Does sinusoidal play a role in physics?

1. Dec 22, 2009

Gerenuk

Does sinusoidal motion play a special role in physics?

Or is it just a mathematical intermediate step or auxiliary?
For example of course you can Fourier transform any function, but you might as well chose not to do so and use the normal local differential equation. This way you never encounter sine-solutions.

2. Dec 22, 2009

mathman

There are many examples of physical processes which can be represented by sine waves. Examples, em waves of a single frequency (radio, tv, light, etc.), single tone sound waves, musical instrument sounds (which contain overtones), electric current (AC), etc.

3. Dec 22, 2009

Gerenuk

These object are simple waves only because you set them up to be a simple waves. Naturally they could be anything. A vibrating string is a combination of sine waves just as an arbitrary function can be split into sine waves. So this is artifical.

Everything "can" be represented by sine waves of course, but the question is where sines waves occur naturally in physics and are not hidden in a mathematical transformation.

Actually one example that came to my mind is temporal sine waves dependence which is an oscillating coil with a weight. Any examples for sine wave dependence in space?

4. Dec 22, 2009

cepheid

Staff Emeritus
I don't know exactly what you're looking for, but if you're looking at an exoplanet in a circular orbit around a star and the system is edge-on, the I think that its radial (line-of-sight) velocity should vary sinusoidally with time.

5. Dec 22, 2009

Gerenuk

That proposal makes me think how to put my question more precise. I see that geometry can yield sine solutions (circle projections). I am looking for a physical process that "naturally" (without "special" conditions) yields a $\sin(kx)$ dependence.

6. Dec 22, 2009

Staff: Mentor

So you're looking for something that must occur as a single sinusoidal wave with a fixed frequency and wavelength, and cannot occur as a superposition of two or more such waves with different frequencies or wavelengths?

This is of course true only to the extent that the coil (spring) obeys Hooke's Law.

A gravity-driven pendulum doesn't meet your criterion because it's only approximately sinusoidal, for small amplitude oscillations.

7. Dec 22, 2009

Gerenuk

Yes (it's a rather philosophical question though)
It has to be only one sine wave, because otherwise mathematically speaking anything could be sine waves by Fourier transform. However, I want to know if something is a single sine wave, rather than an arbitrary function.

8. Dec 22, 2009

Staff: Mentor

That isn't true. These things are directly represented as simple sine waves. They really behave that way.

Every harmonic motion works the same way - some are complex, containing more than one motion at the same time, but a great many are simple, single frequency motions, to a very high precision. All of the examples given so far seem to fit your requirement.

9. Dec 22, 2009

Staff: Mentor

Yes - the moons of Juptier too...

10. Dec 22, 2009

chroot

Staff Emeritus
The sine function is ubiquitous because it is an eigenfunction of the Fourier transform. The Fourier transform itself plays a central role in virtually every physical theory in existence.

- Warren

11. Dec 22, 2009

Gerenuk

As I have explained most of the given examples are pure sine waves, because a human has set them up to be a single wave. I'm looking for an example where a sine wave solution of the form $\sin(kx)$ comes out with all of the allowed boundary conditions.

Radio waves are sine, because the engineer has designed the antenna to produce sine waves. In general an electromagnetic wave can have any shape. The sine decomposition is only a mathematical tool there.

12. Dec 22, 2009

cepheid

Staff Emeritus
Yeah, especially AC currents and voltages, which I was going to point out. In fact an electrical generator naturally produces a purely sinusoidal voltage, does it not?

EDIT: If your previous post is what the question truly is, then the thread title, "Does sinusoidal play a role in physics?" was not very good at all! They are certainly of great importance!

13. Dec 22, 2009

Gerenuk

That's surely true, but this is a mathematical fact. For normal waves "nature" only needs to know about $\square f=0$ and sine waves play no special role for her. She doesn't do the Maths the way we do. Let me put it this way: I can simulate EM waves without knowledge about the sine function.

14. Dec 22, 2009

DrGreg

Just to be pedantic, that's not true, actually. The significance is that the complex sine function $\exp(i\omega t)$ is a periodic eigenfunction of the derivative operator d/dt, and the Fourier transform is the unitary operator via which the d/dt operator is diagonalised. That's probably what you meant to say.

15. Dec 22, 2009

Redbelly98

Staff Emeritus
I think the OP is asking about naturally occurring single-frequency (or nearly so) sine waves.

Sunlight and starlight are ruled out because they are not single frequency. Laser beams and broadcast radio waves are ruled out because they are "man-made", not occurring naturally.

16. Dec 22, 2009

SystemTheory

This clip shows the simple harmonic motion of a weight on a spring as a sinusoidal plot of position over time. If you want to skip the video (you need high speed internet) just visualize the sine trace of simple harmonic motion when plotted over time.

http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmonicmotion.html

A magneto on a small engine produces a sinusoidal voltage and current due to the machine properties, you can see it on an oscilloscope trace. The three phase power transmission system operates on sinusoidal 60hz frequencies. I suppose harmonics could exist in many such systems if one has measurement equipment to detect the oscillations.

Edit: I notice my comments are redundant. I would point out that our clocks are generally built as simple harmonic oscillators and that time and frequency are in some sense physically related. I'm not a math wiz so how that impacts the "natural reality" of sine waves is beyond my reckoning.

Last edited: Dec 22, 2009
17. Dec 22, 2009

Phrak

Gerenuk, I'll make an attempt at restating your question, as I understand it, and you can make corrections if it's close.

Are sine waves in physics our anthropomorthic projection on objective phsyical law, in much the same way that we tend to use, to a great extent, linear equations to describe physical law, or approximate natural phenomena?

Sorry about the stilted language. It's the best I can do at the moment.

18. Dec 23, 2009

Staff: Mentor

1. You're moving the goalposts. Your first post said nothing about requiring the systems to be naturally occurring. Still, many natural examples have been given. You've stated they aren't really simple sine waves, but you are wrong.

2. #1 aside, you've tried to critique the manmade exmples given and you've done so incorrectly:

2a. Contrary to what you claimed in post #3, a vibrating string (or any simple spring-mass system) *is* a single mode of vibration/simple sine wave. In order to make it multi-mode you have to start hanging extra weights on it or use multiple strings/spring constants. In any case, this is both a man-made and a natural example (ie, spider webs and vines can produce both pendulums and vibrating strings) of a single mode of vibration spring mass systesm, operating sinusoidally.

2b. Radio and arbitrary choices: The choice of a sine wave (as opposed to a square or sawtooth) is not arbitrary (nothing an engineer does is arbitrary). Sine waves are chosen precisely because they are most natural. An antenna has a preferrential oscillation frequency and the waveform it best accepts is sinusoidal. Electric power is generated using rotational motion and rotational motion creates sinusoidal AC power. If we chose to make one of those unnatural waves, it requires effort and reduces the efficiency/effectiveness of our devices.

So again, the list is:
Natural: orbital projections and other rotational motion, light (and single frequency examples exist such as emission/absorption spectra - not to mention photons themselves), natural pendulums (a monkey on a vine), natural strings (a vibrating spider web strand), natural simple spring-mass systems (both of the above), waves on water.

Artifical but still meeting the other criteria:
Radio, AC power, artificial spring mass systems (near limitless possibilities and there and many specific examples given), artificial pendulums.

So to sum up, the answer to the question in the OP - and however you want to move the goalposts on it - is a simple yes.

19. Dec 23, 2009

Staff: Mentor

Also it should be pointed out that simple objects ALWAYS have single mode free vibration. So you can't use the fact that a bridge has hundreds of separate natural frequencies to advance your point: the bridge can also be represented as hundreds of individual objects, each with its own single natural frequency.

20. Dec 23, 2009

Gerenuk

Thanks for making a guess :)
Actually I wouldn't compare the linear approximation to my sine wave problem. The sine wave solutions are in fact exact, however pure sine waves you only get when you restrict yourself to special boundary conditions.

I really don't like it, when instead of discussing real content, all someone is after is to be offensive. That is not a political debate. All other guys here do much better than you at understanding the question.
You are right that I didn't specify my question fully in my first post. So what?
I'm not in the mood to explain to you why your style is offensive (even though the content isn't strictly wrong), but prefer to concentrate on other more constructive responses.

A string in general does not exhibit single mode vibration unless you excite it in a very special way. All music instruments show over-tones.
A simple spring-mass system is something completely different and I have already mentioned that it shows a temporal sine dependence, however not a spatial one of the form $\sin(kx)$.

The rest I won't comment on, because you don't seem to be reading all of the posts. Most examples for yours have very simple objections to being answer to my question, but I do not want to repeat all my previous posts again.