| Non-linear phenomenon is so usually for the natural world, non-linear science concerning about non-linear common disciplinary in different study fields of science and technology has become the base of investigation of natural disciplinary. In the previous 18th century, science workers had paid more attention to so many non-linear phenomena in actual world. As the development of modern science and technology, non-linear science has obtained rapidly progress and its essence was discovered and explained continually. The research of the nonlinear theory is a contemporary important subject that is obtained prevalent attention, which is the science on the complexity. Nonlinear wave is an important part of it. Recently, the propagation has become more and more important in a large number of fields of physics and engineering science。The earth is a complicated and changing system, so there have complexity and nonlinear, we can't get all data from earthquakes. How to find the biggest degree to use these data and improve dependability, so as to embody great economical benefit in searching now oil fields, it's necessary for us to study the nonlinear wave. The 2-D nonlinear longitudinal wave in the isotropy solid medium studied in this paper is the finite amplitude wave, which thought about the nonlinear of the motion and medium. The finite element numerical method based on centimo theory and dispart-insert-value One side ,it is conventional energy method. It is transfiguration of Ritz-Galerkin method .On the other hand ,it also has some similitude with difference method . Therefore, The finite element numerical method is the unite of energy method and difference method . The finite element disperse keep the symmetry poditive nature of original problem. Even its freshness matrix is sparsity matrix. All of these excellence make mumerical value compute easy. but wave energy look toward to high frequency space, these lead to strong dissipation in the high frequency, accelerating the nonlinear wave dissipation. In this time, the nonlinear is the mainly characteristic, so we reasonable analyze the medium nonlinear action is most necessary. The finite element method is made use of the nonlinear wave equations, along with increment of degree of freedom in unit , the precision of interpolating function and approximate solution have to certain extent difficulty .Every concrete nonlinear wave equation need concrete interpolating function, variant periphery condition need variant precise approximate function. We can put to use variant approximate function in variant unit. We can utilize variant interpolating function in variant periphery condition. By this method, we can heighten precision and quicken the velocity of convergence. we gained the solution of hyperbolic 2-D nonlinear primary wave equation by finite element numerical method .Based on the proof of existence of the exact value ,we have gained the numerical value,analyzed the stability, convergence, and established the error estimate formula. Firstly, the basic theory of nonlinear science was introduced and its influences on the development of science and technology were showed in this paper. Furthermore, the history and its modern status of solutions on non-linear wave equations were shown here.Secondly, in this paper, Physical and mathematical fundamental of the nonlinear wave is introduced, especially, with regard to the nonlinear wave of elastic solid medium. Form power of point of view, the conception of the strain work is introduced. We built 3-D hyperbolic nonlinear evolution and the general motion equation by making good use of the conservation of mass,moment and energy. Meanwhile, the paper builds three-dimension nonlinear wave model in isotropic medium. Ignoring the transverse wave domino effect, 2-D nonlinear Primary wave equation in isotropy solid medium was gained.Moreover ,existence of the exact value have been rigidly proved with functional analysis. Based on it ,we have gained the numerical value,analyzed the stability, con...
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