Why is the sinusoidal considered the fundamental frequency?

[jaydnul]

What property of a sinusoid makes it so special? I understand Fourier analysis, but really you could do Fourier using any periodic function as the building block.

Sinusoids really do seem to be fundamental though, if you narrow the pass band of a filter with any random signal you will get a sinusoid. If you have a pure sine sound wave, it will pass through any obstacle like walls without losing any fundamental quality, only phase and amplitude changes.

So I am curious what makes it basically the pure representation of frequency, or time for that matter. Is it the fact that the it is infinitely differentiable?

Thanks!

 

[f95toli]

Well, it is convenient to work with as a basis function.
However, you also have that sine (and/or cosine) turn up as the solution to the type of second order ODEs that are extremely common in physics; so in many cases it is not just a convenient basis function but also what you actually see when you do a measurement (e.g. plot the position of a oscillating mass hanging from a spring as a function of time).

 

[DaveE]

I think it ultimately comes down to the relative simplicity of the sinusoids as a mathematical function, what with circles and such. Whether your math problem is simple or complex, sinusoids are a common, simple tool. You absolutely could use other orthogonal functions, like wavelets, for example, which people do, when it suits their problem.

 

[b.shahvir]

Power generation is by rotating magnet (or coil) generators so the output voltage waveform by nature is a sinusoid.

 

[anorlunda]

@f95toli has the best answer. It is not a wave shape we choose, it results from the solutions of ther simplest possible dynamic system there is. Simplicity is very often dominant in physics. When faced with a choice between two models in physics, always bet on the simplest one.

However, you also have that sine (and/or cosine) turn up as the solution to the type of second order ODEs that are extremely common in physics; so in many cases it is not just a convenient basis function but also what you actually see when you do a measurement (e.g. plot the position of a oscillating mass hanging from a spring as a function of time).

 

[tech99]

A sinusoid does not have any harmonics. It is the only waveshape to have only one frequency.

 

[jaydnul]

Interesting, thanks for the replies. I like the simplicity explanation, basically a system that applies force that is inversely proportional to position will result in a sinsuoid.

I guess what I am still confused about is why a sinusoid represents a fundamental tone, like when a bandpass filter is narrowed on an arbitrary signal it would output a SINUSOID at that frequency.

Let me frame the question like this, if you input a step into the passband filter you will see the output shoot up after a certain time T, then shoot back down. From that measured time, you can say that a sinusiod at the frequency 1/T will also be able to pass through the filter. What is it about the sinusoid at each point along the wave shape that allows it to pass through the filter? However a parabolic wave shape would be altered because it contains multiple “fundamental frequencies” that are not in the pass band.

 

[anorlunda]

Does this picture of a low pass filter's frequency response help? The horizontal axis is log of frequency. The vertical axis is attenuation. The simple answer is that higher frequencies are attenuated by the filter, so a waveform with fundamental plus higher frequencies will come out closer to fundamental only.

 

[Keith_McClary]

A related question:
https://math.stackexchange.com/questions/114505/whats-so-special-about-sine-concerning-y-y

 

[DaveE]

Interesting, thanks for the replies. I like the simplicity explanation, basically a system that applies force that is inversely proportional to position will result in a sinsuoid.

I guess what I am still confused about is why a sinusoid represents a fundamental tone, like when a bandpass filter is narrowed on an arbitrary signal it would output a SINUSOID at that frequency.

Let me frame the question like this, if you input a step into the passband filter you will see the output shoot up after a certain time T, then shoot back down. From that measured time, you can say that a sinusiod at the frequency 1/T will also be able to pass through the filter. What is it about the sinusoid at each point along the wave shape that allows it to pass through the filter? However a parabolic wave shape would be altered because it contains multiple “fundamental frequencies” that are not in the pass band.

Because that is the solution to a simple harmonic oscillator, or, if you prefer, the simplest form of a 2nd order differential equation. As stated previously. Basically it goes back to the simplest forms of mathematical problems, which we also see in simple physical systems; like a high Q BPF, mass and spring, etc.

 

[artis]

In fact many mechanical systems for example can only oscillate in a sinusoidal fashion , you can't have a spring and weight oscillate with a square wave fashion, at least not without external additional input energy and mechanism. Nature in many ways is linear , it is a fundamental property of nature and not one that we humans made up.

 

[sophiecentaur]

I guess what I am still confused about is why a sinusoid represents a fundamental tone,

This is not a "why" because it's a definition.
V=A.Cos(2πft). The V varies as the cosine of 2πft and f is the number of cycles per second. The only signal that's more simple would be a steady DC voltage.

There are two ways of describing any variation in time (waveform). You can describe it in the Time Domain, which is what we see on an oscilloscope, The frequency components of a signal can be related to the series of Voltage values of the signal in time.
Or you can describe it in terms of the spectrum of frequencies of the waveform. The Frequency Domain. The frequency components of a signal are assumed to be endless sinusoids. The Fourier transform takes you from one domain to the other.

In practice, we have to cheat and assume that the signal exists over a finite period of time but repeats endlessly. We use the Discrete Fourier transform which works on a finite time for the Time Domain version. Doing this digitally involves a discrete number of samples. A digital Spectrum Analyser does this for you. The DFT produces a Comb of frequency components which we, again, assume to exist for a short time (seconds, say). This approximation can be as good as you want it to be (how long is your measurement taking and how many samples will you take?)

 

[jaydnul]

This is not a "why" because it's a definition.
V=A.Cos(2πft). The V varies as the cosine of 2πft and f is the number of cycles per second. The only signal that's more simple would be a steady DC voltage.

How do you arrive at the conclusion that the cosine waveform is "simple" in the same way a DC voltage is simple?

 

[sophiecentaur]

How do you arrive at the conclusion that the cosine waveform is "simple" in the same way a DC voltage is simple?

Can you think of an easier way to describe a variation that's defined just in terms of a single trig function and a single variable (f)? You can't expect getting to grips with this stuff will be an absolute doddle. Simplicity is relative.

 

[hilbert2]

It's not just that the solutions of the wave equation are as simple as possible when it's a sine wave passing through a medium. The sources of waves are also easiest to model if they can be seen as harmonic oscillators. The most common examples are an LC electrical circuit emitting radio waves or an oscillating metal plate that can be approximated with linear elasticity theory and produces sound waves of an audible frequency.

How do you arrive at the conclusion that the cosine waveform is "simple" in the same way a DC voltage is simple?

A DC voltage is just a special case of AC with zero frequency. And the cosine function is a solution of a very easy differential equation.

 

[sophiecentaur]

How do you arrive at the conclusion that the cosine waveform is "simple" in the same way a DC voltage is simple?

I think you may be over-reacting against this Cos wave, which has served us well for a long time. (For ever, aamof)

Any suitable mathematical function (and it has to involve Maths) has satisfy a number of conditions to make it useful. It has to be continuous, it has to vary over a fixed range of values and 'go on' for ever, Ideally, it will describe the most common forms of 'oscillation' in the simplest possible way. Also, it would be handy if it is possible to differentiate it at all points in time (useful in the Maths).
There are many alternative ways to describe a waveform, For instance, a string of spiky pulses of different heights or even a series of joined up 'box car' levels can build up wave. This is the starting point for Analogue to Digital Conversion and what could be simpler than describing a wave as a set of 'levels' in time? It works in all our digital signal recording etc BUT the details (shapes) of the pulses and steps will affect how easy it is to pass along a wire.

 

[jaydnul]

Ok I see, thanks for the replies. For some reason the explanation that helped click in my brain is that the sine wave is the only oscillation that does not require and instantaneous change in position, velocity, acceleration, jerk, etc. Which in essence is the only possible wave in physics as instantaneous change isn't possible.

Cheers!

 

[Dali]

As many pointed out, it is of course a down to a definition of choice. But it is also easy to get an intuitive understanding of why so many systems in nature are well-described by harmonic sinusoidal oscillations - it is simply down to the fact that all oscillations are around some equilibrium point where the potential energy of the system necessarily has a local minima (otherwise it wouldn't be an equilibrium point!). That means that whatever the potential looks like, the first-order term is zero and the first non-zero term is the square of the displacement from the eqilibrium $\Delta X$:
$$
E(\Delta x) = E_0 + a (\Delta x)^2) + b (\Delta x)^3) + c (\Delta x)^4) + ...
$$
The second order term gives the differential equation for the harmonic oscillator (with $a=\frac k 2$). As long as $\Delta X$ is small third order and higher terms are negligible, and any system in any potential will behave very close to an harmonic oscillator. And, the smaller the deviations from the equilibrium, the better that approximation will be.

 

[cmb]

What property of a sinusoid makes it so special? I understand Fourier analysis, but really you could do Fourier using any periodic function as the building block.

Sinusoids really do seem to be fundamental though, if you narrow the pass band of a filter with any random signal you will get a sinusoid. If you have a pure sine sound wave, it will pass through any obstacle like walls without losing any fundamental quality, only phase and amplitude changes.

So I am curious what makes it basically the pure representation of frequency, or time for that matter. Is it the fact that the it is infinitely differentiable?

Thanks!

'Sine' is the ratio of the magnitude of the vertical ordinate of a point on a circle (wrt an axis which subtends a diameter of the circle horizontally) to the circles radius.

So, as any system 'spins cyclically', as in around 360 degrees of phase, then by definition the sine is the resultant function of a magnitude of displacement.

 

[marcusl]

There is a deeper explanation as well. A linear time-invariant (LTI) system is a system with a linear response which does not vary with time. (Example: if you are going to characterize an audio amplifier, linear means that you aren't overdriving the amp into distortion and time-invariant means that no one is changing the volume, bass or treble knobs during the measurement. The music signal varies but the amplifier ("system") is constant.) Sine and cosine waves are the eigenfunctions of every LTI system. They are the most fundamental excitations of the system, from which all other signals may be constructed; another word for them is "characteristic functions" or "characteristic modes." They also can be used to fully describe the system in mathematical terms. (For example, the frequency response is just the gain and phase at the output of the amplifier when a characteristic function at each frequency is applied to its input).

 

[atyy]

I guess what I am still confused about is why a sinusoid represents a fundamental tone, like when a bandpass filter is narrowed on an arbitrary signal it would output a SINUSOID at that frequency.

There is nothing fundamental about this. Random, white noise can be made from superimposing sinusoids at all frequencies, but with the phase of each sinusoid randomly chosen. When a bandpass filter is narrowed, you are choosing a narrower and narrower frequency range. If you were to choose only one frequency, you would recover a sinusoid, by definition. The only time you would not recover a sinusoid, even from pure noise, would be if there happened not to be any noise at that frequency.

However, as many have pointed out in other posts, the sinusoid is pretty fundamental. In classical physics, the sinusoid is the exact solution to the dynamical equations of motion when the potential is parabolic. However, even when the potential is not parabolic, we expect the parabolic approximation to be useful when there are small oscillations about a trough. For small oscillations about a trough in a potential, one expects the Taylor series approximation of the potential to be useful. When you expand the potential about the trough, the first term in the Taylor series will be zero, by definition of expanding about the trough. The parabola is the second term in the Taylor series approximation, and for small oscillations, it will be the dominant term. The classic sinusoidal motion of a pendulum described in physics textbooks relies on a small oscillation approximation.

This importance of the small oscillation approximation was also given above by @Dali.

https://web.mit.edu/8.01t/www/materials/Presentations/Presentation_W12D2.pdf
https://courses.physics.illinois.edu/phys325/fa2014/discussion/Disc7.pdf
http://web.physics.ucsb.edu/~fratus/phys103/LN/SHM.pdf
https://www.physics.rutgers.edu/~shapiro/507/book6.pdf

 

[sophiecentaur]

white noise can be made from superimposing sinusoids at all frequencies,

That's just re-stating (another example) the equivalence between frequency and time domains.
That, along with a number of earlier posts on this thread only go to show how "a=sin(b)" is a lot easier to write down than other ever complicated ways to describe a basic varying and unending signal. Stating a trig function in terms of an infinite series just adds complication, however valid the Math is.

If you think of the cosine function as being the simplest mathematical function (statement?) that describes how the simplest (linear, of course) mechanical system behaves in time then that is sufficient to make it a "fundamental" description of varying systems. And it works for variations in time and in space - champion.

 

[sophiecentaur]

The sine function passes through the origin. That could make it a bit special. Jus' sayin'.