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.
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/)
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.
Simple: a sine wave in time, t, is the imaginary part of:
$$e^{j \omega t}$$
where ω is the angular frequency.
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.
See also this answer: https://electronics.stackexchange.com/a/368217/39297
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.
Start with this:
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.
Ask yourself (or the student(s)):
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.
Here is a another explanation:
Adapated quote:
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.
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"
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.
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).
An orthogonal projection of a point moving with constant angular speed and direction along a circle, plotted against time.
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.
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.
