Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/25

is negative where the substance is diluted or rarefied and its density is less than normal.

Likewise, for the second wave in which all the quantities depend upon the combination $x+ct$, that is, for a wave propagating to the left, in the direction of decreasing $x$, we get

$u=-\varepsilon {\frac {c}{\rho _{0}}}=-(\rho -\rho _{0}){\frac {c}{\rho _{0}}}.$

(II-13)

In both cases the velocity of motion is directed towards the direction of wave propagation where the substance is compressed. If at an initial instant there is assigned an arbitrary distribution of density and an arbitrary distribution of velocity of motion in space

$t=0;\quad \rho =\rho (x);\quad \varepsilon =\varepsilon (x)=\rho (x)-\rho _{0};\quad u=u(x),$

(II-14)

then for the two waves looked for: the first $\varepsilon _{1}=\varepsilon _{1}(x-ct)$, $u=u(x-ct)$ and the second $\varepsilon _{1}(x+ct)$, $u_{2}=u_{2}(x+ct)$, we obtain two equations

$\varepsilon _{1}(x)+\varepsilon _{2}(x)=\rho (x)-\rho _{0}=\varepsilon (x),$

(II-15)

$u_{1}(x)+u_{2}(x)={\frac {\varepsilon _{1}(x)c}{\rho _{0}}}-{\frac {\varepsilon _{2}(x)c}{\rho _{0}}}=u(x).$

(II-16)

The second equation, (II-16), is obtained by applying (II-10) to $\varepsilon _{1}$ and $u_{1}$, and (II-13) to $\varepsilon _{2}$ and $u_{2}$. Then we immediately obtain

${\begin{aligned}\varepsilon _{1}(x-ct)&={\frac {1}{2}}\varepsilon (x-ct)+{\frac {\rho _{0}}{2c}}u(x-ct);\\u_{1}(x-ct)&={\frac {c}{2\rho _{0}}}\varepsilon (x-ct)+{\frac {1}{2}}u(x-ct);\\\varepsilon _{2}(x-ct)&={\frac {1}{2}}\varepsilon (x+ct)-{\frac {\rho _{0}}{2c}}u(x+ct);\\u_{2}(x+ct)&={\frac {c}{2\rho _{0}}}\varepsilon (x+ct)+{\frac {1}{2}}u(x+ct).\end{aligned}}$

(II-17)

It is not difficult also to study the reflection of an arbitrary perturbation from a motionless (stationary) wall. To find a solution for the propagating perturbation $\varepsilon _{1}(x-ct)$, $u_{1}(x-ct)$, we add a wave which seemingly arrives from the other side of the wall and propagates in the inverse direction, that is, a counterwave $\varepsilon _{2}(x+ct)$, $u_{2}(x+ct)$.