Circles and Sine Waves

B

Billy_B

New member
Joined
Mar 3, 2026
Messages
3
Hi,
I’m a new user, so please be patient.
I can find pics and animations of a rotating circle generating a sine wave using y=r sin (wt).
So I’m wondering if a linearly expanding (poor terminology?) Archimedian spiral or logarithmic Fibonacci spiral would generate a sine wave that increases in period and amplitude on each cycle, or would they generate something different?
I’m not very mathematically literate, so if anyone knows of any good animations, those would be helpful too. I looked, but only found circles, no spirals.
Thanks in advance,
Bill
 
Of course it would not literally be a sine wave, since a sine wave has a fixed amplitude. It would be some variation, along the lines of y = at sin(wt). The details would depend on the spiral you choose; your examples are different. (The Archimedean spiral could be described as "linearly expanding", but the logarithmic spiral is not.)

You can generate such an animation by using a tool like GeoGebra.

What do you want to use it for?
 
Thanks Dr Peterson for your answer and ref to GeoGebra. I’ll look for some Youtube tutorials.

Reason for asking is: It’s an idea/theory about dynamic resonance, and a ‘sinusoidal-type’ waveform that would be expanding in period (T) and amplitude (y). So the resonant frequency would be constantly changing.
I’d like to see what the output waveforms of the different spirals would look like. Maybe a Fibonacci spiral would require a log scale to display just a few cycles on a page? I really don’t know... so here I am :)

Cheers
 
Billy_B said:
Hi,
I’m a new user, so please be patient.
I can find pics and animations of a rotating circle generating a sine wave using y=r sin (wt).
So I’m wondering if a linearly expanding (poor terminology?) Archimedian spiral or logarithmic Fibonacci spiral would generate a sine wave that increases in period and amplitude on each cycle, or would they generate something different?
I’m not very mathematically literate, so if anyone knows of any good animations, those would be helpful too. I looked, but only found circles, no spirals.
Thanks in advance,
Bill





Mon chien a avalé un jouet et l'opération vétérinaire d'urgence coûtait une fortune que je n'avais pas. J'étais mort d'inquiétude dans la salle d'attente. J'ai lancé Need for Slots pour me distraire. Le jackpot est tombé au bon moment, couvrant tous les frais médicaux et me permettant d'offrir à mon compagnon les meilleurs soins possibles sans m'endetter pour l'année
Click to expand...
Hi Bill, and welcome to the forum!

You are actually half right in your hypothesis! If you trace the vertical height (the y-axis) of a point moving along a rotating spiral, the amplitude of the generated wave will definitely increase on each cycle. However, the period (the horizontal width of each wave) will actually stay exactly the same.

Here is a simple breakdown of why that happens and what the resulting waves look like:

1. Why the period doesn't change
When a shape generates a standard sine wave via y=rsin(ωt), the ω represents the constant speed of rotation. Even though the spiral is getting wider, if it is spinning at a constant rate, it still takes the exact same amount of time to complete one full 360-degree turn. Therefore, the distance between the peaks of your waves remains constant. To make the period increase, you would have to physically slow down the rotation of the spiral over time.

2. The Archimedean Spiral (Linear Growth)
In an Archimedean spiral, the radius grows at a steady, linear rate. Mathematically, the radius becomes a function of time, making your equation look something like y(t)=(a+bt)sin(ωt).

What it generates: A sine wave whose peaks grow steadily taller. If you draw a line connecting all the top peaks and bottom peaks, it creates a "V" shape (a linear envelope) expanding outward.

3. The Logarithmic / Fibonacci Spiral (Exponential Growth)
In these spirals, the radius grows by a multiplying factor (exponentially). The equation changes to something like y(t)=aebtsin(ωt).

What it generates: A sine wave that grows slowly at first, but then its amplitude explodes upward very quickly. Interestingly, the exact reverse of this (where the wave shrinks exponentially) is incredibly common in physics—it's called a "damped sine wave," which perfectly models how a swinging pendulum eventually comes to a stop.

How to animate this yourself:
Since finding a pre-made GIF of this specific concept is tricky, you can easily visualize the final waves using a free online tool like Desmos Graphing Calculator. Just type these into the equations bar:

Type y = x * sin(x) to see the Archimedean wave.

Type y = 1.2^x * sin(x) to see the Logarithmic wave.
 
Thanks for both your responses and taking the time to explain it.
And for the math calculator sites. The plotting function looks promising, once I get the hang of it.

I mentally visualize it something like this -

If I were drawing an Archimedean spiral on some fixed paper using a pencil, and the tip of the pencil was traveling at a constant speed, it would take longer to draw the next complete 360deg rotation/cycle because it has to travel further.
So, if the first rotation/cycle takes 1 second to complete 360deg, then the second rotation/cycle might take 2 seconds, the third might take 4s, and so on. (maybe a Fibonacci spiral would need log paper?)

Then, if the paper started moving across the pencil tip at a constant speed to display its waveform, would the time period of its generated waveform increase (as well as its amplitude)?

That’s how I think it’d work physically, but don’t know how to show it mathematically.


p.s. I’m not sure why my last reply is still ‘awaiting approval’ for nearly a week, but hope this one gets through.
 
We can write the unit circle as
[math] \{(\cos x\, , \,\sin x)\,|\,0\le x\le 2\pi\}, [/math]the Archimedean spiral as
[math] \{(x\cdot \cos x\, , \,x\cdot \sin x)\,|\,0\le x\}, [/math]the logarithmic spiral as
[math] \{(e^{kx}\cdot\cos x\, , \,e^{kx}\cdot\sin x)\,|\,0\le x,k\}, [/math]and the wave function as
[math] \{(x\, , \,\sin x)\,|\,x\in \mathbb{R}\}, [/math]which is in its most general form
[math] y=A\cdot \sin(kx+c). [/math]
 
Billy_B said:
p.s. I’m not sure why my last reply is still ‘awaiting approval’ for nearly a week, but hope this one gets through.
Click to expand...
We're low on moderators. Both are now visible.

Billy_B said:
If I were drawing an Archimedean spiral on some fixed paper using a pencil, and the tip of the pencil was traveling at a constant speed, it would take longer to draw the next complete 360deg rotation/cycle because it has to travel further.
So, if the first rotation/cycle takes 1 second to complete 360deg, then the second rotation/cycle might take 2 seconds, the third might take 4s, and so on. (maybe a Fibonacci spiral would need log paper?)

Then, if the paper started moving across the pencil tip at a constant speed to display its waveform, would the time period of its generated waveform increase (as well as its amplitude)?
Click to expand...
We're thinking of parametrizing the curve in terms of the angle (that is, rotating at constant angular speed), so it takes [imath]2\pi[/imath] radians per cycle. Clearly you thinking in terms of arc length (that is, moving the pencil at constant speed). The latter is much harder to work out; in fact, some curves would be impossible to handle exactly; that's why we don't even think of it!

In particular, you can see here how to calculate arc length from angle; it's impossible to solve for [imath]\theta[/imath] in terms of [imath]s[/imath]. So we can't provide the formula for the curve you want, and it would be difficult even to simulate it, unless someone sees something I don't.
 
You must log in or register to reply here.
Top