The Path of Light Using Calculus of Variations

1. Mar 12, 2012

pelmel92

1. The problem statement, all variables and given/known data

A rectangular solid of height h
increases in density as its height
increases, so the index of refraction of
the solid increases with height
according to:
n(y) = 1.20(2y + 1)
where y is the distance, in meters,
from the origin (see diagram). A beam
of light traveling in air (n = 1.00) in the
x-y plane strikes the bottom of the
tank at the origin, making an angle of
incidence with the normal of θ1. Assume:
• n varies only with y, not with x or z.
• The light travels in the x-y plane.
• The block is wide, so the light leaves through the top and not through a side.

a) Use Fermat’s principle and Euler’s equation to find the equation for the path of the
light y(x) in the tank. There will be two undetermined constants.
HINT: If the equations become too complicated (and they will), remember we learned
some alternative ways to solve calculus of variations problems.
HINT: The answer should come out as a hyperbolic function.

2. Relevant equations

Euler's equations for extremum: f - y'(∂f/∂y') = constant
n=c/v

3. The attempt at a solution

I'm looking to use Fermat's least time principle, I've tried to find the minimum of an integrated functional equal to time t.
Currently I have it as:

t = ∫ ds/v = ∫(1.20/c)(2y+1)√(1+y'^2)dx

which would make my functional:

f=(2y+1)√(1+y'^2)

and using the Euler equations above I have from that:

(2y+1)(1+y'^2)^.5 - y'^2(2y+1)(1+y'^2)^-.5 = (2y+1)(1+y'^2)^-.5 = constant

and that's where I'm stuck...

(2y+1)(1+y'^2)^-.5 = constant seems like it should be easy enough to integrate with dy and dx somehow, but try as I might I'm totally stumped.

Somehow I'm supposed to get a hyperbolic function for y(x), which leads me to believe I'm approaching this whole problem the wrong way.