We give here an existence theorem, see Theorem 1.1,for the solution of the system of equations (1.1)that describes the motion of a compressible inviscid fluid in the half space. Moreover, we establish some sharp estimates, see Theorem 3.2,for the solution of the linear second order hyperbolic mixed problem (3.1)in terms of suitable norms of the coefficients. These estimates play a main rule here and in reference [BV3],where a first proof of Hadamard's classical well-posedness for the above nonlinear system of equations is given; see also [BV4].Here, we adapt and simplify the method followed in our previous paper [BV1].