Simple wave

Let ${\displaystyle \rho }$ be the gas density, ${\displaystyle u}$ the velocity, ${\displaystyle p}$ the pressure and ${\displaystyle c={\sqrt {(\partial p/\partial \rho )_{s}}}}$ the speed of sound. In isentropic flows, entropy ${\displaystyle s}$ is constant and if the initial state of the gas is homogenous, then entropy is a constant everywhere at all times and therefore the pressure is a function only of ${\displaystyle \rho }$, i.e., ${\displaystyle p=p(\rho )}$ In simple waves, all dependent variables are just function of any one of the dependent variables (this is certainly the case in one-dimensional sound waves) and therefore we can assume the velocity to be also a function only of ${\displaystyle \rho }$. i.e., ${\displaystyle u=u(\rho ).}$ This latter property is the cause of origin of the name simple wave, although the wave is nonlinear.

From the one-dimensional Euler equations, we have

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial (\rho u)}{\partial x}}=0}$

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}=0}$

which, because ${\displaystyle u=u(\rho )}$, can be written as

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {d(\rho u)}{d\rho }}{\frac {\partial \rho }{\partial x}}=0}$

${\displaystyle {\frac {\partial u}{\partial t}}+\left(u+{\frac {1}{\rho }}{\frac {dp}{du}}\right){\frac {\partial u}{\partial x}}=0.}$

Further, since (remember that the time derivative of a function ${\displaystyle f(x,t)}$ integrated along a curve ${\displaystyle x=\varphi (t)}$ is given by ${\displaystyle (df/dt)_{\varphi }=\partial f/\partial t+(dx/dt)_{\varphi }\partial f/\partial x}$)

${\displaystyle {\frac {\partial \rho /\partial t}{\partial \rho /\partial x}}=-\left({\frac {\partial x}{\partial t}}\right)_{\rho },\quad {\frac {\partial u/\partial t}{\partial u/\partial x}}=-\left({\frac {\partial x}{\partial t}}\right)_{u},}$

the two equations lead to

${\displaystyle \left({\frac {\partial x}{\partial t}}\right)_{\rho }={\frac {d(\rho u)}{d\rho }}=u+\rho {\frac {du}{d\rho }},\quad \left({\frac {\partial x}{\partial t}}\right)_{u}=u+{\frac {1}{\rho }}{\frac {dp}{du}}.}$

However, since ${\displaystyle \rho }$ determines ${\displaystyle u}$ and therefore the above derivatives must be equal so that ${\displaystyle \rho du/d\rho =(1/\rho )dp/du=(c^{2}/\rho )d\rho /du}$. Thus, we obtain ${\displaystyle du/d\rho =\pm c/\rho }$, whence

${\displaystyle u=\pm \int {\frac {c}{\rho }}d\rho =\pm \int {\frac {dp}{\rho c}}.}$

This equation provides the required relation ${\displaystyle u=u(\rho )}$ or, ${\displaystyle c=c(u)}$ or, ${\displaystyle u=u(p)}$ etc. The above equation is just a statement that either the ${\displaystyle J_{+}}$ or the ${\displaystyle J_{-}}$ Riemann invariant is constant.

Thus, we obtain

${\displaystyle \left({\frac {\partial x}{\partial t}}\right)_{u}=u\pm c(u)}$,

which upon integration becomes

${\displaystyle x=t[u\pm c(u)]+f(u)}$

where ${\displaystyle f(u)}$ is an arbitrary function. This equation indicates that the characteristics in the ${\displaystyle x}$-${\displaystyle t}$ plane are just straight lines. When ${\displaystyle f(u)=0}$ and when consequently length scale and time scale associated with the initial function disappears, the problem is self-similar and the solution depends only on the ratio ${\displaystyle x/t}$. This particular case is referred as the centred simple wave.

Similar to the unsteady one-dimensional waves, simple waves in steady two-dimensional system cab be derived. In this case, the solution is given by

${\displaystyle y=xf_{1}(p)+f_{2}(p)}$

where ${\displaystyle f_{1}(p)=(\partial y/\partial x)_{p}}$ and ${\displaystyle f_{2}(p)}$ is an arbitrary function of pressure. The characteristics in the ${\displaystyle x}$-${\displaystyle y}$ plane are straight lines. Similarly, the case corresponding to ${\displaystyle f_{2}(p)=0}$ is referred as the centred simple wave; the Prandtl–Meyer expansion fan is a special case of this centred wave.