The amount of material used in the wall is proportional to the product of its length and thickness.
Suppose the wall is shaped like a sine wave and consider a section of wall 2π long.
If the wall is in the shape of a sin(θ), then we need to find the arc length of this curve.
This works out to the following integral.
The parameter a is the amplitude of the sine wave.
If a = 0, we have a flat wave, i.
a straight wall, as so the length of this segment is 2π = 6.
If a = 1, the integral is 7.
So a section of wall is 22% longer, but uses 50% less material per unit length as a wall two bricks thick.
The integral above cannot be computed in closed form in terms of elementary functions, so this would make a good homework exercise for a class covering numerical integration.
The integral can be computed in terms of special functions.
It equals 4 E(-a²) where E is the “complete elliptic integral of the second kind.
” This function is implemented as EllipticE in Mathematica and as scipy.
ellipe in Python.
As the amplitude a increases, the arc length of a section of wall increases.
You could solve for the value of a to give you whatever arc length you like.
For example, if a = 1.
4422 then the length is twice that of a straight line.
So a crinkle crankle wall with amplitude 1.
4422 uses about as many bricks as a straight wall twice as thick.
*** Photo “Crinkle crankle wall, Fulbourn” by Bob Jones is licensed under CC BY-SA 2.
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