Global uniformly expanding finite amplitude waves (Q913122)

Consider globally defined, uniformly expanding wave solutions of certain finite amplitude wave equations \[ u_ t+(c_ 0+\beta u)u_ r+c_ 0u A'(r)/(2A)=0, \] where u is the particle velocity, r is the range measured along the channel, and \(c_ 0\) is the ambient sound speed, A(r) is a cross sectional area. This paper discusses three cases. For \(0<k<2\), all solutions have a weak discontinuity along the characteristic \(r=c_ 0t\) and all solutions vanish for \(r\leq c_ 0t\). For \(k=2\), everywhere smooth solutions exist, but these vanish too in \(r<c_ 0t\). For \(k>2\) smooth solutions exist and they may not vanish for \(r<c_ 0t\).