Eikonal equation with n(y)

1. Nov 13, 2009

avisha03

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
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).

2. Nov 13, 2009

tiny-tim

Welcome to PF!

Hi avisha03! Welcome to PF!

(have a square-root: √and a theta: θ and try using the X2 tag just above the Reply box )
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.

3. Nov 14, 2009

avisha03

What do you mean by hyperbolic trig substitution?
The integral result is logarithmic, isn't it?

4. Nov 14, 2009

tiny-tim

Hi avisha03!

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