Some Variants of Water Wave Dispersion Equation, Formulated with Small Amplitude Wave Assumption

— This research formulates some dispersion equations with formulating procedure similar to the one in formulating dispersion equation of the small amplitude and long wave theory, i.e. by applying velocity potential equation on the Bernoulli surface equation and Kinematic Free Surface Boundary Condition equation. Furthermore, this research uses non-linear term of the Bernoulli equation, whereas the Kinematic Free Surface Boundary Condition equation is applied with two scenarios, i.e. neglected non-linear term and not-neglected non-linear term Wave length from various dispersion equations that are obtained are then compared with breaker length of the breaker index equation. This research aims only to show that using similar governing equations can be obtained some dispersions equations to produce different wave length.


INTRODUCTION
Wave length is an important parameter of a water wave. Various phenomena in a water wave that are determined by wave length are among others shoaling and breaking, refraction and diffraction, wave force on a structure, and sediment transportation by a wave. Therefore, a dispersion equation that produces an appropriate wave length is needed. Dean (1991) formulated dispersion equation an equation to calculate wave length, using two basic equations, i.e. Bernoulli equation at the surface and Kinematic Free Surface Boundary Condition (KFSBC) equation. In both equations, the non-linear term is neglected, by applying an assumption of small amplitude and long wave. In this research, dispersion equation is formulated using similar governing equation, i.e. Bernoulli surface equation and KFSBC equation by keep applying the small amplitude wave assumption but without applying the long wave assumption. The non-linear term at the Bernoulli equation is still used, whereas the KFSBC is applied in two scenarios, i.e. the neglected non-linear term and the not neglected non-linear term. In the scenario where KFSBC equation is not neglected, the formulation is applied with two different approaches.
The wave length from the resulting dispersion equation is compared with breaker length calculated by breaker index equations from Komar and Gaughan (1972), Mc. Cowan (1894) and Miche (1944). Breaker height is calculated using equation from Komar and Gaughan (1972), using input breaker height breaker depth is calculated withMc.Cowan (1894) equation, using input breaker height and breaker depth, breaker length is calculated using Miche (1944) The  and  functions have an intersection point  where the two functions have similar values, so at that  point the velocity potential equation can be written  By applying an assumption that the wave amplitude is very small then , where is the water surface elevation vis-à-vis still water level, will also be very small, so that the surface pressure can be considered as equal on the entire surface and if the reference is the atmospheric pressure then = 0. By applying an assumption that the wave amplitude is very small, then the velocity of particles and are also very small number so that in Bernoulli equation the second term is much smaller than the third term and therefore can be neglected.
Substitute the potential flow equation, water surface equation is obtained, i.e., For a very small wave amplitude , then ℎ ≪ 1, therefore the last equation becomes, has an average value against time that is equal to zero, then ( ) = 0, the water surface equation becomes, where is the wave amplitude. (6) can be written to be an equation for , i.e.

Applying the Surface Kinematic Boundary Condition
The next constant to be formulated its equation is wave number , which will be formulated using surface kinematic boundary condition equation, i.e.,

International Journal of Advanced Engineering Research and Science (IJAERS)
[ Substitute (11) and (6), = , the following is obtained The water surface elevation of (6) at the characteristic point is, The result of wave length calculation with (14), (20), (21) and with the system of equation (22) and (24) (14), (20), (21) and ((22)+(24)) using a wave with wave period = 8seconds and wave amplitude = 1.0 m. The result of the calculation shows that 14 is the longest, whereas, 21 and 22+24 is more or less equal although 21 is relatively shorter. In addition, there is a constraint at ((22)+(24)), i.e. it cannot be used in a shallow water. Henceforth, ((22)+(24)) can no longer be used.
Wave length calculation is then done with wave period = 8seconds and wave amplitude = 1.0 m in a shallow water, with the result of the calculation as presented in  In the deep water, the difference between 14 and 21 could reach 9.5 %, whereas in shallow water the difference could reach 44 %. Furthermore, the effect of wave amplitude A on (21) will be studied using a wave with wave period of = 8 sec., with various wave amplitudes, i.e. 0.20 m, 0.60 m and 1.0 m, with the result of the calculation as presented in table (3). It shows that the larger the wave amplitude the shorter the wave length. It can be concluded that wave amplitude is to shorten the wave length. To see the effect of the difference in wave length, particle velocity in the direction of horizontal is used with the result of the calculation as presented in table (4), using a wave with wave period = 8 sec., wave amplitude = 1.0 m, and velocity calculated at = −0.25 h. The calculation of the wave number is done using (14), (20) and (21). In table (4) t 14is the particle velocity calculated using wave number from (14), and so forth,  is breaker length. Breaker height was obtained from (24) whereas breaker depth ℎ was obtained from (25).
The calculation of breaker height with (24) requires an input of deep water wave height 0 and wave period for the calculation of deep water wave length 0 . Those two parameters were obtained by applyingWiegel equation (1949,1964 (14), (20) and (21), with the result as presented in table (6), where on the table, 14 is wave length of (14), 20 is wave length of (20) and 21 is wave length of(21).  VI. CONCLUSION As a conclusion, from a governing equation can be obtained e some dispersion equations that produce various wave lengths. The higher the level of the precision, the shorter the wave length. Even using an assumption of small amplitude wave, dispersion equation with wave amplitude as its parameter can be resulted. The influence of wave amplitude is to shorten the wave length. Variety of dispersion equations producing variety of wave lengths require a criteria on the appropriate wave length. One of the criteria that can be used is critical wave steepness, Further research needed is formulating dispersion without applying an assumption of small amplitude wave and by taking into account the criteria of critical wave steepness.