One-dimensional inviscid flow
Acoustic Wave
In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continuity equation for one-dimensional flows between 1 and 2. If we could change frame of reference and follow the sound wave, we would see fluid approaching the wave with the propagation speed of the wave, $a$, and behind the wave, the fluid would have a slightly modified speed, $a+da$. There would also be a slight in all other flow properties. Let's apply the one-dimensional continuity equation between station 1 and station 2.
\[\rho_1 u_1=\rho_2 u_2\]\\
\[\rho a=(\rho+d\rho)(a+da)\]\\
\[\cancel{\rho a}=\cancel{\rho a} + \rho da + ad\rho +\underbrace{d\rho da}_{\sim 0} \Rightarrow\]\\
\begin{equation} a=-\rho\frac{da}{d\rho} \label{eq:speedofsound:a} \end{equation}\\
\noindent The one-dimensional momentum equation between station 1 and station 2 gives\\
\[\rho_1 u_1^2+p_1=\rho_2 u_2^2+p_2\]\\
\[\rho a^2+p=(\rho+d\rho)(a+da)^2+(p+dp)\]\\
\[\cancel{\rho a^2}+\cancel{p}=\cancel{\rho a^2}+2\rho ada+\underbrace{\rho da^2}_{\sim 0}+d\rho a^2+\underbrace{2d\rho ada}_{\sim 0}+\underbrace{d\rho da^2}_{\sim 0}+\cancel{p}+dp \Rightarrow\]\\
\[dp=-2\rho ada-d\rho a^2 \Rightarrow\]\\
\[da=-\frac{dp+d\rho a^2}{2\rho a}=-\frac{d\rho}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right) \Rightarrow\]\\
\begin{equation} \frac{da}{d\rho}=-\frac{1}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right) \label{eq:speedofsound:b} \end{equation}\\
\noindent Eqn. \ref{eq:speedofsound:b} in \ref{eq:speedofsound:a} gives\\
\[a=\frac{1}{2a}\left(\frac{dp}{d\rho}+a^2\right) \Rightarrow \]\\
\begin{equation} a^2=\frac{dp}{d\rho} \label{eq:speedofsound:c} \end{equation}\\
\noindent Sound wave:
\begin{itemize} \item gradients are small \item irreversible (dissipative effects are negligible) \item no heat addition \end{itemize}
\noindent Thus, the change of flow properties as the sound wave passes can be assumed to be an isentropic process\\
\begin{equation} a^2=\left(\frac{dp}{d\rho}\right)_s \label{eq:speedofsound:c} \end{equation}\\
\begin{equation} a=\sqrt{\left(\frac{dp}{d\rho}\right)_s}=\sqrt{\frac{1}{\rho \tau_s}} \label{eq:speedofsound:d} \end{equation}\\
\noindent where $\tau_s$ is the compressibility of the gas. Eqn. \ref{eq:speedofsound:d} is valid for all gases. It can be seen from the equation, that truly incompressible flow ($\tau_s=0$) would imply infinite speed of sound. \\
\noindent Since the process is isentropic, we can use the isentropic relations if we also assume the gas to be calorically perfect\\
\[\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^\gamma \Rightarrow p=C\rho^\gamma\]\\
\[a^2=\left(\frac{dp}{d\rho}\right)_s=\gamma C\rho^{\gamma-1}=\gamma \underbrace{\left[C\rho^\gamma\right]}_{=p}\rho^{-1}=\frac{\gamma p}{\rho}\Rightarrow\]\\
\begin{equation}
a=\sqrt{\frac{\gamma p}{\rho}}
\label{eq:speedofsound:e}
\end{equation}\\
or
\begin{equation} a=\sqrt{\gamma RT} \label{eq:speedofsound:e} \end{equation}\\
\noindent From the relation above, it is obvious that the local speed of sound is related to the temperature of the flow, which in turn is a measure of the motion of elementary particles (atoms and/or molecules) of the fluid at a specific location. This stems from the fact that sound waves are propagated via interaction of these elementary particles. Since information in a flow is propagated via molecular interaction the relation between the speed at which this information is conveyed and the speed of the flow has important physical implications. Figure~\ref{fig:speed:of:sound} compares three sound wave patterns generated by a a beacon. In the left picture, the sound transmitter is stationary and thus the acoustic waves are centered around the transmitter. In the middle image, the transmitter is moving to the left at a speed less than the speed of sound and thus the transmitter will always be within all sound wave circles but it will be off-centered with a bias in the direction that the transmitter is moving. In the right image the transmitter is moving faster than the speed of sound and thus it will always be located outside of all acoustic waves. In a supersonic flow, no information can travel upstream and therefore there is no way for the flow to adjust to downstream obstacles. This is compensated for by the introduction of shocks in the flow. Over a shock flow properties changes discontinuity. An example is given in Figure~\ref{fig:supersonic:flow}.
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter03/pdf/speed-of-sound.pdf} \caption{Acoustic signature of a moving transmitter} \label{fig:speed:of:sound} \end{center} \end{figure}
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter03/pdf/supersonic-flow.pdf} \caption{Physical consequences of the speed of sound} \label{fig:supersonic:flow} \end{center} \end{figure}