- #1

Trevorman

- 22

- 2

## 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

The answer should be

## \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!