我們的論文主要在探討一些退化擬線性波動方程的解的性質。首先我們先探討線性退化波動方程，我們由d'Almbert formula 得到了解具有 L1-stability的性質。而在非線性的例子當中，我們由雙曲線型守恆律的 Lax method 及 Glimm method 得到了柯西黎曼問題在第一階段的估計解。並且在我們的論文當中，我們將會由一些例子，來探討退化擬線性波動方程的估計解的總變異量是否會接近無限大。 In this paper we consider the Cauchy problem of some degenerate quasilinear wave equations. We first study the behavior of solutions to the linear degenerate wave equation. We obtain the -stability of solutions for the linear case just by the d'Almbert formula. To the nonlinear degenerate case, the Lax method and Glimm method in hyperbolic systems of conservation laws are used to construct the approximate solution of Cauchy problem in the first time step. As we demonstrate in this paper, the total variation of approximate solution may go to infinity due to the degeneracy of equation. We will do the case study for the behavior of solutions for some particular case of degenerate quasilinear wave equations.