Assume
that a person is standing at one end of a long airport runway.
A vertical temperature gradient in the air has
resulted in the index of refraction of the air above the runway to
vary with height y according to
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Solution: Click For PDF Version We will set up a differential equation that describes the change in angle of refraction with height in a medium in which refractive index of the medium is continuously changing with height.
Let
Retaining
terms up to first order in
The solution of this differential equation is
It is given that the refractive index of the medium is given by the function
Using the boundary condition
for the ray that is tangential to the runway and passes by the person at the height of the eyes. Thus the light ray is fixed by the requirement that the value of the constant in the equation
is
The variation of the angle of refraction along the ray is given by the function
The angle
We are given that
As
The length of the runway seen by the person is 752.7 m.
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,
where
is the index of refraction at the runway surface and
.
That person’s eyes are at a height
above the runway. We
have to find the horizontal distance d that person sees for the
runway.
be the refractive index of the medium at height y;
see the figure. According to Snell’s law, we have
,
we get
.
,
of the ray from the normal at height h
of the observer is
