\frac{x}{t}\: ;\: w_-,w_+\right)$ of (\[riemann-burgers\]) is explicitly given by $$\label{rarefaction-burgers}
w^r \left( \, \frac{x}{t}\: ;\: w_-,w_+\right) := 
  \left\{\begin{array}{ll}
  w_{-} & \bigl(\, x \leq w_{-}t \, \bigr),\\[5pt]
  \displaystyle{ \frac{x}{t} } & \bigl(\, w_{-} t \leq x \leq w_{+} t \, \bigr),\\[5pt]
  w_+ & \bigl(\, x\geq w_{+} t \, \bigr).
  \end{array}\right.$$ Next, under the condition $f''(u)>0\ (u\in \mathbb{R})$ and $u_-<u_+$, the rarefaction wave solution $u=u^r\left( \, \frac{x}{t}\: ;\: u_-,u_+\right)$ of the Riemann problem (1.2) for hyperbolic conservation law is exactly given by $$u^r\left( \, \frac{x}{t} \: ; \:  u_-,u_+\right) 
= (\lambda)^{-1}\biggl( w^r\left( \, \frac{x}{t} \: ; \:  \lambda_-,\lambda_+\right)\biggr)$$ which is nothing but (1.6), where $\lambda_\pm := \lambda(u_\pm) = f'(u_\pm)$. We define a smooth approximation of $w^r(\, \frac{x}{t}\: ;\: w_-,w_+)$ by the unique classical solution $$w=w(\, t,x\: ;\: w_-,w_+)\in \mathscr{B}^{\infty }( \, [\, 0,\infty )\times \mathbb{R})$$ to the Cauchy problem for the following non-viscous Burgers equation $$\begin{aligned}
\label{smoothappm}
\left\{\begin{array}{l}
 \partial _t w 
 + \displaystyle{ \partial _x \left( \, \frac{1}{2} \, w^2 \right) } =0 
 \, \, \; \; \quad \qquad \qquad \qquad \qquad \qquad  
 (\ t>0,\,x\in \mathbb{R}),\\[7pt]
 w(0,x) 
 = w_0(x) 
 := \displaystyle{ \frac{w_-+w_+}{2} + \frac{w_+-w_-}{2}\tanh x } 
 \qquad \quad \; \:  (x\in \mathbb{R}),
\end{array}
\right.\end{aligned}$$ By using the method of characteristics, we get 