= Solution
Let $c_L=(\gamma p_L/\rho_L)^{1/2}$ and $c_R=(\gamma p_R/\rho_R)^{1/2}$, with positive pressures and densities. A nontrivial single <rarefaction wave> has constant <specific entropy> and constant <Riemann invariant>, so for one family label $\sigma=\pm1$ the necessary matching conditions are
$$
\boxed{\frac{p_L}{\rho_L^\gamma}=\frac{p_R}{\rho_R^\gamma}=K,\qquad u_R-u_L=\frac{2\sigma(c_R-c_L)}{\gamma-1},\qquad \sigma(c_R-c_L)>0.}
$$
The final inequality says that the fan opens rather than compresses. Its edge speeds are $\xi_L=u_L+\sigma c_L$ and $\xi_R=u_R+\sigma c_R$, and their difference is $\sigma(c_R-c_L)(\gamma+1)/(\gamma-1)>0$. The profiles in part (a) then interpolate the two end states and supply a sufficient construction. For the left family, $c_R<c_L$, hence $\rho_R<\rho_L$ and $p_R<p_L$; for the right family all three inequalities reverse. In both cases $u_R>u_L$. These are the <single-rarefaction matching conditions for an ideal-gas Riemann problem>.
For a concrete nontrivial example, take $\gamma=5/3$ and
$$
(\rho_L,p_L,u_L)=(1,3/5,0),\qquad (\rho_R,p_R,u_R)=(1/8,3/160,3/2).
$$
The <adiabatic sound speeds> are $c_L=1$ and $c_R=1/2$, with common $K=3/5$ and left-family invariant $u+3c=3$. The fan edges are $\xi_L=-1$ and $\xi_R=1$. The complete <self-similar solution> is uniform at these stated left and right values outside the fan, while inside $-1<\xi<1$,
$$
\boxed{u=\frac34(\xi+1),\qquad c=\frac{3-\xi}{4},\qquad \rho=c^3,\qquad p=\frac35c^5.}
$$
It matches both states continuously, has $u'=3/4=2/(\gamma+1)$, and has constant <specific entropy>. For fixed $x\ne0$, its $t\downarrow0$ limit is the required initial step.
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