# Arc length constant, despite varying the period via varying amplitude

• I
In summary, the conversation discusses a theoretical problem involving a non-stretchable string tied between two endpoints and forming a sinusoidal wave. The question is whether there is a simple way to calculate the amplitude of the wave for different integer values. The conversation concludes with an approximation method using the parametric form of the curve.
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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.

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.

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

## What is the arc length constant?

The arc length constant is a mathematical constant that represents the distance along a curve from one point to another. It is used to measure the length of a curve despite any variations in its period or amplitude.

## How is the arc length constant calculated?

The arc length constant is calculated using the formula L = ∫√(1 + (dy/dx)^2) dx, where L is the arc length, dy/dx is the derivative of the curve, and the integral is taken over the interval of the curve.

## Why is the arc length constant important?

The arc length constant is important because it allows us to measure the length of a curve accurately, even when the curve has varying periods and amplitudes. This is useful in many fields, such as engineering, physics, and mathematics.

## Can the arc length constant be changed?

No, the arc length constant is a fixed value that cannot be changed. It is a fundamental property of a curve and is not affected by any variations in its period or amplitude.

## How does the arc length constant relate to the period and amplitude of a curve?

The arc length constant is independent of the period and amplitude of a curve. This means that regardless of how much the period or amplitude is varied, the arc length constant remains the same.

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