# Why is sine wave preferred over other waveforms?

Why did scientists chose to go with sine wave to represent alternating current and not other waveforms like triangle and square?

What advantage does sine offers above other waveforms in representing current and voltage?

• No one "chose" those wave forms, its what naturally appears in generators. – PlasmaHH Feb 5 '15 at 9:39
• I suggest you have a look at how these things work: en.wikipedia.org/wiki/Single-phase_generator and if you can construct one that gives me a triangle or square wave, I would like to have one please. – PlasmaHH Feb 5 '15 at 9:47
• Fourier figured out that any signal/waveform can be described as a number of sines superimposed. – HKOB Feb 5 '15 at 11:34
• @PlasmaHH It is possible to build generators for waveforms other than sine. Just look at the back EMF of a BLDC, which is trapezoidal (in the common case). But yes, without additional effort, a sine wave is just what you get easily. – Roland Mieslinger Feb 5 '15 at 12:53
• @Plutoniumsmuggler That's exactly what I said! You claimed that every function can be represented as a Fourier series; I corrected this to every periodic function. (And, actually, you probably need to restrict even further, including some suitable notion of continuity and differentiability.) – David Richerby Feb 6 '15 at 8:49

Circular motion produces a sine wave naturally: -

It's just a very natural and fundamental thing to do and trying to produce waveforms that are different is either more complicated or leads to unwanted side effects.

Up and down motion (in nature) produces a sine wave against time: -

• Nice piccys Andy, SHM rules. (+1) – JIm Dearden Feb 5 '15 at 12:49
• harmonic oscillation FTW – user20088 Feb 5 '15 at 19:57
• IIRC the spring motion only is approximately by a sine wave, and the approximation is good only for small deflections. But the rotational case is exactly the reason alternating current is sinusoidal. +1 – Ben Voigt Feb 6 '15 at 0:35
• If I may, I'd like to add that since sinusoid are fundamental, you could build other waveforms out of those; Fourier series and transform, anybody ? – Sergiy Kolodyazhnyy Feb 7 '15 at 16:19
• Sinusoids are also special in that they differentiate & integrate into other sinusoids. – Roman Starkov Feb 11 '15 at 19:08

Cosine and sine waves (actually their constituents in the form of complex exponentials) are the Eigenfunctions of linear, time-invariant systems, having a time-dependent system response of \begin{align}f\bigl(a(t)+b(t),t_0\bigr)&= f\bigl(a(t),t_0\bigr)+f\bigl(b(t),t_0\bigr)&&\text{linearity}\\ f\bigl(a(t+h),t_0\bigr)&=f\bigl(a(t),t_0+h\bigr)&&\text{time invariance}\end{align} If you build any network from linear passive components (resistors, inductors, capacitors on this StackExchange) and feed it with a continuous sinoidal signal, then any point in the network will deliver a continuous sinoidal signal of possibly different phase and magnitude.

No other waveform shape will generally be preserved since the response will be different for different input frequencies, so if you decompose some input into its sinoidal components of unique frequency, check the individual responses of the network to those, and reassemble the resulting sinoidal signals, the result will generally not have the same relations between its sinoidal components as originally.

So Fourier analysis is pretty important: passive networks respond straightforwardly to sinoidal signals, so decomposing everything into sinoids and back is an important tool for analyzing circuitry.

• Isn't this a circular argument? If you decomposed the input into some other kind of component (triangle waves for example) you'd get different results. – Random832 Feb 5 '15 at 16:22
• @Random832 No, sine wave input to a passive R-C-L network always gives sine wave output (attenuated & phase shifted by a different amount depending on frequency.) To see why, see the mechanical resonance shown in Andy Aka's answer, of which electrical resonance is a direct analogue. Triangle input doesnt give triangle output. Fourier analysis tells us a triangle wave is composed of the following amplitudes, frequencies: a,f a/3,3f, a/5,5f etc. If we decompose the triangle into these sine waves and analyse them separately, we can add them together and see what waveform the circuit will produce. – Level River St Feb 5 '15 at 16:55
• @Random832 If you try to analyse the input and output of a R-C-L system with triangle waves for example, you would find non-linear response. With sine/cosine waves, you get linear response, that is important. – Aron Feb 6 '15 at 10:51
• @Aron: Related to that is the fact that adding together two sine waves with the same frequency but a phase that differs by an amount smaller than 180 degrees will yield one sine wave of the same frequency and an intermediate phase. Adding together two matching-frequency-different-phase signals of most other kinds of wave, however, will yield a wave shape that is not similar to the original. – supercat May 24 '18 at 19:25

Things oscillate according to sine and cosine. Mechanical, electrical, acoustical, you name it. Hang a mass on a spring and it will bounce up and down at its resonant frequency according to the sine function. An LC circuit will behave the same way, just with currents and voltages instead of velocity and force.

A sinewave consists of a single frequency component, and other waveforms can be built up from adding up multiple different sinewaves. You can see the frequency components in a signal by looking at it on a spectrum analyzer. Since a spectrum analyzer sweeps a narrow filter over the frequency range you're looking at, you will see a peak at each frequency that the signal contains. For a sinewave, you will see 1 peak. For a square wave, you will see peaks a f, 3f, 5f, 7f, etc.

Sine and cosine are also the projection of things that rotate. Take an AC generator, for example. An AC generator spins a magnet around next to a coil of wire. As the magnet rotates, the field that impinges upon the coil due to the magnet will vary according to the sine of the shaft angle, generating a voltage across the coil that is also proportional to the sine function.

• Thank you @alex.forencich so sine and cosine is in the fundamental actions around us right. – Rookie91 Feb 5 '15 at 9:57
• Perhaps you could include in your answer that higher frequency waves are generally undesirable, since this leads to more capacitive and inductive losses, as well as more noise (since more higher frequencies are present) that needs to be filtered out by power supplies (for example in your hi-fi setup). – Sanchises Feb 5 '15 at 13:44
• As a note: sine and cosine are so fundamental because they appear naturally in differential equations, and many facets of the universe are well modeled by differential equations (including E&M, springs, and more) – Cort Ammon Feb 5 '15 at 15:20
• on the second point - the concept of frequency components (vs periodicity) really only makes sense when you start with an orthogonal set of waveforms to use as a reference - i think a sine wave can be viewed with various frequency components of triangle waves - the sine wave is special there because of linearity properties, so that we can decompose a signal into sines and apply that to a pasive network (a linear system) – user3125280 Feb 6 '15 at 8:26
• Just because you can decompose a waveform into a set of a different waveform does not mean that this other waveform is somehow more 'fundamental'. It is certainly possible to decompose sinewaves in to something else. However, electronic circuits do behave in terms of oscillations and sinewaves. If you build a 100 Hz low pass filter and put a 50 Hz square wave into it, you will get a 50 Hz sinewave on the other side. Not a square wave or a triangle wave. This is why sine waves are fundamental. – alex.forencich Feb 6 '15 at 8:52

On a more mathematical and physical sense why sine and cosine happen to be the fundamentals of waves can have its roots on the Pythagorean theorem and calculus.

Pythagorean theorem gave us this gem, with sines and cosines:

$$\mathrm{sin}^2(t) + \mathrm{cos}^2(t) = 1, t \in \mathbb{R}$$

This made sines and cosines cancel each other out in the inverse-square laws that scatter around in the entire physics world.

And with calculus we have this:

$$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{sin}x = \mathrm{cos}x$$

$$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{cos}x = -\mathrm{sin}x$$

This means that any form of calculus operation would preserve sines and cosines if there is perfectly one of them.

For example, when we solve the instantaneous position of object in Hooke's law (similar form everywhere too) we have this:

$$-kx = F = m\frac{\mathrm{d}^2}{\mathrm{d}t^2}x$$

And the solution happens to be a linear function of $x=\mathrm{sin}(t)$.

• +0.(9) ; also, IMO it's worth noting that solving most of the commonly used differential equations (wave equations, string equations, fluid equations) requires x=e^(lambda*t) substitution, which later creates a solution that can be made into x = A*sin(lambda*t) + B*cos(lambda*t) form, essentially forcing a sine/cosine expansion in the solutions of such equations. – user20088 Feb 5 '15 at 20:03
• @vaxquis The $x=A\mathrm{sin}(\lambda t)+B\mathrm{cos}(\lambda t)$ can be folded into one $x=f(\mathrm{sin}(g(t)))$ where f and g are linear functions. – Maxthon Chan Feb 6 '15 at 5:49
• yes, exactly. They can, as well, be expressed as cosine; I just pointed that out since it IMO clearly shows that all three forms (sine, cosine, sine+cosine) are equivalent and, in fact, are used interchangeably, depending on needs and context, as can be seen, e.g. on en.wikipedia.org/wiki/Harmonic_oscillator or en.wikipedia.org/wiki/Wave_equation . – user20088 Feb 6 '15 at 13:10

Scientists did not chose the sine wave, that's what they got from an AC generator. In AC generator, sine wave is generated due to the rotor motion inside a magnetic field. There is no easy way to make it otherwise. See this figure in Wikipedia. http://en.wikipedia.org/wiki/Single-phase_generator#Revolving_armature

Sine waves contain only one frequency. A square or triangle wave is a sum of infinite amount of sine waves that are harmonics of the fundamental frequency.

The derivative of a perfect square wave (has zero rise/fall time) is infinite when it changes from low to high or vice versa. The derivative of a perfect triangle wave is infinite at the top and bottom.

One practical consequence of this is that it is harder to transfer a square/triangle signal, say over a cable compared to a signal that is only a sine wave.

Another consequence is that a square wave tends to generate much more radiated noise compared to a sine wave. Because it contains a lot of harmonics, those harmonics may radiate. A typical example is the clock to an SDRAM on a PCB. If not routed with care it will generate a lot of radiated emission. This may cause failures in EMC testing.

A sine wave may also radiate, but then only the sine wave frequency would radiate out.

• You could argue that square waves contain only one frequency. A sine wave is a sum of infinite amount of square waves. – jinawee Feb 7 '15 at 13:19
• @jinawee You could, but there are other things that make sinewaves the "fundamental" wave type. For example, it's the only one that differentiates into itself (disregarding the phase shift). Although the physical explanation about oscillating springed systems is the one I like best. – Roman Starkov Feb 11 '15 at 19:05
• @jinawee, would you prove that, please? – Eric Best Mar 28 '15 at 13:19
• @EricBest I don't know the proof, but I was referring to Walsh functions en.wikipedia.org/wiki/Walsh_function which are a Hilbert basis on the interval [0,1]. Of course some subtetlies may arise such as equality up to a set of measure zero or stuff like that. – jinawee Mar 28 '15 at 16:50
• @jinawee: Putting one sine wave through a linear system will yield either one sine wave of the same frequency, or DC (which may be viewed as one sine wave of the same frequency but zero amplitude). Putting a sum of sine waves through such a system will yield the same result as putting each wave through individually and adding the outputs. The combination of these two properties is unique to sine waves. – supercat May 25 '18 at 19:24

First of all, the sine and cosine functions are uniformly continuous(so there are no discontinuous points anywhere in their domain) and infinitely differentiable on the entire Real line. They are also easily computed by means of a Taylor series expansion.

These properties are especially useful in defining the Fourier series expansion of periodic functions on the real line. So non-sinusoidal waveforms such as the square, sawtooth, and triangle waves can be represented as an infinite sum of sine functions. Ergo, the sine wave forms the basis of Harmonic Analysis and is the most mathematically simple waveform to describe.

We always like to work with linear mathematical models of physical realities because of it simplicity to work with. Sinusoidal functions are 'eigenfunctions' of linear systems.

This means that if the input is $\sin(t)$
the output is of the form $A\cdot\sin(t + \phi)$

The function stays the same and is only scaled in amplitude and shifted in time. This gives us a good idea what happens to the signal if it propagates through the system.

• Thank you @Axel Vanraes for your valuable input.I appreciate it very much. – Rookie91 Apr 22 '15 at 11:18

Sine/Cosine are solutions of second order linear differential equations.

sin'=cos, cos'=-sin

Basic electronic elements as inductors and capacitors produces either an integration of a differentiation of current to tension.

By decomposing arbitrary signals into sine waves, the differential equations can be analysed easily.

One way to look at it, in a nutshell, is that a harmonic series of sine and cosine functions forms an orthogonal basis of a linear vector space of real-valued functions on a finite time interval. Thus a function on a time interval can be represented as a linear combination of harmonically related sine and cosine functions.

Of course you could use some other set of functions (e.g. particular wavelets) as long as they'd form a valid basis set, and decompose the function of interest that way. Sometimes such decompositions may be useful, but so far we only know of specialized applications for them.

Taking a geometrical analogy: you could use a non-ortogonoal basis to describe the components of a vector. For example, a vector in an orthonormal basis may have components of [1,8,-4]. In some other, non-orthonormal basis, it may have components of [21,-43,12]`. Whether this set of components is easier or harder to interpret than the usual orthonormal basis depends on what you're trying to do.

1. less losses
2. less number of harmonics
3. no interference with communication line
4. very less distrosional effect
5. the machine run their efficiency
6. very very little transient behavior in case L and C