Analytical and Numerical Modeling of Tsunami Wave Propagation for double layer state in Bore

Tsunami wave enters into the river bore in the landslide. Tsunami wave propagation are described in two-layer states. The velocity and amplitude of the tsunami wave propagation are calculated using the double layer. The numerical and analytical solutions are given for the nonlinear equation of motion of the wave propagation in a bore.


Introduction
Since the 2004 Indian Ocean tsunami disasters, tsunami related research gain momentum among the countries bordering the Indian Ocean such as India, Sri Lanka and Indonesia. The disasters was one of the deadliest natural disasters in human history, more than 2, 30,000 people were killed or missing nearly in 14 countries if Indian Ocean border and it was most deadliest earth quake tsunami was ever recorded in past 25decades or 250 years [ Figure 1].
Tsunami ( Japanese word in the meaning of harbor waves) waves or some called as a seismic sea waves, is a series of waves generated by the displacement of the large amount of water, due natural disasters such as Earthquakes, volcanic eruptions and land sliding. Underwater nuclear explosions is one of artificial reason behind the generation of tsunami waves. Sea shores are less than 25 feet within the 1-2 km are at greater risk even before the warning is issued. Tsunami waves and the receding water are more dangerous to the building in the run-up zone. After tsunami, the drinking water was completely contaminated. Tsunami waves are often referred to as tidal waves (sea or ocean waves), but the major differences between this two are tsunami waves are Earthquakes, volcanic eruptions and land sliding, whereas, tidal waves are generated by wind and gravitation force of moon. Another important different is the size of tsunami waves incomparably bigger than the tidal waves. Out in the depths of the ocean, the high of tsunami are not increasing drastically. When the travels to land, the size rapidly increase to very high value. The speed and height of tsunami waves inversely proportional to the depth of the ocean and it is independent to the distance from the source of the wave. The velocity of Tsunami waves is very high, it may travel as fast as a jet plane (1000 km /hrs) over deep waters, only slowing down when reaching shallow waters.

The motion of the wave fluid as a layer
From the sea floor, Tsunami waves are originated. The arbitrary change of free wave surface and the depth of the ocean are considered as a main variable for equation of motion and the continuity of wave.
The bottom landslide generates water motion from the bottom the ocean. The wave motion is assembled using the variables space, time, density and the layer of the pressure in the defined region. The General equation of wave motion and their continuity are expressed of a two-layer water surface. The single layer water surface flow also established with same parameters.
The amplitude of wave surface for their different states are 1 and 2 , the coninuty of wave surface break between the different states. Generated wave density of lower state and upper state 1 and 2 . The motion of the water is the vertical of the surface of the ocean and also the surface of the ocean is considered as homogeneous. The equation of the wave motion in the upper direction as = − (2.1) The sum of the depth of the defined region in the two states with the amplitude 1 and 2 for the water surface = 1 ( , , ) The wave surface pressure ( 1 ) is same as the pressure in the outer surface of ocean 0 and pressure on the surface of water by wave motion in any depth ( ) is considered as . For the upper surface state For the lower surface state The combined upper surface equation of continuity wave is The overall depth of the upper state is (2.7) The water wave motion for the upper state in axis direction can be expressed as the upper state in axis direction is The water wave motion for the upper state in axis direction can be expressed as the lower state in axis direction is Velocity of the upper state and lower state are denoted as 1 and 2 The circumference of the upper and lower state is 1 and 2 The stress of the water surface along the x direction and y direction as , and , 1 ` √`2 +`2 and 1 ` √`2 +`2 are the elements of the upper state water surface pressure.

Tsunami wave generation for single layer state
For the single state tsunami wave generation by seafloor displacement after the first state of water wave, the convening equation are expressed as Where 1 = 1 + 1 − 2 = = + − is the overall depth of the water surface to the seafloor movement.
The wave motion equation for the upper state of the water surface in the axis direction is defined as Lower state of the water surface in the axis direction is defined as The surface of water is considered in the wind free surface and the friction is omitted in horizontal direction. The tsunami wave is considered along the surface plane direction friction and the small waves occurred in the ocean surface without surface plane direction friction.

Tsunami wave generation for double layer state
The two-state water movement in the wave propagation after the seafloor movement. The equation of the double layer state in the upper and lower state is expressed as

Non-linear shallow water equation approximation
The water wave equation for shallow water surface of the ocean is explained with the motion in the wave channel The resistance of the motion of wave in the sea floor gives the velocity as nonlinear functions The value of `r` taken in the range of (1/100) *(4) based on the coefficient of the resistance of the wave motion The numerical approximation w.r.to to time is given as The distance ℎ assigned for equidistance of the cells for each layer state. The vertical wave motion applied in the first order equations. This method gives the stable solution for the second and higher order with the initial derivative. The initial derivative is constructed using the three -point cell and the four-point cell approach. The depth of the ocean changes due to their rate of water displacement from the seafloor and the density of the water. The sea level N is kept unchanged w.r.to time. The level depends on the variable assigned to each cell of states using the points between and + 1

NUMERICAL AND ANALYTICAL COMPARISON OF WAVE PROPAGATION IN BORE
The numerical and analytical solutions of the wave propagation along the bore are experimented. In this the linear and nonlinear equation parameters are used to calculate the depth, amplitude and velocity of the wave propagation in bore. From the results of the amplitude and velocity the distance of the wave generation and wavelength are estimated. Considered the sea level and the traveling of wave are constant for the analytical approach, = 300 + 4.5 × 10 −4 , = + . The co ordinates applied in this methodology are considered as rectangular form with major axis along the river bore channel. The Equations of wave motion in the bore are considered as continuity for the resistance of the wave movement is given as The Velocity of the bore is not considered for the equation of the ocean depth The solution of the above equation is given as The amplitude of the wave for the defined state is Η and = 2 / 2 .
The wave propagation for the time period 10 sec and amplitude for 150 meters the tsunami wave in to the bore at the 100-meter depth. The solution for both numerical and analytical for the bore in the lower state is given below.
The approximate numerical solution calculated for the non-linear equation for every 0.2 seconds and upper state is at the distance of 40 meters.