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11.4: Modelling the eye–revisited

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Let me return to my model of the eye. With the function Pn(cosθ) as the solution to the angular equation, we find that the solutions to the radial equation are

R=Arn+Brn1.

The singular part is not acceptable, so once again we find that the solution takes the form

u(r,θ)=n=0AnrnPn(cosθ)

We now need to impose the boundary condition that the temperature is 20 C in an opening angle of 45, and 36 elsewhere. This leads to the equation

n=0AncnPn(cosθ)={200<θ<π/436π/4<θ<π

This leads to the integral, after once again changing to x=cosθ,

An=2n+12[1136Pn(x)dx112216Pn(x)dx].

These integrals can easily be evaluated, and a sketch for the temperature can be found in figure 11.4.1.

eye.png

Figure 11.4.1: A cross-section of the temperature in the eye. We have summed over the first 40 Legendre polynomials.

Notice that we need to integrate over x=cosθ to obtain the coefficients An. The integration over θ in spherical coordinates is π0sinθdθ=111dx, and so automatically implies that cosθ is the right variable to use, as also follows from the orthogonality of Pn(x).


This page titled 11.4: Modelling the eye–revisited is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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