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I recently read the following post:

What is the length of a sine wave from $0$ to $2\pi$?

This covers the equation y = sin(x) with no particular units. To describe any sine wave, we use the equation y = Asin(wx + phi) where A is the amplitude, w is the angular frequency (in rad/s) and phi is the phase angle. Now, since the solution involves an integral and once you integrate over a whole period or even quarter period, the phi doesn't matter, so we can simplify to y=Asin(wx)

For the unitless unity case, the final result was 4*sqrt(2)*E(1/2) = 2.432*pi, or 7.640 (rounded), where E is the "complete elliptic integral function of the second kind" which is a function available in many popular math programs (MATlab, Mathematica, Maple, etc)

Now, if I have a sine wave of known amplitude and frequency, how does that affect the solution (i.e. what do I plug into the function "E" then multiply by to get the result I want)?

Or, alternately: If I have an ellipse of length L and width W, what is the input m for the function to calculate E(m)?

Or, alternately: If I have an ellipse with foci at x1,0 and x2,0, what is the input m for the function to calculate E(m)?

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Trashman
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1 Answers1

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The circumference of an ellipse is

$$ 4aE(\sqrt{1-(b/a)^2}) $$

where $a$ and $b$ are the semi-major and -minor axes. So for an amplitude $A$ and frequency $\omega$ this maps to:

$$E(k) \to \frac{2\pi}{\omega}E(k\sqrt{1-(A\omega/2\pi)^2})$$

lemon
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  • Is that "k" supposed to be there on the right hand side of the second formula? I was thinking we need to substitute 1−(b/a)2−−−−−−−−√ in for k... or I am reading this wrong? – Trashman Jun 26 '14 at 21:02