= Solution
For $U<0$ the same positive algebraic root remains real and satisfies $0<V<c_0$, because $V^2-c_0^2=(\gamma+1)UV/2<0$. The jump relations then give a <density> decrease, $\rho_1/\rho_0=V/(V-U)<1$, and a <pressure> decrease. Where both states remain thermodynamically positive, this is an expansion <shock>: its upstream <shock>-frame flow is subsonic and its <entropy> change has the forbidden sign. At sufficiently large negative $U$ the formula even predicts nonpositive <pressure>, an additional reason it cannot represent a physical gas state.
The observed outgoing wave is instead a continuous isentropic <rarefaction wave>. Its plus-family rays spread from the initial piston corner rather than coalescing. The fan head enters the undisturbed gas at $x=c_0t$; for moderate recession the tail is $x=(U+c_1)t$, with the uniform gas next to the piston moving at $U$ and $c_1=c_0+(\gamma-1)U/2>0$. The appropriate <Riemann invariant> is $u-2c/(\gamma-1)=-2c_0/(\gamma-1)$, so inside the fan, with $\xi=x/t$,
$$
u=\frac{2(\xi-c_0)}{\gamma+1},\qquad c=\frac{2c_0+(\gamma-1)\xi}{\gamma+1}.
$$
These expressions show the spreading characteristics in the right panel of the preceding figure. If $U\leq-2c_0/(\gamma-1)$, the gas cannot follow the piston with positive <sound speed>; a <vacuum> gap forms, bounded by the limiting <rarefaction wave> edge. Thus the fan description also identifies the strong-recession limit.
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