Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/24

state $(\rho _{1},p_{1},u_{1})$ at a point $x_{1}$, then at the following instant $t_{2}$ this same state will occur at that point $x_{2}$ where the variable $x-ct$ (upon which depend all the quantities $\rho _{1},p_{1},u_{1}$ of the solution under investigation) has the same value

$x_{2}-ct_{2}=x_{1}-ct_{1},$

(II-7)

$x_{2}=x_{1}+c(t_{2}-t_{1}).$

(II-6)

The assigned state propagates in the direction of increasing $x$ at a velocity $c$, q.e.d.

By substituting this type of solution into the fundamental equations, we can readily find for this wave from (II-3)$^{3a}$

$-c\varepsilon '=-\rho _{0}u',$

(II-6)

where the prime denotes the differentiation of function (11-6) with respect to the variable $x-ct$. If we assume at high values of $x$, i.e., way ahead in an unperturbed (undisturbed) gas, $u=0$, $\varepsilon =0$, and $\rho =\rho _{0}$ we find for a wave propagating to the right.

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

(II-10)

The instant pressure value is also linearly connected with density and velocity:

$p-p_{0}={\frac {\partial p}{\partial \rho }}(\rho -\rho _{0})=\rho _{0}uc,$

(II-11)

Let us point out specifically that pressure is proportional to the first degree of velocity in sound; according to Bernoulli's theorem, in a steady flow we should have a considerably smaller change in pressure:

$p=p_{0}-{\frac {\rho _{0}u^{2}}{2}}.$

(II-12)

Thus we draw extremely important conclusions from formulas (II-10) and (II-11): In a wave which propagates to the right, i.e., in the direction of increasing values of the coordinate $x$, the mass rate of motion $u$ is positive where the substance is compressed, and