How long is a piece of string

Question

If your string is a single sine wave øf wavelength $2\pi$?

I ask because intuitively it would be $2\pi$, but obviously that disagrees with logic, if the wave is length $2\pi$ then the actual string has tø be greater than $2\pi$ surely.

Answer 1

The length of the curve $y = f(x)$, $a \le x \le b$, is $\int_a^b \sqrt{1+f'(x)^2}\; dx$. In this case you get an elliptic integral which can't be done in terms of elementary functions. In Maple's notation, the answer is $4\,\sqrt {2}{\it EllipticE} \left(\sqrt {2}/2 \right) $. Mathematica, which uses a different convention, would call it $4 \sqrt{2} E(1/2)$.

Answer 2

Continuing on Robert's answer, consider the graph of $\sin(\theta)$ over the unit circle in $\mathbb{R}^{2}.$ This just looks like an ellipse that is on a $45^{\circ}$ angle to the plane, passing through the points $(1,0)$ and $(-1,0).$ We know that at $(0,1),$ which corresponds to $\theta=\pi/2,$ $\sin(\theta)=1,$ so the major axis of this ellipse has length $2\sqrt{2},$ and the minor axis has length $2.$ This gives a picture to help understand Robert's answer.