In 1978 Nishida considered the first order quasi-linear system \(u_ t+p(v)_ x=-2\alpha u\), \(v_ t-u_ x=0\) (\(\alpha\geq 0)\), which describes the propagation of one-dimensional waves in a compressible fluid with dissipation [T. Nishida, Publ. Math. D’Orsay No.7802, 123 p. (1978; Zbl 0392.76065)]. Nishida proved that, under appropriate assumptions of the regularity of both p(v) and the initial data, the system admits smooth global solution. Subsequently M. Slemrod [Arch. Ration. Mech. Anal. 68, 211-225 (1978; Zbl 0393.76004)] proved that if the initial data are not small enough, than global solutions no longer exist. But in some cases, it is interesting to see whether smooth global solutions might exist with large values of data or better, with large oscillations of the initial data. In order to answer this question, the original system must be diagonalized in the form \(s_ t+\mu s_ x=- \alpha (r+s),\) \(r_ t+\lambda r_ x=-\alpha (r+s),\) where r and s are the Riemann invariants, and \(\lambda =\lambda (r-s)<0<\mu =\mu (r-s)\). Then, if the initial data \(r_ 0(x)\), \(s_ 0(x)\) satisfy certain inequalities and certain conditions of regularity integrated by the constitutive assumption \(p(u)=k^ 2/v\), the system admits smooth global solutions.