the following formula $$\begin{aligned}
 \left\{\begin{array} {l}
 w(t,x)=w_0\bigl(x_0(t,x)\bigr)=
 \displaystyle{ \frac{\lambda_-+\lambda_+}{2} } 
+ \displaystyle{ \frac{\lambda_+-\lambda_-}{2}\tanh \bigl( x_0(t,x) \bigr)} ,\\[7pt]
 x=x_0(t,x)+w_0\bigl(x_0(t,x)\bigr)\,t.
 \end{array}
  \right.\,\end{aligned}$$ We also note the assumption of the flux function $f$ to be $\lambda'(u)\left( =\frac{\mathrm{d}^2f}{\mathrm{d}u^2}(u)\right)>0$. Now we summarize the results for the smooth approximation $w(\, t,x\: ;\: w_-,w_+)$ in the next lemma. Since the proof is given by the direct calculation as in [@matsu-nishi1], we omit it.

[**Lemma 2.1.**]{}

We define the approximation for the rarefaction wave $u^r\left( \, \frac{x}{t}\: ;\: u_-,u_+\right)$ by $$U^r(\, t,x\: ; \: u_-,u_+) := (\lambda)^{-1} \bigl( w(\, t,x\: ;\: \lambda_-,\lambda_+)\bigr).$$

Then we have the next lemma as in the previous works (cf. [@hashimoto-matsumura], [@liu-matsumura-nishihara], [@matsu-nishi1], [@matsumura-yoshida]).

[**Lemma 2.2.**]{}

Because the proofs of (1) to (4) are given in [@matsu-nishi1], (5) to (7) are in [@matsumura-yoshida] and (8) is in [@yoshida], we omit the proofs here.

We also prepare the next lemma for the properties of the contact wave for $p$-Laplacian type viscosity $U\Bigl(\, \frac{x}{t^{\frac{1}{p+1}}}\,;\,u_- ,u_+ \Bigr)$ defined by (1.11). In the following, we abbreviate “contact wave for $p$-Laplacian type viscosity” to “viscous contact wave”. Substituting (1.12) into (1.13), we rewrite the viscous contact wave as $$\begin{aligned}
\begin{aligned}
U(t,x)&=U\left(\, \frac{x}{t^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right) \\
      &=u_{+}-{\mathlarger{\mathlarger{\int}}}^{\infty}_{x}
             \frac{1}{t^{\frac{1}{p+1}}}\, 
             