# Finding The Equation For An Unusual Sine Wave

#### JinE

Hello everyone.

https://math.stackexchange.com/questions/3383264/finding-an-expression-for-a-tricky-sine-wave

Sadly despite offering a 'bounty' for it, I've not gotten very far. Does anyone have any ideas? (I didn't redo the post here because it would just take too much time, see my link....) (BTW, sorry my post may be hard to read as it was put together in a piecemeal manner.)

#### DarnItJimImAnEngineer

I'm not sure I understand the question. You have a particular image of part of a curve, and you're trying to find an analytic expression that maps that image as closely as possible?

Where did the curve come from? What makes you think it's going to fit a distorted sine wave, as opposed to a spline fit or Bessel function or something else? Do you have any information about the curve other than that picture?

#### JinE

Re; Sine Wave

You totally get what I'm trying to do. Thanks for the time.

Good Qs.

The origins / uses of the wave would take up too much space to explain. But, in short, here's some more info.

1. It was drawn by a mathematician

2. It's nothing like a spline fit or some such.

3. It's either a sine wave or 'sine-like.'

4. The Bessel function idea is good; but to me the differences between them don't seem too great, not mathematically but practically; i.e., I think you could express it with either, perhaps...

Thank you!

#### tahirimanov19

Instead of drawing, take something real to analyze, like rivers.

#### JinE

Re; Sine Wave

Thank you for the input. I'm not sure I understand?

#### romsek

Math Team
looks like a pretty standard damped sine wave to me.

you have three parameters to estimate
a) base frequency
b) damping factor
c) phase factor

basically

$f(t) = e^{-\alpha t}\sin(\omega t + \phi)$

A damped sine wave has it's apparent frequency shifted towards
lower frequencies as it's amplitude decays.

#### DarnItJimImAnEngineer

$f(t) = e^{-\alpha t}\sin(\omega t + \phi)$

A damped sine wave has it's apparent frequency shifted towards
lower frequencies as it's amplitude decays.
What are you talking about? No, it doesn't.
Peaks of a damped sine wave still occur at regular intervals ($2\pi/\omega$).

Exponential decay in the amplitude looks pretty close to what is shown, but the frequency is also decaying somehow.

Assuming it's "sinelike," it's probably going to be of the form
$y(t) = A(t) \sin(\omega(t)t+\phi)$, where we need to determine the form of $A(t)$ and $\omega(t)$.

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#### JinE

Re; Re; Sine Wave

DarnItJimImAnEngineer, Good observations; agreed! If you aren't on the other forum, I can't give you reputation points for working on this. But, if you do or give it someone who can, I'd be happy to try and repay you somehow, for instance, I'm really good with CAD in all the major programs and have a number of other skills. Thanks once more!

#### DarnItJimImAnEngineer

I'll tell you how I would go about it. The only place we can easily decouple the oscillating and decaying components is at the peaks. If you assume the bottom point is (f = 0, t = 0), I would carefully measure the f values and t values at each peak (the local minima and maxima, which occur each half-cycle). Then I would plot f(n) versus n, where n is the number of cycles. I would probably plot on a semi-log plot; this makes exponential relations easier to find. I might plot f(t), as well. Then I would try plotting t(n). The problem is, six points is very sparse for trying to determine curve fits, especially if we don't know what type of functional form to expect. It's a starting point, though.