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Eikonal equation with n(y)

  1. Nov 13, 2009 #1
    1. The problem statement, all variables and given/known data

    A duck is walking in a flat desert (yes, the proff. is creative...). According to atmospheric conditions the Refractive index is n(y)=1.0006-0.0003y. y in KM.
    An observer is on the ground, 26.2 KM far. Where will he see the duck? Show the gragh of the ray path.


    2. Relevant equations
    d(ne)/ds= grad n
    e- the ray path direction

    3. The attempt at a solution
    I used the eikonal equation, and some matematical manipulations to get the following equation:
    dy/dx= sqrt((a-by)^2/c^2-1)
    (a=1.0006, b=0.0003, c=1.0006*cos(theta_0))
    In order to try and find y as a function of x, I tried to solve the latter eq. by seperating it, and using the variable replacment: m=a-by
    I got:
    ln(m+sqrt(m^2-c^2))=exp(-bx/c)
    now I dont know wheather my way till now is correct, and how to display y as y(x).
    Thanks for your help!
     
  2. jcsd
  3. Nov 13, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi avisha03! Welcome to PF! :smile:

    (have a square-root: √and a theta: θ and try using the X2 tag just above the Reply box :wink:)
    You mean dy/dx= √((a-by)/c)2 - 1) ?

    (I haven't checked it, but assuming that it's correct …)

    Easier to put n = (a-by)/c, dn = -bdy/c

    dn/√(n2 - 1) = -(b/c)dx

    and then use a hyperbolic trig substitution. :wink:
     
  4. Nov 14, 2009 #3
    Thanks for your help & advices. :)
    What do you mean by hyperbolic trig substitution?
    The integral result is logarithmic, isn't it?
     
  5. Nov 14, 2009 #4

    tiny-tim

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    Hi avisha03! :smile:

    Haven't you come across cosh and sinh? if not, see http://en.wikipedia.org/wiki/Sinh#Standard_Integrals"

    (See also standard integrals in the PF Library)

    There are tables in which you can look up, for example, cosh(x) and cosh-1(x).
    Yes, but, as you've noticed, it's difficult to invert in that form (ie ln(x + √(x2 + a2))) :wink:
     
    Last edited by a moderator: Apr 24, 2017
  6. Nov 14, 2009 #5
    I do know cosh & sinh, but since I'm not a native english speaker- I wanted to make sure. Thanks alot, anyway.
     
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