Arc length constant, despite varying the period via varying amplitude

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.
  • #1
nomadreid
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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.
 
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  • #2
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.
 
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  • #3
Thanks very much, Haborix. That is an excellent low-complexity approximation.
 

Related to Arc length constant, despite varying the period via varying amplitude

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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