The calculation of was done using water wave surface equation (1). With wave parameter input at breaker point, i.e.
, ℎ , and , the elevation of wave crest , the elevation of wave trough were calculated, then = − . From the example of the result of shoalingbreaking in fig.3, breaker height = 2.522 m was obtained, with the value of = 1.798, where ≠ 2 . The wave profile at breaker point is presented in Fig.4.  /dx.doi.org/10.22161/ijaers.6759  ISSN: 2349-6495(P) | 2456-1908(O) which shows the almost perfect cnoidal shaped of wave profile.

Fig. 4. Wave profile at breaker point
The value of = 1.798is quite close to = 2 .
Therefore in Hutahaean (2019b), model was found to be quite close with the breaker from empirical equation formulated from laboratory observation.

Some of the Model Results
In this section, model was done for several wave periods, with deep water wave amplitude from (6), and the result of the calculation is in Table (5) and Table (    It shows that the numerical result is very close to or similar to the analytical result which also proved that = 1.80is a good value.

VI. VI CONCLUS ION
Wave profile at breaker point is cnoidal-typed wave, where wave crest is not symmetrical with wave trough.
Interpreting the result of shoaling-breaking model should perform analysis on water wave surface profile. As an approximation,a criteriathat at breaker point the relation = 1.8 applies with wave steepness = 0.572 can be used.
Water wave surface equation that was used can produce wave types that corresponds to the Wilson criteria (1963), and compatible enough with the shoaling-breaking model that was used. However, further research is still needed because adjustment should still be done on wave contant of the shoaling breaking model. Further research is still needed in shoaling-breaking model as well as water wave surface equation.
The calculation of wave force in a structure, should take into account the cnoidal-shaped wave profile, so that a better estimation on the location point of wave force is needed. Furthermore, the planning of coastal construction elevation should take into account the asymmetry wave profile so that a parameter as big as half the wave height cannot be used.