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Deformation of bichromatic wave groups based on third order side band solution of Benjamin-Bona-Mahony equation

This paper concerns on propagation of Benjamin Bona Mahony (BBM) wave groups. The previous results, experimental, analytical and numerical, show that nonlinear effects will deform wave groups and may lead to large waves with wave heights larger than twice the original input; the deformation may show itself as peaking and splitting. To investigate this, especially at which location the waves will achieve their maximum amplitude, and to determine the amplitude amplification factor, a concept called Maximal Temporal Amplitude (MTA) is applied. This quantity is a tool that can be used to measure the maximum amplitude of the waves over time. In this paper we will use Benjamin-Bona-Mahony (BBM) model and third order side band approximation theory to investigate the peaking and splitting phenomena of the wave groups which is initially in bichromatic signal. The bichromatic signal here is a signal that is described by superposition of two monochromatic signals with the same value in amplitude but slightly different in frequencies. We present that the waves undergo deformation in their propagation.


Introduction
This research is motivated by the vast interest in studying the behavior of extreme wave. Investigation on extreme wave, also known as freak, rogue, or giant wave, is an active study in the fluid dynamic realm. A wave is called extreme when its height exceeds the significant wave height of measured wave train by factor more than 2.2 [1]. The occurrence of this wave is unpredictable while it is a major threat to the marine transportation and offshore facility due to its impact causing severe damage and loss [2,3]. The study of extreme wave is important in order to investigate its characteristics and to generate it at a certain position in a water tank on its application in the hydrodynamic laboratory activity. The wave is generated to examine the endurance of a model of a marine object in order to prepare it to the worst scenario that may happen in the ocean. Some studies about extreme wave have been conducted experimentally [4,5,6], analytically [7,8,9], and numerically [10,11,12].
Extreme wave can occur in coastal area, shallow water, as well as deep water. One of the physical mechanism of extreme wave generation is modulational instability which is also known as Benjamin- [14,15,16,17] and Kadomtsev-Petviashvili (KP) [18] wave group. The equation has been studied for broad applications in fluid dynamic, quantum electronics, optics, and plasma physics [19,14,20,21,18]. One of NLS' solutions, Soliton on Finite Background, shows that the waves experience modulational instability during its propagation and is a well approximation to the Benjamin-Feir type waves [22,23]. The nonlinear evolution of the trichromatic signal derived from Korteweg-de Vries (KdV) equation was also explained in [7]. Instability also happens to the bichromatic wave type, a superposition of two monochromatic waves with same amplitude and slightly different frequencies. This phenomenon had been investigated numerically and experimentally in [6] and [24]. In [25,26], bichromatic signal is implemented to generate extreme waves by applying third and fifth order theories of KdV equation. The results show that the waves' height keeps increasing until it reaches an extreme position (a spatial position where the wave height reach its maximum). The higher the order, the higher the maximum amplitude. However, the maximum wave heights are still far below the maximum height of wave generated by numerical software HUBRIS which is governed from Laplace equation [24], whereas the extreme positions conform. The same signal is also applied to the Boussinesq equation, which exhibits an amplification on its amplitude with maximum height and extreme position meet the HUBRIS' result [8]. The extreme position can be determined by using a quantity called Maximal Temporal Amplitude (MTA). MTA measures the maximum value of the wave height at every spatial position during the observation time [25,26,8].
This paper focuses on the propagation of wave group governed from the Benjamin-Bona-Mahony (BBM) equation. The solution will be presented as an asymptotic expansion up to the third order. It will be determined a solution as wave groups which represent bichromatic signal as the boundary condition at x = 0. The bichromatic signal is implemented to observe the wave deformation along the propagation. The extreme position and amplitude amplification will be investigated through the wave's MTA.

Third Order Theory of BBM Model
Benjamin-Bona-Mahony (BBM) equation is known as the regularized long wave equation. It was proposed as the a revised model of KdV equation, at which BBM can better explain the physical interpretation of the propagating wave. The model is given by the following normalized equation as what stated in [27], where η is elevation, x and t are respectively space and time variables. The solution of (1) will be developed in to the asymptotic expansion up to the third order: To anticipate the secular terms which may occur, we expand the wave number into the asymptotic series based on Linstedt-Poincare method: for each p = 1, 2. By substituting (2) and (3) into (1),we obtain an equation equivalent to the linear form of equation (1) for the first order equation O(ε). For this linear equation, we choose an ansatz where a p represents amplitude, θ p = ω p t−k p x+ψ p for i = 1, 2., ω p is frequency, k p is wavenumber, ψ p is phase shift, and c.c. is the complex conjugate. Equation (4) On the second order O(ε 2 ), there appears a resonant term which yields the same solution as the homogeneous solution of the equation (1). To eliminate this term, we set k where p needs to be assigned to treat another resonant term. The value of the correction wave number reads where t = (p mod 2) + 1, for p = 1, 2. Hence, the equation of O(ε 3 ) gives a solution a p a q a r B − e i(θp+θq−θr) + c.c.
where Furthermore, in order to acquire the bichromatic signal at x = 0, the second and third order solution needs to be compensated by their free waves. Free waves for second and third order side band are given by where ϕ(θ p ) = ω p t − Ω −1 (ω p )x + ψ p . Therefore, the final solution is
For the third order solution, the equation (8) where ∆θ = θ 1 − θ 2 . Equation (14) is waves consisting frequencies whose value are far from the frequencies of bichromatic waves (4), so are the first four terms in the equation (14). These waves will not be considered since they will not affect the shift of the wave train. Meanwhile, the last two terms of equation (14) have frequencies which are close to the frequencies of bichromatic waves (4). These waves will interact with the bichromatic waves creating a wave train and causing deformation along the propagation. These waves will be denoted as side band waves. The third order side band wave and its free wave are stated as follow, The bichromatic waves η (1) and third order side band η (3) sb form wave groups which have envelopes with wave number κ. These waves will have the same maximum amplitude's position. Therefore, the extreme position may be determined through the superposition of these waves with the side band free wave η (3) sb,f w . While the envelope of side band η (3) sb,f w has wave number K = 1 2 (Ω −1 (ω 1 + ∆ω) − Ω −1 (ω 2 − ∆ω)) where ∆ω = ω 1 − ω 2 , the wave length of the envelope is λ = 2π/| (κ − K) /2|. Thus, the position of the amplitude of the wave train is x max = 2π/| (κ − K) |. However, this value only applicable when λ is the wave length of the MTA.

Maximal Temporal Amplitude and Amplitude Amplification
In this section, the evolution of the wave train will be observed. First, all the variables and parameters need to be transformed to the physical measure. The BBM equation presented in (1) is in nondimensional form with the following scaling factors for the variables as well as the corresponding frequency and wave number: The extreme position and amplitude amplification will be observed through the Maximal Temporal Amplitude (MTA). MTA is defined as [25] M T A(x) = max t η(x, t).
We consider a water tank with water depth 5 m and gravitational acceleration 9.8 m/s 2 . Taking q = 0.4 m gives the initial amplitude of the bichromatic signal at 0.8 m, ω 1,lab = 0.57 rad/s, ω 2,lab = 0.53 rad/s gives corresponding wave numbers k    figure 3. At the initial position, the signal is formed by two linear signals with slightly different frequencies ∆ω = 0.04 rad/s and the same amplitude. Furthermore, at the extreme position, a pair of side bands with smaller amplitudes have appeared. The side bands are contributed by the third order solutions with supporting waves' frequencies ω 1 + ∆ω = 0.61 rad/s and ω 2 − ∆ω = 0.49 rad/s. However, although the wave train deforms and peaks, its amplitude amplification factor is not enough to satisfy the definition of an extreme wave as stated in [1]. The maximum position and peaking are affected by the value of frequency difference and the initial amplitude. Table 1 presents the maximum position where the wave achieves the maximum peaking and the amplitude amplification factor (AAF) for some values of frequency difference. The closer the both frequencies, the higher the AAF, but the farther the maximum position. This fact however can be a problem in generating an extreme wave with the highest amplification in a wave tank because the extreme position may exceed the tank's spatial limitation. Meanwhile,

Conclusion
We have derived a solution of BBM equation on the basis of bichromatic wave group to investigate the occurrence of extreme wave. The initially bichromatic signal experiences focusing phenomena which results a peaking on its amplitude. The extreme position of the the maximum peaking can be determined analytically for a certain case. However, the amplification factor and the extreme position can be observed by using the MTA. The experimental results of the MTA suggest that the frequency difference value will affect the maximum position and the amplitude amplification. The closer the frequencies of the bichromatic waves, the higher the amplitude amplification, yet the farther the maximum position. Meanwhile, the initial amplitude change does not cause significant shift in the maximum position but it does affect the amplification factor. Higher amplitude may trigger higher AAF and vice versa.