Supersonic Speeds, Shock Waves — Question 9

Question 9

Faster trailing compression characteristics steepen a pulse

In a supplied one-dimensional weakly nonlinear model, a rightward characteristic carrying particle velocity $u$ moves at $c+\kappa u$. Let $c=340\,\mathrm{m/s}$ and $\kappa=1.2$. At $t=0$ a leading characteristic is at $x=d=0.50\,\mathrm m$ with $u_1=1.0\,\mathrm{m/s}$, while a trailing one is at $x=0$ with $u_2=3.0\,\mathrm{m/s}$. Treat these values as constant until intersection.

1. Find the catch-up time and intersection position.

2. Explain what the intersection means physically and why the linear superposition model cannot continue through it unchanged.

Question 9 - Solution

Faster trailing compression characteristics steepen a pulse

(a) Method and calculation

The characteristic speeds are $v_1=341.2$ and $v_2=343.6\,\mathrm{m/s}$. Their paths meet when$$d+v_1t=v_2t,\qquad t_s=\frac d{\kappa(u_2-u_1)}=\frac{0.50}{2.4}=0.20833\,\mathrm s.$$$$x_s=v_2t_s=71.5833\,\mathrm m.$$

(b) Interpretation and further calculation

The higher-velocity trailing compression catches the weaker leading one, making the profile steeper. After characteristic crossing, this smooth-wave construction would assign different $u$ values to one event, so it ceases to be a single-valued physical solution. A shock with an appropriate dissipative/conservation description replaces the unresolved steep region. If $u_2\le u_1$, this pair does not catch up for positive time.

Physical check

The catch-up speed is the difference $\kappa(u_2-u_1)$, not the large common sound speed. The latter mainly sets the distance traveled before steepening.