# Does sinusoidal play a role in physics?

• Gerenuk
In summary, sinusoidal motion plays a significant role in physics, occurring naturally in many physical processes such as electromagnetic waves, sound waves, and AC current. The sine function is an eigenfunction of the Fourier transform, making it a useful mathematical tool in understanding these phenomena. While some examples of sine waves may be artificially created by humans, there are also many cases where they arise naturally and play a crucial role in the behavior of a system.
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.

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.

mathman said:
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.
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?

Gerenuk said:
Any examples for sine wave dependence in space?

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.

cepheid said:
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.
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.

Gerenuk said:
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.

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?

an oscillating coil with a weight

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.

jtbell said:
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?
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.

Gerenuk said:
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.
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.

cepheid said:
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.
Yes - the moons of Juptier too...

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

russ_watters said:
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.
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.

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

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!

chroot said:
The sine function is ubiquitous because it is an eigenfunction of the Fourier transform.
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.

chroot said:
The sine function is ubiquitous because it is an eigenfunction of the Fourier transform.
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.

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.

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.

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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.

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.

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.

Phrak said:
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?
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.

russ_watters said:
1. You're moving the goalposts. Your first post said nothing about requiring the systems to be naturally occurring.
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.

russ_watters said:
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.
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.

not a spatial one of the form

Standing waves are natural and I believe would form a spatial sine wave under some conditions. You are saying the boundary conditions are set to give a sine wave, as in math, that is true. In physics one might say that nature sets the boundary conditions and "does the math" that produces the standing sinusoidal wave.

I am reminded of one of my favorite quotes of Albert Einstein, "In as much as the laws of mathematics are certain, they do not apply to reality. And in as much as the laws of mathematics apply to reality, they are not certain."

I think it is reasonable to question the correspondence between math and reality, however, in this case, I would argue that the math of sines and cosines and the natural logarithms are based on historical development of very good models of natural systems. A man made system is a natural process too, by the way.

SystemTheory said:
Standing waves are natural and I believe would form a spatial sine wave under some conditions. You are saying the boundary conditions are set to give a sine wave, as in math, that is true.

Well, even standing waves are a superposition of a set of sines waves. Some sine waves will be naturally stronger of course.

I am sure there are others, but the 21 cm hydrogen line is a very pure single-frequency sinusoid.

That said, I agree with russ_waters in general. Sinusoids are all over, and the separation between man-made and natural is itself man-made.

DaleSpam said:
I am sure there are others, but the 21 cm hydrogen line is a very pure single-frequency sinusoid.
Yes, the quantisation character of QM makes the hydrogen line similar to the example of the mass and the spring.
But the EM wave itself again could have any shape and the emission process is like a special boundary condition? But I'm not sure. In any case that is a good answer - not what I was looking for, but some input that makes me think of what I'm search for.

DaleSpam said:
Sinusoids are all over, and the separation between man-made and natural is itself man-made.
The separation is suggested in this thread. "Natural" might not be word to define it fully.

espen180 said:
The wavefunction of a particle in an infinitely deep potential well at any instant in time is single mode sinusoidal.
But wait... isn't the general wave function a superposition of all energy states so again not a single sine wave?

Gerenuk said:
the emission process is like a special boundary condition
If you exclude the groud state of the most common element in the universe as a "special boundary condition" then I think you are committing the logical fallacy known as "begging the question".

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My 2 pence

To produce a perfect sine curve requires a a perfect SHO or a perfect circular motion. The perfect sine curve would then obviously be a product of that perfect motion. Such a motion doesn't exist.

And I'm no expert on quantum mechanics, but I'd say that a perfect SHO or circular orbit doesn't exist even in that realm. So the pure sine is just an abstract mathematical construct that enables us to simplify and model the world. The maths we use in physics is an oversimple model, and as such doesn't accurately describe reality, because it's too accurate. The pure sine doesn't exist. Because the perfect circle doesn't exist.

Um.. all radiation has bandwidth, including the 21 cm hydrogen line. In bulk this is due to doppler broadening as a result of random velocity of a gas. For a single H2, the uncertainty in decay time is less than infinite; the uncertainty in frequency and energy is greater than zero.

I thought the hydrogen line could be a good principle, but then again there is no guarantee that the hydrogen is in a single energy state. It could well be a superposition. Only due to the entropy law it turns out to be mostly in the ground state.

However, if photons are quantized, they would be perfect sine waves. Now that is beyond my scope to understand whether photons and their sine wave field is a "mathematical" construct only.

DaleSpam said:
If you exclude the groud state of the most common element in the universe as a "special boundary condition" then I think you are committing the logical fallacy known as "begging the question".
No, you are doing a mistake in believing you know exactly what the question is. I haven't defined it to all extend, but you shouldn't believe you know exactly what it is. It's hard to state and as I get more answers, I try to pose it more precisely.

Gerenuk said:
No, you are doing a mistake in believing you know exactly what the question is. I haven't defined it to all extend
This is Physics Forums, not Psychics Forums. It is obvious that you are just re-defining the question as we go along in order to reach the conclusion you want, which is clearly "begging the question".

DaleSpam said:
This is Physics Forums, not Psychics Forums. It is obvious that you are just re-defining the question as we go along in order to reach the conclusion you want, which is clearly "begging the question".
If you don't understand the question or if you find it ill-defined you don't have to make unhelpful comments. I'm not going to start a new thread each time I know how to pose my question more clear. And if you feel offended that your comprehension of the question and the answer don't fit what I was looking for, then sorry.

How about you properly define the question then. If you want "naturally occurring single mode sines" there are plenty, many of which have been mentioned in this thread. Whenever a suggestion is made, you either play the "manmade" card or the "superposition" card.

At least compile a complete list of requirements.

espen180 said:
Whenever a suggestion is made, you either play the "manmade" card or the "superposition" card.
At least compile a complete list of requirements.
Some people showed real competence by asking questions or making suggestions how to reformulate my question. Whereas others find it hard to have a two-sided discussion. Instead they try to boost their self-esteem by reciting some textbook stuff they have collected without trying to understand if the information fits the question. And then they try to prove that their answer is correct, but it's the question which is wrong and doesn't fit the answer. Whenever I have an objection to the answer, they ignore it, i.e. they don't object my objection which at least would be a start for a discussion.
It is bad science thinking that the answer to any question must be in ones head already. Sometimes one has to make some own thoughts.

And sorry if I still cannot define the question completely. Yet, some people helped me to make it more precise. I wish there were more of them.

Gerenuk said:
If you don't understand the question or if you find it ill-defined you don't have to make unhelpful comments.
Then let me be helpful by answering the question. The answer to your question is "No, sinusoidal functions do not play a role in physics" where the meaning of "play a role in physics" is narrowly (and poorly) defined so as to make the answer "no".

That is the essence of the fallacy "begging the question". Sorry that you think it is unhelpful to identify errors in logic.

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