# 11.3.1: Introduction

- Page ID
- 791

People had recognized for several hundred years that sound is a variation of pressure. This velocity is referred to as the speed of sound and is discussed first.

\[
\rho\, c = (\rho + d\rho)\,(c-dU)

\label{gd:sd:eq:cvMass1}

\]

\[
\rho d U = c \, d\rho \Longrightarrow dU = \dfrac{c d \rho }{ \rho}

\label{gd:sd:eq:cvMass2}

\]
From the energy equation (Bernoulli's equation), assuming isentropic flow and neglecting the gravity results

\[
\dfrac{ \left( c - dU\right)^2 - c^{2} }{ 2} + \dfrac{dP }{ \rho} = 0

\label{gd:sd:eq:cvEnergy1}

\]
neglecting second term (\(dU^2\)) yield

\[
-c dU + \dfrac{dP }{ \rho} = 0

\label{gd:sd:eq:cvCombined}

\]

Substituting the expression for \(dU\) from equation (2) into equation (4) yields

Sound Speed

\[
\label{gd:sd:eq:cvCombined2}

c^{2} \left( { d\rho \over \rho } \right) = {dP \over \rho}

\Longrightarrow

c^2 = \dfrac{dP }{ d\rho}

\]

An expression is needed to represent the right hand side of equation (5). For an ideal gas, \(P\) is a function of two independent variables. Here, it is considered that \(P= P(\rho, s)\) where \(s\) is the entropy. The full differential of the pressure can be expressed as follows:

\[
dP = \left. \dfrac{\partial P}{\partial \rho} \right|_{s} d\rho

+ \left. {\dfrac{\partial P}{\partial s} } \right|_{\rho} ds

\label{gd:sd:eq:insontropic1}

\]

\[
\dfrac{\partial P}{\partial \rho} = \left. \dfrac{\partial P}{\partial \rho} \right|_{s}

\label{gd:sd:eq:insontropic2}

\]
Note that the equation (5) can be obtained by utilizing the momentum equation instead of the energy equation.

Example 11.1

Demonstrate that equation (5) can be derived from the momentum equation.

Solution 11.1

The momentum equation written for the control volume shown in Figure 11.2 is

\[
\overbrace{(P + dP) - P}^{\sum F} =

\overbrace{(\rho +d\rho)(c - dU)^{2} - \rho \, c^2}^{\int_{cs} U\,(\rho\, U\, dA)}

\label{gd:sd:eq:cvMomentumEx}

\]

\[
dP = (\rho + d\rho) \left( c^{2} - \cancelto{\sim 0}{2\,c\,dU} +

\cancelto{\sim 0}{dU^{2}} \right)

- \rho c^{2}

\label{gd:sd:eq:mem}

\]
And finally it becomes

\[
dP = c ^{2} \, d \rho

\label{gd:sd:eq:memSimple}

\]
This yields the same equation as (5).

## Contributors and Attributions

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.