This problem is about mathematics models.( from haberman book)

Solution: [1] The wave velocity 1:: dqr'dp, is given by r_=a {ln{prnax}l In{p]I1}I. In particular, for p=pg =1.-"2 prnax, we have c=c0=ailn 21). Thus, the value p=pD metres in time following the characteristic:- 3: =CD tLI'E x9 {I} Since pi-l x43,}=pg for the given initial data. Substituting t= 2 hours in this last formula} gives the desired answer. So, we have:- x = EEC. 1i2xjtns]. 71.]. Experiments in the Lincoln Tunnel (combined with the theoretical work dis- cussed in exercise 63.7) suggest that the traic ow is approximately 9(9) = \"P [1'1 (permit) _ In P] (where a and pm\" are known constants). Suppose the initial density p(x, 0) varies linearly from bumper-to-bumper trafc (behind it = ax\") to no name (ahead of x = 0} as sketched in Fig. 71-6. Two hours later. where does P = pron}!!! Figure '3'1-5. 71.9. Show that p = f (x q'(p)t) satises equation 71.1 for anyr function fl Note that initially p = f (x). Briey explain how this solution was obtained