“If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.” ~ Nikola Tesla

## Definition

A sine wave, or sinusoid, is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the trigonometric function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields.

Its most basic form as a function of time (t) is y(t) = Asin(2πft + φ) = Asin(ωt + φ)

Where:

• A, amplitude, the peak deviation of the function from zero.
• f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
• ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second
• φ, phase, specifies (in radians) where in its cycle the oscillation is at t = 0.

When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.1

A sine wave, as illustrated above, has the following characteristics:

### Amplitude (A)

Amplitude is the peak deviation of the function from zero.

Two waves may have the same wavelength, but the crest of one may rise higher above the reference line than the crest of another. The height of a wave crest above the reference line is called the amplitude of the wave. The amplitude of a wave gives a relative indication of the amount of energy the wave transmits – in other words the signal’s strength..

### Angular Frequency (ω)

Angular frequency is the rate of change of the function argument in units of radians per second. Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π. This figure uses the symbol ν, rather than f to denote frequency.4

Angular frequency ω is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. One revolution is equal to 2π radians.

### Angular Wavenumber (k)

In theoretical physics, a wave number defined as the number of radians per unit distance, sometimes called “angular wavenumber”. It is a scalar quantity represented by k and the mathematical representation is given as follows: 7

Where:

• k is the wave number
• λ is the wavelength

### Cycle

A continuous sine wave is composed of several repeating parts called “cycles”. Points ABCDE comprise one complete cycle having a maximum value above, and a maximum value below, the reference line. The maximum value above the line is referred to as the TOP or CREST, and the maximum value below the line is called the BOTTOM or TROUGH, as depicted in the figure. Therefore, one cycle has one crest and one trough.

### Frequency (f)

Ordinary frequency, the number of oscillations (cycles) that occur each second of time. Frequency is measured in hertz (Hz) which is equal to one event (cycle) per second.5

When a continuous series of waves passes through a medium (like air), a certain number of individual waves pass a given point in a specific amount of time. The number of cycles of a continuous wave per unit of time is called the frequency of the wave and is measured in Hertz. One Hertz (abbreviated Hz) is one cycle per second. Therefore, if 5 waves pass a point in one second, the frequency of the wave is 5 cycles per second or 5 Hz.3

### Period (T)

For cyclical phenomena such as oscillations, waves, or for examples of simple harmonic motion, the term frequency is defined as the number of cycles (see Cycle definition above) or vibrations per unit of time. The conventional symbol for frequency is f; the Greek letter ν (nu) is also used. The period T is the time taken to complete one cycle of an oscillation. The relation between the frequency and the period is given by the equation:5

### Phase (Φ)

Phase specifies (in radians) where in its cycle the oscillation is at t=0.

In physics and mathematics, the phase of a periodic function of some real variable (such as time) is an angle-like quantity representing the fraction of the cycle covered up to . It is denoted  and expressed in such a scale that it varies by one full turn as the variable goes through each period (and goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or 2π as the variable completes a full period.6

### Wavelength (λ)

The wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests (on top), or troughs (on bottom), or corresponding zero crossings.

A wavelength is the distance in space occupied by one cycle of a radio wave at any given instant. If the wave could be frozen in place and measured, the wavelength would be the distance from the leading edge of one cycle to the corresponding point on the next cycle. Wavelengths vary from a few hundredths of an inch at extremely high frequencies to many miles at extremely low frequencies; however, common practice is to express wavelengths in meters. The distance between A and E is one wavelength.

## Who

Most of us are introduced to sine and cosine in Geometry class. We’re taught that sine and cosine are functions whose primary use is the determination of unknown lengths and angles when working with right triangles. I think it’s better to think of sine waves as expressions of pure periodic movement, and to focus on the deep relationship between sine waves and circles.8

## What

Understanding the relationship between the sine curve and the unit circle is a basic trigonometric concept which you need to understand and starting with the sine wave equation:

y(t) = Asin(2πft + φ) = Asin(ωt + φ)

In this equation, f is the frequency in cycles per second. Thus, f = 5 cps means the point goes around the circle 5 times every second. The value ft is then the number of cycles that elapse in time t. For example, if f = 100 cps and t = 2 seconds, we’d know that ft = 200 cycles: the wave would have gone through 200 cycles in those 2 seconds.

This leaves us with noticing that the ω above in the right hand formulation equals 2πf in the central formulation. We understand then that ω is the angular frequency, which is the rate of change of the function argument in radians per second. Thus ω is simply taken to mean the number of times around the circle in time t.

The value of φ is just the angle at which the unit radius starts (i.e., the angle at t=0). In many applications of interest, we’ll start the radius at an angle of 0°, so will equal 0, and our equation will be . For these purposes we can ignore the phase shift part, and we can also understand A to simply mean the size of the wave (i.e., its maximum displacement from zero), with 1 being its maximum in terms of the unit circle.

We see that a sine wave is a function that shows the position of the angular frequency of the wave at time t, expressed in radians, and offset by a phase shift (if present). For these purposes we can ignore the phase shift part, and we can also understand A to simply mean the size of the wave, with 1 being its maximum in terms of the unit circle.9

## Why

Practically everything in reality oscillates. All electromagnetic energy, including visible light, microwaves, radio waves, and x-rays, can be represented by a sine wave. At the lowest level, even matter oscillates like a wave, but for macroscopic objects, these oscillations are so minimal are to be impossible to measure. Sound waves can be represented as sine waves, and the up-and-down waves on an oscilloscope may be the most widely known representation of these waves. The study of sine and related functions is the most basic kind of higher (post-algebra) mathematics.

Besides appearing in sound waves, light waves, and ocean waves, the sine wave is also very important in electronics, as it can be used to model the intensity of an alternating current. The current of a direct current full-wave rectification system, used to convert AC into DC, can be modeled using an absolute value sine wave, where the wave is similar to a normal sine wave because the value always stays above the x-axis, but has twice as many peaks. Along with the sine wave is its cousin, the cosine wave, which is exactly the same except displaced to the right by half a cycle.2

All waves can be represented by a periodic function and any repeated shape represents a periodic function. Most periodic functions, both in the real world and in theory, are pretty complicated, at least mathematically. There’s no rule that waves have to be sinusoidal in nature. Some waves show different waveforms, and many of waves aren’t even continuous and hence aren’t described by sine functions. But it’s useful to model most continuous waves in this manner because trigonometric functions are the easiest periodic functions (easiest to use). It is all about modeling data.

In order to understand in a simple way, the way the waves, that you are considering, behave in a similar way to the functions of a sine wave.

They have properties similar to them and so the sine and cosine waves are the easiest of the periodic functions to represent them. It’s not that exactly that shape will apply to waves. It’s not a compulsion that waves are represented only in the form of sine curves. There are many waves which don’t have a shape like that, especially triangular waves and rectangular waves. But, this periodic function makes representation a lot more easier.

It is much easier to understand the properties of waves when we use sine and cosine graphs to describe them.

They closely represent these waves in their propagation.

## How

I don’t show you how to add sine waves or do a Fourier analysis to study how general functions can be decomposed into trigonometric or exponential functions with definite frequencies. Many of the References and Additional Reading websites and Videos will assist you with that.

⭐ Please see the Mathematical Mysteries page for a simple understanding of Fourier Series.

As some professors say: “It is intuitively obvious to even the most casual observer.”

## References

1 “Sine Wave – Wikipedia”. 2014. en.wikipedia.org. https://en.wikipedia.org/wiki/Sine_wave.

2 “What Is A Sine Wave? (With Pictures)”. 2011. https://www.wise-geek.com/what-is-a-sine-wave.htm.

4 “Angular Frequency – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Angular_frequency.

5 “Frequency – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Frequency.

6 “Phase (Waves) – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Phase_(waves).

7 “Wavenumber – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Wavenumber.

8 “Circles Sines And Signals – Sine And Cosine”. 2021. jackschaedler.github.io. https://jackschaedler.github.io/circles-sines-signals/sincos.html.

9 “On The Mathematics of the Sine Wave”. 2021. Earlham College. http://legacy.earlham.edu/~tobeyfo/musictechnology/2_SineWaveMath_edit.html.

10 “Why Can A Wave Be Expressed With A Sine Function?” Physics Stack Exchange. https://physics.stackexchange.com/questions/406469/why-can-a-wave-be-expressed-with-a-sine-function.

“2.1.2 Properties Of Sine Waves – Digital Sound & Music”. 2021. digitalsoundandmusic.com. http://digitalsoundandmusic.com/2-1-2-properties-of-sine-waves/.

“Circles Sines And Signals – Introduction”. 2021. jackschaedler.github.io. https://jackschaedler.github.io/circles-sines-signals/index.html.

“Roll Your Own Math – Sine & Cosine”. 2017. F1LT3R. https://f1lt3r.io/roll-your-own-math-sine-cosine.

“Why Sine Waves? – Audio Precision”. 2021. Audio Precision. https://www.ap.com/blog/why-sine-waves/.

## Interactive Sine Wave Demonstrations

“Addition Of Sine Waves”. 2021. Geogebra. https://www.geogebra.org/m/BOMfKCIK.

“Derivatives Of The Sum Of Two Sines – Wolfram Demonstrations Project”. 2021. demonstrations.wolfram.com. https://demonstrations.wolfram.com/DerivativesOfTheSumOfTwoSines/.

“Sum Of Sines – Wolfram Demonstrations Project”. 2021. demonstrations.wolfram.com. https://demonstrations.wolfram.com/SumOfSine.

“Sums Of Sine Waves With Several Step Sizes (Sawtooth Or Square Approximations) – Wolfram Demonstrations Project”. 2021. demonstrations.wolfram.com. https://demonstrations.wolfram.com/SumsOfSineWavesWithSeveralStepSizesSawtoothOrSquareApproxima/.

## Videos

“Stanford Engineering Everywhere | EE261 – The Fourier Transform And Its Applications”. 2021. see.stanford.edu. https://see.stanford.edu/Course/EE261.

Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood provides an overview of the course, then begins lecturing on Fourier series. The Fourier transform is a tool for solving physical problems. In this course the emphasis is on relating the theoretical principles to solving practical engineering and science problems.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.