Optics and waves, mirage. University physics

[YogiBear]

Mirage: we consider the x-y plane describing vertical y and horizontal x directions, with an inhomogeneous index of refraction n(y). In this case, using calculus of variations, Fermat’s principle for the trajectory of a ray of light may be re-written as n(y)/√1+(dy/dx)^2 = A. Where A is a real constant, to be determined by boundary conditions. We consider the index of refraction to be n(y) = n0(1 + αy) where n0 and α are real parameters.

Show that the trajectory y(x) of a ray of light is given by
y = − 1/α + A/n0α*cosh [ (n0α/A) * (x − x0) ]

Limits for integration are not given.

What i have done so far: Well i separated variables and then used f(x) = cosh^-1(x) f'(x) = 1/(x^2 -a^2)^1/2 to get pretty close to the solution. However i don't see where n0α/A comes from within the cosh bracket. Also I used x and x0, y and 0, as limits for integration. Huge thanks in advance

 

[rude man]

Hi Yogibear, I got

y(x) = [exp(-abk - abx)+exp(abk +abx) - 2 b]/2ab

where and b are functions of one or more α, A and/or n₀, and k is the constant of integration. I have to adhere to the rules of this forum and not give the explicit functions. If you want more help you'll have to show us more math detail. E.g. what is f(x) and how did you get it?

Note that you can substitute k' = e^k. Looks like you might get a cosh term out of that ...

 

[YogiBear]

Hi Yogibear, I got

y(x) = [exp(-abk - abx)+exp(abk +abx) - 2 b]/2ab

where and b are functions of one or more α, A and/or n₀, and k is the constant of integration. I have to adhere to the rules of this forum and not give the explicit functions. If you want more help you'll have to show us more math detail. E.g. what is f(x) and how did you get it?

Note that you can substitute k' = e^k. Looks like you might get a cosh term out of that ...

Thanks, I was actually able to solve that part using t = n(y)/A substitution.
However now i have to do the following: "Determine the trajectory y(x) of a ray of light that just grazes the ground at x = xg as shown in the figure (i.e. determine expressions for A and x0 for this ray). Figure, in this case is just a generic x/y plane with y=x^2 graph on it touching x-axis at xg. I have no idea where to start. :/