Arc length constant, despite varying the period via varying amplitude

[nomadreid]

This is not a school problem, just my own mucking about, but since it has the form of a problem, I am willing to shift it to the "homework problems" rubric.
If there is a theoretical string (no thickness, etc.) that is non-stretchable tied to two endpoints and is long enough to be able to form a (taut) sin wave (say, y= sin x from x=0 to pi), then the same string makes a new sin wave y= A sin (nx) for an integer n, is there any relatively simple closed-form way to calculate A (as a function of n)? For example, a brute force attack to compare n=1, A=1 to n=2, gives the ratio A=
(2/5)^½E(½)/E(4/5), where E is the elliptical integral of the second kind with parameter m=k², making it rather more complicated than desired. If there is no simpler alternative, OK, but it would be nice if there were. Thanks.

 

[Haborix]

Without some approximations I think you are stuck with the elliptic integrals. It is a matter of taste, but you can get a "nice" result assuming $A$ is small. Call the fixed total length of the curve $L$, and let the curve have the parametric form $\vec{r}(t)=(t,A\sin(nt))$. I get for the length of the curve:

$L={\sqrt{1+A^2n^2}\over n}\mathbb{E}\left[n\pi,{A^2n^2\over{1+A^2n^2}}\right]$.

Expanding to order $A^2$ gives

$L=\pi+{\pi\over 4}n^2A^2 \Rightarrow A={2\over n}\sqrt{{L\over \pi}-1}$.

It seems like the approximation is self-consistent is some predictable ways: when the length is close to $\pi$, then curve must be close to straight (i.e. $A$ small), and if you add more wiggles (higher $n$), then the amplitude must get smaller for a given length of rope.

 

[nomadreid]

Thanks very much, Haborix. That is an excellent low-complexity approximation.