§30 General solution

Having found values for the pressures $p_3^\star$ and $p_4^\star$ as a function of the initial conditions $p_1$, $p_2$, $\ensuremath{\mathit{V}}_1$ and $\ensuremath{\mathit{V}}_c$, the problem is completely solved. Indeed, using the shock adiabatic relation $\ensuremath{\mathit{V}}_2$ is known. With this, the values of $\ensuremath{\mathit{V}}_{3'}^\star$, $\ensuremath{\mathit{V}}_{3}^\star$, $\ensuremath{\mathit{V}}_4^\star$ and $\ensuremath{\mathit{V}}_{4'}^\star$ are determined by means of eq.(28.12), eq.(28.13), eq.(29.9) and eq.(29.10) respectively. The values for all the pressures and specific volumes are obtained using eqs.(28.6)-(28.8) and eqs.(29.4)-(29.6). The velocities of the flow, as defined in fig.(V.1), are calculated either by mass flux conservation on crossing a shock, or by the formula for the velocity discontinuity in eq.(27.6). The hydrodynamical values of the pressure $p_R$ and density $\rho_R$ inside the rarefaction wave come from eqs.(27.2)-(27.5).

In order to analyse the variations of the hydrodynamical quantities as a function of position and time, let us now describe the problem in a system of reference in which the gas far away to the right of the cloud is always at rest, as illustrated in fig.(V.4). Let $x_{tl}$ and $x_{tr}$ be the coordinates of the left and right tangential discontinuities, $x_{sl}$ and $x_{sr}$ the coordinates of the reflected and transmitted shocks produced after the first initial discontinuity, $\chi_{sr}$ the position of the transmitted shock after the second initial discontinuity and $x_a$ and $x_b$ the left and right weak discontinuities which bound the rarefaction wave. The new velocities are defined by Galilean transformations:

$$u_2 = \ensuremath{v}_2 + \ensuremath{v}_c,$$ (30.1)

$$u_{sl} = \ensuremath{v}_{sl} - \ensuremath{v}_c,$$ (30.2)

$$u_{sr} = \ensuremath{v}_{sr} + \ensuremath{v}_c,$$ (30.3)

$$\nu_R = w_1 - w_R,$$ (30.4)

$$\nu_{sr} = w_1 + w_{sr}.$$ (30.5)

Figure V.4: Description of the problem of a collision of a shock with a cloud in a system of reference in which the gas far away to the right (at $x \! = \! \infty$) is always at rest. Originally a shock is travelling to the right and hits a tangential discontinuity (top panel). This produces a discontinuity in the initial conditions so a reflected and transmitted shock are produced; the gas in the cloud begins to accelerate (middle panel). Eventually the transmitted shock into the cloud collides with its right boundary producing a ``reflected'' rarefaction wave bounded by two weak discontinuities (region R) and a transmitted shock into the external medium (lower panel). In this system of reference every single discontinuity produced by means of the interaction move to the right, except for the reflected shock produced after the first collision. Arrows represent the direction of motion of various boundaries and direction of flow. Numbers label different regions of the flow. Dashed lines represent shock waves, dash-dotted ones weak discontinuities and continuous ones tangential discontinuities.

The direction of motion of the flow is shown in fig.(V.4) and it follows from eq.(28.17), eq.(28.20) and eq.(30.2) that $u_{sl}$ points to the left in this system of reference. Since, in the same frame, $\ensuremath{v}_c$ and $w_1$ point to the right, continuity across a weak discontinuity demands $\nu_R$ to behave in the same way.

The tangential discontinuities and the shocks produced by the initial discontinuities move with constant velocity throughout the gas. This implies that the time at which the first initial discontinuity takes place is:

$$t_0 = - { \frac{\Delta}{ u_{sr} } },$$ (30.6)

where $\Delta$ represents the initial width of the cloud. Hence, the positions of all different discontinuities for $t_0 \! < \! t \! < 0$ are:

$$x_{sr} = u_{sr}t,$$ (30.7)

$$x_{sl} = { -\Delta - u_{sl} (t - t_0) },$$ (30.8)

$$x_{tl} = { -\Delta + \ensuremath{v}_c ( t - t_0) }.$$ (30.9)

and for $0 \! < \! t\! < \! \tau$, eqs.(30.8)-(30.9) are valid together with

$$x_{a} = { -t \left( \frac{\kappa_c+1}{2} w_3 + c_4 \right) + w_1 t },$$ (30.10)

$$x_b = \left( w_1 - c_4 \right) t,$$ (30.11)

$$\chi_{sr} = \nu_{sr} t,$$ (30.12)

$$x_{tr} = w_1 t.$$ (30.13)

The time $\tau$ at which the left tangential discontinuity collides with the left boundary of the rarefaction wave is given by $x_{tl} \! = \! x_a$, and thus:

$$\tau c_{3'} = { {\ensuremath{v}_c t_0} + \Delta}.$$ (30.14)

Fig.(V.5) shows the variation of the pressure and density as a function of time and position in a system of reference in which the gas far away to the right of the cloud is at rest.

The width of the cloud varies with time, and it follows from eq.(30.9) and eq.(30.13) that this variation is given by:

$$\bar{X}(t) = \Theta(t) w_1 t + \Delta - \ensuremath{v}_c (t-t_0),$$ (30.15)

where $\Theta(t)$ is the Heaviside step function. This linear relation is plotted in fig.(V.6).