# Gas Dynamic to Acoustic wave equation

• Trevorman
In summary, the conversation discusses deriving the one-dimensional wave equation in the acoustical limit from the given formulas, with the assumption that the flow is isotropic. The compressible Euler equations are also mentioned, along with the relationship between c^2, gamma, p, and rho.
Trevorman

## Homework Statement

Derive from the formulas
##\frac{D^\pm}{Dt}(u \pm F) = 0##

where
##\frac{D^\pm}{Dt} = \frac{\partial}{dt} + ( u \pm c) \frac{\partial}{\partial x}##
the one-dimensional wave equation in the acoustical limit.

\begin{cases}
u << c\\
c \approx c0 = const\\
F = \frac{2c}{\gamma-1}
\end{cases}

## Homework Equations

## \frac{1}{c_0^2} \frac{\partial^2 p}{\partial t^2} = \frac{\partial^2 p}{\partial x^2}##
where
##c_0 = \sqrt{\frac{\gamma p_0}{\rho_0}}##

## The Attempt at a Solution

Expanding
##\frac{D}{Dt} = \frac{\partial}{\partial t}+\frac{\partial u}{\partial x} \pm \frac{\partial c}{\partial x}##

Now combining equations
##\left( \frac{\partial}{\partial t}+\frac{\partial u}{\partial x} \pm\frac{\partial c}{\partial x} \right) \cdot (u \pm F) = 0 \Rightarrow##
##\Rightarrow\dot{u} \pm\dot{F} + uu^\prime + Fu^\prime + uc^\prime +Fc^\prime = 0 ##

Since ##c = c_0 = const## and ##u<<c \Rightarrow uc^\prime \approx 0 ##, the quadratic terms are neglected. ##\dot{F} = 0## ,and substituting F
##\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{2c}{\gamma-1} \frac{\partial c }{\partial x} = 0##

This should be done by substituting

##c=c_0 + c^\prime##
##u=u^\prime##
##p = p_0 + p^\prime##
##\rho = \rho_0 + \rho^\prime##
and by neglecting small terms as ##u^\prime c^\prime##

I do not know how to proceed with this example.

Thank you!

This is nothing but the characteristic form of the 1-D gas equations. No perturbations are necessary.
1) Write out the compressible Euler equations in 1D
2) Note that c^2=\gamma p/rho
3) The flow you're looking for is isotropic so p=A\rho^{\gamma}
4) Play.

## 1. What is the Gas Dynamic to Acoustic wave equation?

The Gas Dynamic to Acoustic wave equation is a mathematical representation of the relationship between gas dynamics and acoustic waves. It describes how pressure and velocity changes in a gas medium can produce sound waves.

## 2. How is the Gas Dynamic to Acoustic wave equation derived?

The equation is derived from the Navier-Stokes equations, which describe the motion of fluids. It involves simplifying assumptions, such as assuming the gas is compressible and inviscid.

## 3. What are the applications of the Gas Dynamic to Acoustic wave equation?

The equation has various applications in fields such as aerospace engineering, acoustics, and meteorology. It is used to study the propagation of sound in gases, design of aircraft engines, and predict the behavior of shockwaves.

## 4. What are the limitations of the Gas Dynamic to Acoustic wave equation?

One of the main limitations is that it is a linear equation, meaning it can only accurately predict the behavior of small disturbances in a gas. It also does not account for factors such as turbulence and viscosity.

## 5. How is the Gas Dynamic to Acoustic wave equation used in engineering?

The equation is used in the design and analysis of various engineering systems, such as gas turbines and aircraft engines. It helps engineers understand the effects of gas dynamics on the production and propagation of sound waves, allowing for more efficient and safe designs.

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