# What is a sine wave?

This came up when a student asked me. A simple question one might think. Except... how to define one without tautology? That is, without using the word "sine" (or cosine for that matter). Wikipedia does not help, although the moving disc might be of relevence.

In short, I suspect his teacher has given him a severely hard problem, though I may be wrong.

This came up as part of an electronics course. So presumably any answers can be derived from the characteristics of various components/circuits.

• I'm voting to close this question as off-topic because this questions is not related to electronics design, but mathematics. – Michel Keijzers May 23 '18 at 11:13
• @MichelKeijzers I disagree because this came up as part of an electronics course. So presumably any answers can be derived from the characteristics of various components/circuits. – Dirk Bruere May 23 '18 at 11:15
• I’m not sure what kind of answer you’re expecting. For me the sine function is just a mathematical representation of many physical phenomena that involves oscillation. Any oscillation can be constructed as a linear combination of sine functions, what make sines a basis for the vector space of all periodic functions. – PDuarte May 23 '18 at 11:21
• @DirkBruere For an electronics student the sine concept should come from mathematics class, not electronics. It should have been made clear when he/she was studying trigonometry. I feel you’re trying to explain basic concepts in higher domains, which is not very effective in pedagogy. – PDuarte May 23 '18 at 11:33
• It is the shadow of a helix that is lit from the side. – Dampmaskin May 23 '18 at 11:47 simulate this circuit – Schematic created using CircuitLab

Say:

we have the inductor L1. We charge C1 separately, and then quickly connect it as shown, so that the top side of this circuit is at +1V potential relative to the lower side.

What will happen next?

Clever students will say: yeah, well, it's a fast change of voltage across L1, so it will take some time until things look more "DC-y", and current starts flowing through L1 and discharge C1, so that the overall potential will be 0V.

But what about the magnetic field in the inductor

Oh yeah, that now stores the energy from the capacitor

So the current flow will stop forever once the voltage across C1 (and L1) is 0 V?

No, the magnetic field energy has to go somewhere. So the Capacitor charges again.

Can we put formulas to that? Yes, we can; enter the differential equations describing current and voltage across capacitors and inductors. Show that you need a function whose second derivative is itself, negated.

Now comes the hard part, and I'm afraid you'll be able to do nothing about it: You need to say: hey, this is a sine, it fulfills that condition.

• That's the one I though of first. I think it would be a good EE student answer. But I long ago learned to answer what the teacher expects... – Dirk Bruere May 23 '18 at 12:23
• Despite popular opinion, I am going to mark this as the answer because it is the kind of answer that would be best for an EE student to offer to their teacher. As people have commented, this is an EE site and not a mathematics one. However, I really like the rotating vector explanation – Dirk Bruere May 24 '18 at 16:13

One way would be to describe a sinewave with respect to the unit circle. The radius obviously draws a circle BUT the x and y co-ordinates trace out the familiar waveforms.

This also helps with pictorially explaining Eulers formula:

$e^{i x} = cos(x)+ i\cdot sin(x)$

where the special case of $x = \pi$ yields Eulers identity: $e^{i \pi} + 1 = 0$ (source: https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/)

• And the x and y co-ordinates of a point on a circle are deeply related to the definitions of cos and sin. If you know what a sine function looks like when graphed, you already know what a sine wave is. – Monty Harder May 23 '18 at 18:58
• Rephrasing this answer into a definition: "A sine wave is a shape or signal that can be modeled by a function that maps a real number $x$ to the real magnitude of the imaginary part of $e^{ix}$. Such a function is called the/a sine function and is denoted by $\sin (x)$." – Todd Wilcox May 23 '18 at 20:07
• @ToddWilcox that definition is very useful! So simple. (My trig teacher was an assistant coach with no business teaching mathematics and the damage has been enduring;) – DukeZhou May 23 '18 at 20:44
• @ToddWilcox i dont really think that is a good answer, since that is a just the same reasoning as the circle. It just follows from basic trigonometry which is defined as projections of unit circles. If we use that definition then the question is what is e and what is imaginary numbers. – joojaa May 24 '18 at 5:54
• @joojaa Remember, the central aspect of the original question is how to define sine without referring to sine. Personally, I feel like a definition of sine based on triangles requires a lot of explanation and diagrams, and then you have to leave triangles behind and re-define it with the unit circle. Assuming a certain amount of sophistication in mathematics (e.g., already knowing what sine is), a definition based on Euler's formula seems like one of the more elegant answers. My goal was a definition that was simple, rigorous, and textual. I think I found one that fits those criteria. – Todd Wilcox May 24 '18 at 13:25 The easiest explanation I find is encapsulated in the moving image above. It's all about right angle triangles existing inside a circle.

Picture taken from here. See also Why is a sine wave preferred over other waveforms.

• I'd describe it as the vertical component of the rotating vector (and cosine as the horizontal) myself, but same principle. – Baldrickk May 23 '18 at 12:18
• beat me in posting such a concept (wasn't there when I was writing) – JonRB May 23 '18 at 12:22
• +1 - SOH CAH TOA! – David K May 23 '18 at 12:45
• @DavidK I always prefered "Smiles Of Happiness, Come After Having, Tankards Of Ale" – JonRB May 23 '18 at 13:10
• Saints On High CAn Have Tea Or Alcohol. – Leon Heller May 23 '18 at 13:50

Simple: a sine wave in time, t, is the imaginary part of:

$$e^{j \omega t}$$

where ω is the angular frequency.

• +1 this is the most fundamental piece of mathematics in all electrical engineering. Given the question was from a student, you might want to elaborate though. – Jon May 23 '18 at 14:33
• I will let my assistant Dave Tweed fill in the details. – Mr Central May 23 '18 at 14:47
• I've love to watch a student, who upon being given this definition, tries to "imagine" part of e^jwt! – Cort Ammon May 23 '18 at 18:58
• @CortAmmon I know what you mean, but it helps to know ℯʲʷᵗ that describes a sine wave, and then try to puzzle out how it means that. – DukeZhou May 23 '18 at 19:43
• It might help to clarify that EEs denote the imaginary unit with $j$, while mathematicians denote it with $i$. – Todd Wilcox May 23 '18 at 20:13

Many problems in physics can be formulated as second order linear differential equations with constant coefficients.

For continuous ("harmonic" oscillations) without dampening, the movement can be described simply as a differential equation of a function and its second derivative. Without dampening, with f typically being a function of time, you get something like this:

$$af''+f=0$$

You could define the sine function as f, the general solution to this equation. It is possible to show that it is the only general solution to this problem.

Here's your straightforward definition: a solution, and a good model, for describing common phenomena.

• Can I ask the meaning of the '' in this context? I found it used in relation to the the double prime... Is this the correct usage here, relating to time? – DukeZhou May 23 '18 at 19:47
• @DukeZhou It is the second derivative with respect to the aforementioned independent variable, which is time in this case. – Todd Wilcox May 23 '18 at 20:14
• bonus answer (posted as comment, since it is a bonus): in the transitory case, you have exponential terms (decreasing exponential in case of dampening). If you rewrite the problem using exponentials taking into account the fact that $$sin(t)=\Im(e^{jwt})$$ you can find a solution using only exponentials, which generalizes to a solution of $$af′′+bf′+f=0$$ for any real numbers a, b – Florian Castellane May 24 '18 at 8:09
• Another way of phrasing this answer: a sine wave is the position of an object moving in such a way that its position is always opposite to its acceleration (with appropriate units). Incidentally, technically, it is not correct that a sine wave is the general solution to your differential equation; it is only a particular solution. (My re-phrasing sneakily says this, but in an obscure way.) – LSpice May 27 '18 at 22:38

Easy. Start at steam locomotives. Sine is the position of its piston relative to the angle of the wheel.* You can go look at one in a museum: trig in living color. For instance look at the linkage at 3:00 and 9:00 positions (90 and 270 on the sinewave, where it is flat) and you see where the piston has a problem: it can't apply any force. That's why the mechanism is duplicated on the other side, 90 degrees out of phase. That piston is at the peak of its leverage.

The concept works even better with 3 (60 degrees out of phase), which steam locomotives did when they could (UK, Shay) and that concept is used today in 3-phase power.

And AC generators do the same thing, as the DC magnetic field on the rotor sweeps across the non-moving field windings. A generator is driven, but a single phase motor can get stuck at top dead center just like a single piston steam engine. That is solved by a special starter winding. Three phase Motors don't have that problem.

This concept comes up over and over in mechanical design and thus electronic design. As others have pointed out, it pops up a lot in nature. Note also that if position is a sine wave, velocity is a sine wave, acceleration is also a sine wave, jerk (dA) is a sine wave too, it's sinewaves all the way down. The "perfect rectangle " of motion.

* now the steam locomotive main rod does jar it slightly off a pure sine wave, but this is a fairly long rod (unlike in your car engine) and so the difference is operationally negligible, and of no concern to locomotive builders.

DaveTweed: not a dup because I go straight for the real world application.

• Thanks for breaking this down in terms of old school engineering! (I find myself often having to point out that computers predate integrated circuits :) – DukeZhou May 23 '18 at 19:56
• @DukeZhou And predating electronic/electromechanical/mechanical computers was the human computer, who performed computations manually. – JAB May 23 '18 at 22:04
• And then you add reversing valve gear, with a bit of "lead" to compensate for the valves not being perfect. Yay, more trig! – AaronD May 25 '18 at 16:51

Here is a another explanation:

sine waves

A sine wave is a repetitive change or motion which, when plotted as a graph, has the same shape as the sine function.

A quote more directed to electronics:

The electrical power in your house is AC or Alternating Current. The direction of current flow reverses 50 or 60 times per second depending on where you live. If you plot the voltage against time, you would find it is also a sine wave, because it is derived from a rotating generator.

In the link also physics examples can be found for sine waves regarding amplitude, period and frequency.

For example, a weight suspended by a spring. As it bounces up and down, its motion, when graphed over time, is a sine wave.

• But you are now back to using tautology again. – Dirk Bruere May 23 '18 at 11:25
• @DirkBruere No he's not, a sine and a sine wave are different things. If you're asking about the definition of a sine, that's completely off topic. Other answers are just trying to say "a sine is the solution to the differential equation associated with a harmonic oscillator, here are a few places where you'll find a harmonic oscillator in electronics". Fact of the matter is that a sine can be defined in many ways, all of them axiomatically in mathematics. A sine wave can only be defined as in this answer. – DonFusili May 23 '18 at 11:59
• @DonFusili Thanks for the remark, I couldn't express it more clearly. – Michel Keijzers May 23 '18 at 12:09
• Somehow I don't think he would get much in the way of credit for that answer, even though it is accurate – Dirk Bruere May 23 '18 at 12:25
• My sense is that a game sum for certain types of games can also be a expressed as sine wave, until the outcome is determined (score flipping between - and +, where player one is + and player two is -) – DukeZhou May 23 '18 at 19:50

The answer given by Florian Castellane shows that the sine wave is the solution for a very basic differential equation. But that answer may be difficult to understand if one hasn't studied differential equations.

When we write:

$a \cdot f'' + f = 0$, or alternatively, $f'' = -\frac{1}{a}\cdot f$

the f is some variable we are measuring, and f'' is its second derivative.

This differential equation appears in very many places in physics:

• Springs: f is position, f' is velocity and f'' is acceleration, and the equation above means: The acceleration is linearly related to position. This is the same as the equation for a spring and mass, where acceleration is given by force $F = kx$.

• Electronics: f is voltage, f' is current and f'' is the rate of change of current. This is the same as the equation for inductors, where rate of change of current is given by $\frac{dI}{dt}=\frac{1}{L}\cdot v$.

But there happens to be also another source of sine waves, and that is anything related to circular rotation. The principle of this is shown well in Andy aka's answer. Circular rotation causes sine waves in e.g. electric generators, and also in our own solar system.

• This. In the context of electrical engineering, the most natural explanation is that it is the solution to a system with a value's second derivative inversely proportional to its current value. – MooseBoys May 24 '18 at 17:43
• @jpa, your "another source", circular motion, is also a place where the same differential equation appears in physics, right? So it could just be a third bullet. Similar to the case with springs, f is vertical component of position, f' is vertical component of velocity, and f'' is vertical component of acceleration. The acceleration is linearly related to position, even if the mechanics are different from those of springs. – LarsH May 25 '18 at 14:19
• @LarsH Yeah, mathematically. But intuitively that seems more like the consequence than the reason. – jpa May 25 '18 at 17:58
• OK. I didn't realize you meant your bullet points to be limited to certain patterns of causation. – LarsH May 25 '18 at 20:05

A sinewave is a waveform that can be expressed in the form $A\sin(\omega t + \varphi)$ (or equivilently with cos or as the real or imaginary part of a complex exponential)

But that is somewhat tautological, what makes sin special? why do we consider sinewaves to be "pure" frequencies.

And the answer to that is how it behaves under differentiation.

$$\frac{d}{dt}A\sin(\omega t + \varphi) = A\omega\cos(\omega t + \varphi) = A\omega\sin(\omega t + \varphi + \frac{\pi}{2})$$

So the derivative of a sinewave is a sinewave at the same frequency. Sure it's phase shifted and has a different amplitude but it's the same frequency and the same shape.

Aside from the arbitary constant the same holds true for integration.

\begin{alignat}{2}\int A\sin(\omega t + \varphi)dt & = -\frac{A}{\omega}\cos(\omega t + \varphi) + C \\ & = \phantom{-}\frac{A}{\omega}\cos(\omega t + \varphi + \pi) +C \\ & = \phantom{-}\frac{A}{\omega}\sin(\omega t + \varphi + \frac{3\pi}{2}) +C \end{alignat}

Sinewaves are the only real periodic functions for which this holds true. All other real periodic functions will change shape when they are differentiated or integrated.

So we can say

"a sinewave is a periodic signal that keeps it's shape and frequency when differentiated or integrated"

• $A\cos(\omega t + \varphi)$ too. It's still called "sine wave" not "cosine wave". – Long Pham May 23 '18 at 14:59
• Yeah, cos is just a phase-shifted version of sin. So the same applies to it. – Peter Green May 23 '18 at 15:11
• Another related issue is that adding Asin(ωt+φ) to the input of any linear filter will add X(ω)sin(ωt+Y(ω)) to the output, for some filter-specific functions X(ω) and Y(ω). A sine wave's shape is invariant not just with respect to integration and differentiation, but to any kind of linear filtering. [A fact which could be useful if one didn't know about the relationship between integration/differentiation and linear filters]. – supercat May 23 '18 at 20:44

Many systems in physics allow for the sudden and surprising appearance of sine waves. When you were young, for example, you've seen ripples in steady water, the motion of a swing after you pushed and let it go, and you've tried bending a stiff ruler and then releasing it. These things, although different, share a common property: they wiggle, or swing, or... vibrate or.. more generally, they go back and forth. Years pass by, then you found yourself in an engineering class, where you study what's really going on with these wiggling stuff you've been observing, only to find out that they wiggle in the same manner! And that is, surprise, surprise, the sine wave. It is the quintessential wave, because its existence in nature is of great significance. Who knows, what if ripples in steady water were square waves, what if the swing's motion takes the form of a square wave, and etc. etc., then the square wave would be the quintessential waveform, it just happens that this isn't true and the sine wave manifests itself in the universe so much.

What's really intriguing is that the sine wave originates from triangles and circles. Now, without knowledge of mathematics, it's really hard to connect the dots from there to manifestations of the sine wave in water, swings, rulers, etc., but the point is that the derivative of a sine wave, is a sine wave, and that is found through the geometry of the circle and the right triangle. And physical systems can be modeled through differential equations, which gives rise to the certainty that sine waves exist in these systems (also don't forget exponentials; their existence in nature is of great significance too; they have a strangely deep connection with sine waves, which is ultimately revealed in Euler's formula).

Another thing about the sine wave is that they can "pass through" some systems quite nicely. Have a sinusoidal input to an LTI system (such as a system built purely of ideal resistors, capacitors, and inductors) and you will get a sinusoidal output (specifically one that preserves the frequency of the input). In other words, the sinusoidal waveform is the only unique waveform that doesn't change its shape through an LTI system. Take a look at this lecture.

And the sad thing about sine waves is, they technically don't exist. Sine waves you get out of nature have some deformations, distortions, noise, and ideal passive components too, don't exist. The best these can get is just close approximations of the sine wave. However if someone is so delicate to advance mathematics such that it takes account these imperfections, then measurements can get more and more precise (which could be limited to the atomic level due to quantum mechanics and all that mumbo jumbo).

• The sine wave often comes from differential equations rather than lines and circles, and there the exponetial fromulation is more apt, it just happens that the sine function is simpler expression. than complex exponentiation. – Jasen May 25 '18 at 10:47
• I was talking about the definition of the sin (and maybe cos) function, the fundamental component of the sine wave. I made a little mistake by not mentioning that. – mjtsquared May 25 '18 at 11:52

An orthogonal projection of a point moving with constant angular speed and direction along a circle, plotted against time.

• – Daniel May 23 '18 at 17:54
• Finally! It seems everybody forgot geometry these days! – Jose Manuel Gomez Alvarez May 28 '18 at 12:16

The easiest way to picture it is it's a projection of a helix onto a plane containing the centerline of the helix. If you put a standard helical spring on an overhead projector, it will project a sine wave. (Rotate to correct the phase accordingly, if you're that much of a purist. :-)

I try to concretize it a bit, by suggesting the idea of building an old school "Plotter" device...something that can roll a sheet of paper forward and back, then has a pen and an arm that can only move on one axis. If you try to get someone to think about building such a machine, then you can easily get them to think about programming it to draw lines and squares. It's also relatively easy to get them to think about drawing a diamond, when they are moving the paper and pen at the same speed.

Then if they start thinking about what it takes to draw a circle, they have to think about what's different from drawing the diamond. They have to speed up and then slow down the arm's movement, and go the other way.

I feel like making it concrete in this way kind of demystifies the graphs.

Imagine an spinning disc. Orient it vertically. Put a glob of chewing gum somewhere on the edge. Look at from the side. place old-fashioned photo paper behind it, and a light in front of it. pull the paper at a constant rate, develop it, and you will see a sine wave.

The sine wave is the basic solution to the simple harmonic motion problem. This is the diff eq y =- k dy^2/dx^2.

If you're dealing with engineering students/someone who's had their first year (semester, whatever) of calculus, you could say that a sine function is a function whose derivative is itself shifted back 90 degrees. In other words, the rate at which it changes position is the same as the rate at which it changes velocity, although not at the same time.

One way to describe what is special about a sine wave is that it is a "pure" frequency. Any analytic repeating function can be described as a combination of sine wave. Sine waves are the building blocks that such functions can be decomposed into.

Sines are also the "natural" waveform that something oscillating produces. Imagine a mass dangling at the end of a spring. Once you get it going, it will bob up and down. With a perfect spring, that vertical movement as a function of time is a sine. In the real world, it will be a sine that decays slowly in amplitude due to the spring dissipating a little energy every time it is flexed.

This same effect can be seen in electronics with a capacitor and inductor in parallel. If you charge up the cap, then close a switch so that the inductor and cap are in parallel, the energy sloshes back and forth between the two indefinitely if they were ideal. Both the voltage and the current are sines, but 90° out of phase with each other. Just like with the spring and mass, in the real world both will actually decay in amplitude over time because some energy is dissipated in the components due to them not being ideal. I go into more detail about such a inductor and capacitor circuit here.

• As discussed in comments on another answer that makes the same argument, you can decompose into infinite sums of square or triangle waves. But the math won't be as nice, and this is where the specialness of sin comes in. – Peter Cordes May 23 '18 at 21:13
• And BTW, the physics term for an ideal oscillator with a proportional to -x is a simple harmonic oscillator, which produces simple harmonic motion. Springs, pendulums (with small amplitude so sin(theta)~=theta), etc. – Peter Cordes May 23 '18 at 21:17
• @Peter: Yes, I agree with both your points. I deliberately left such things out of the answer to keep it simple and in more lay terms. Someone that is asking what a sine wave is isn't likely to understand answers with a lot of math. Given the level of the question, I felt that simplicity of the answer was more important than getting into all the details. – Olin Lathrop May 23 '18 at 21:20
• Ok right, but I don't think you avoid the tautology (or make a correct argument) if you phrase it this way. The reason why sine waves are the natural thing for decomposing signals is a bunch of complicated math. It's a useful thing to know and point out about signals, and I guess about sine waves, but it kind of follows from other factors, like the sin/cos derivative thing (same signal with different phase). Maybe you could say that decomposing into sine waves is natural because that's the sum of simple harmonic oscillators, to sidestep the math and connect the two parts of your answer. – Peter Cordes May 23 '18 at 21:27
• @PeterCordes: Passing a sine wave through any linear filter will yield either DC or a wave with the same shape and frequency. Passing most non-sinusoidal wave forms through most linear filters will yield results that include frequencies that were absent from the original. If one views an oscillator as group of filters configured in a ring, the only periodic waveforms an oscillator can support are those which, when passed through all the filters, will yield the original waveform. While some linear filters may preserve certain non-sinusoidal waveforms, ... – supercat May 25 '18 at 19:17

Think of any type of waveform(square, triangular, sawtooth, pulse ) analogue or digital. All waveforms are made of large number of a kind of wave added together(with different frequencies, amplitudes and phases). This kind is known as the sine wave.

• You could just as well decompose all other waves into sums of triangle waves, or sums of square waves. The math wouldn't be as nice, because sin is special. But why is sin special? You're not really avoiding a tautology. – Peter Cordes May 23 '18 at 16:36
• @PeterCordes: The answer should go further to note that a sine wave is the only kind of wave where linear filtering cannot change the set of frequencies that are present in a passed-through signal (except by eliminating anything other than DC). If one passes a square wave or triangle wave with period 3 through the linear filter function F(f(t))=f(t-1)-f(t)+f(t+1), the result will be a square wave or triangle with period 1 (3x the frequency). – supercat May 23 '18 at 21:07
• @supercat your proposed filter will not give triangle/square wave for a triangle/square input. See input and output. – Ruslan May 24 '18 at 12:56
• @Ruslan: Sorry--I should have made all three terms positive when using a period of 3; the formula I gave would have been correct for a period of 6. In either case, it adds together three signals that are phase shifted by 120 degrees. Such a filter won't preserve the shape of all waveforms, but it does preserve the shape of a number of waveforms including triangle wave, square wave, sawtooth. – supercat May 24 '18 at 14:56