How Does Atmospheric Refraction Affect the Visibility of a Duck in a Desert?

[avisha03]

Homework Statement

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.

Homework Equations

d(ne)/ds= grad n
e- the ray path direction

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 don't know wheather my way till now is correct, and how to display y as y(x).
Thanks for your help!

 

[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 )

… 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 don't know wheather my way till now is correct, and how to display y as y(x).
Thanks for your help!

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.

 

[avisha03]

Thanks for your help & advices. :)
What do you mean by hyperbolic trig substitution?
The integral result is logarithmic, isn't it?

 

[tiny-tim]

Thanks for your help & advices. :)
What do you mean by hyperbolic trig substitution?

Hi avisha03!

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).

The integral result is logarithmic, isn't it?

Yes, but, as you've noticed, it's difficult to invert in that form (ie ln(x + √(x² + a²)))

 

[avisha03]

I do know cosh & sinh, but since I'm not a native english speaker- I wanted to make sure. Thanks a lot, anyway.