Eiffel Tower Equation

Introduction:

At any height on the Tower, the moment of the weight of the higher part of the Tower, up to
the top, is equal to the moment of the strongest wind on this same part.

Writing the differential equation of this equilibrium allows us to find the "harmonious equation"
that describes the shape of the Tower.

Writing the equation:

Let f(x) be the half-width of the Tower at A. The moment of
the weight of the Tower relative to point A is equal to P(x)·f(x).
 
 
 
 
 

Let us consider a slice of the Eiffel Tower located at a distance t
from the top of the Tower, its thickness being equal to dt.

Viewed from the top, this slice looks like a square whose
width is 2·f(t).

The forces applying on this slice are:

1. its weight dP(t), which is proportional to its volume:

dP(t)=k·4·f(t)²·dt
2. the horizontal wind dV(t), which is proportional to the the

slice's surface to the wind:
dV(t)=K·2·f(t)·dt

Writing the differential equation:

These 2 moments relative to point A sould be equal:

 
 
 

Let a be  . The function , which gives the width of the Eiffel Tower as a function
of the distance from top, is a solution of the following equation:

How can function f(x) be written as a combination of usual functions ?

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