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Gas Dynamic to Acoustic wave equation

  1. May 17, 2015 #1
    1. The problem statement, all variables and given/known data
    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}

    2. Relevant 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}}##
    3. 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!
     
  2. jcsd
  3. May 19, 2015 #2

    hunt_mat

    User Avatar
    Homework Helper

    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.
     
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