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1D wave equation variables

  1. Nov 8, 2012 #1
    The 1D wave equations
    [tex]p_{x}'' - (1/c_{0}^2)p_{t}'' = 0[/tex]
    [tex]u_{x}'' - (1/c_{0}^2)u_{t}'' = 0[/tex]
    [tex]ρ_{x}'' - (1/c_{0}^2)ρ_{t}'' = 0[/tex]

    and linearised continuity and momentum equations
    [tex]ρ_{t}' = -ρ_{0}u_{x}'[/tex], [tex]ρ_{0}u_{t}'=-p_{x}[/tex]

    how may one derive the following two equations?

    [tex]u=p/ρ_{0}c_{0}[/tex], [tex]ρ=p/c_{0}^2[/tex]

    My notes jump from the first equation to the last two.

    Thanks for any input.
  2. jcsd
  3. Nov 24, 2012 #2
  4. Nov 25, 2012 #3


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    That should read
    [tex]ρ_{0}u_{t}'=-p'_{x}[/tex] yes?
    I'll use the subscripts only, discarding the ', doubling the subscript for 2nd derivative.
    Don't know whether this helps, but investigating uxt gives c02pxx = c02ρtt = ptt = ρxx.
  5. Dec 24, 2012 #4

    That's right, a single/double prime denotes a single/double derivative.

    Could you suggest how to obtain [tex]u = p/\rho_{0}c_{0}[/tex]?

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