# Waves

Sinusoidal waves
Fields: Algebra and Geometry
Image Created By: Xah Lee
Website: Xah Lee Web

Sinusoidal waves

This main image is called "Flame."
This image shows the graph of $\sin(x \cdot \sin y)=\cos(y \cdot \cos x)\,$. It is the composition of complicated sinusoidal waves.

# Basic Description

Wave Mathematics - Trigonometric Functions: Waves are familiar to us from the ocean, the study of sound, earthquakes, and other natural phenomena. But as any surfer can tell you, ocean waves come in very different sizes, as can all waves. To fully understand waves, we need to understand measurements associated with these waves, such as frequency, wavelength, and amplitude.

While these measurements help describe waves, they do not help us make predictions about wave behavior. In order to do that, we need to look at waves more abstractly, which we can do using a mathematical formula. It is possible to look at waves mathematically because a wave's shape repeats itself over a consistent interval of time and distance. This behavior mirrors the repetition of the circle. Imagine drawing a circle on a piece of paper. Now imagine drawing that same shape while your friend slowly pulled the piece of paper out from under your pencil - the line you would have drawn traces out the shape of a wave. One rotation around the circle completes one cycle of rising and falling in the wave, as seen in the picture below.

cycle

Mathematicians use the sine function (Sine) to express the shape of a wave. The mathematical equation representing the simplest wave looks like this:

$y=\sin x$

## Basic Graph of the Sine Function

figure 1

Figure 1 shows the graph of the function, which is defined as $y=\sin x$.

The graph repeats itself as it moves along the x-axis. The cycles of this regular repetition are called the period; we use the notation $T$. This graph repeats every $6.28$ units or $2\pi$ radians. It ranges from -1 to 1; half of this distance is called the amplitude, we use notation $\varphi\,$ to represent it. Because $\varphi\,$ is 0, there is no shift from the origin. The graph is symmetric around the point $(0,0)$. So the graph of $f(x)=\sin x\,$ has a period of $2\pi$ and an amplitude of 1.

## History of the Sine Wave

Trigonometry is a field of mathematics first compiled in the 2nd century BCE by the Greek mathematician Hipparchus. The history of trigonometry and of trigonometric functions follows the general lines of the history of mathematics. All trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) can all be simply defined in terms of a single function sine. Sine, as associated with trigonometry, began in early civilization as a very important measuring aid. When the function concept was introduced in about 1700 along with calculus and analytic geometry, sine became a function and has little to do with triangles anymore. The sine function appears unexpectedly throughout analysis, because in essence it captures the idea of a wave, a fundamental concept in physics.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry, Algebra, Trigonometry and a little Physics

## Sinusoidal Waves

A Sine Wave or Sinusoidal Wave, which describes a smooth repetitive [...]

## Sinusoidal Waves

A Sine Wave or Sinusoidal Wave, which describes a smooth repetitive oscillation, is one of the Basic Trigonometric Functions and a periodic function. The basic form of the function is $y(t) = A \cdot \sin(\omega t + \varphi)\,$.

Sinusoidal waves are based on trigonomitric functions.

### General Form of Sine Function

The general form of a Sine Function is $y(x,t) = A\cdot \sin(kx + \omega t- \varphi ) + D\,$.

• $A\,$, amplitude of vibration, measures the peak of the deviation of the function from the center position.
• $k$, wave number, also called the propagation constant, this useful quantity is defined as $2 \pi$ divided by the wavelength, so the SI units are radians per meter, and is also related to the angular frequency: $k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }$.
• $c$, wave speed is the speed of propagation.
• $f$, frequency is the number of cycles in a unit of time. The SI unit of frequncy is the hertz (Hz).
• $\lambda$, wavelength measures the distance between any two points at corresponding positions on successive repetitions in the wave, so e.g. from one crest or trough to the next, in SI units of meters.
• $x$, spatial dimension, also called position, measures the horizontal position of the wave.
• $\omega\,$, angular frequency, measures the frequency of the function appearing in each unit, and is $2 \pi$ times the frequency, in SI units of radians per second.
• $\varphi\,$, phase shift, measures the phase shift from the origin.
• $D$, non-zero center amplitude, also called DC offset, gives the vertical position of the wave.
• $T$, period, an essential concept of the sine wave, measures how long it takes the wave to complete one cycle. $T=\frac{1}{f}=\frac{2 \pi}{\omega}$.

The general form gives a sine wave for a single dimension, thus the generalized equation given above gives the height of the wave at a position x at time t along a single line. This could, for example, be considered as the value of a wave along a wire.

In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wave number k are interpreted as vectors, and their product as a dot product. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.

For more techniques related to the sine function, see also the page "Law of Sines".

Useful formulas transformed by a general sine function:

• $y(x,t)=A \sin \omega (t - \frac{x}{c}) = A \sin 2 \pi f (t- \frac{x}{c})$
• $y(x,t)=A \sin 2 \pi(\frac{t}{T}- \frac{x}{cT})$
• $y(x,t)=A \sin(\omega t- k x)$

### Periodic & Odd

The sine function has a number of properties that result from it being periodic and odd. The basic sine function is periodic with a period of $2 \pi$, which implies that

$\sin(x) = \sin(x + 2 \pi)$

or more generally,

$\sin(x) = \sin(x + 2 \pi k), k \in Z$

$Z$ means integers. Since the period of the function $y=\sin x$ is $2 \pi$, the function $y=\sin(x + 2 \pi)$ is shifted to the left by $2 \pi$. Since the graph of the sine function is periodic and symmetric, the two function's graphs will coincide with each other.

The function is odd; therefore,

$\sin(-x) = -\sin(x)$

Both functions, $\sin(-x)$ and $-\sin(x)$ are created from the original function $y=\sin x$ by flipping it upside down.

### Explore The Graph of Sine Function

We use a the Sine function $f(x)=2\sin(\frac{1}{2}x+\frac{1}{6}\pi)$ as an example. In the following, we use the red curve to represent this original function.

From changing amplitude, perod, and phase shift, we can explore the properties of a general sine function.

Change of The Amplitude

The light green curve, which is "shorter" than the original curve, represents the function $g_1(x)=\sin(\frac{1}{2}x+\frac{1}{6}\pi)$.

Notice how high and how low the graph goes; in mathematics this is called range, but in trigonometric funtions it also called amplitude. By changing the amplitude from 2 to 1, the peak of the graph changes from 2 to 1, and the minimum changes from -2 to -1. What do you think will happen when the sign of $A$ is changed to a negative? The graph will flip over upside down.

Change of The Period

The blue curve represents the function $g_2(x)=2\sin(x+\frac{1}{6}\pi)\,$, which is "wider" than the original function.

From the image, it is obvious that the period of the new function is twice that of the original function. For the red curve, there are two periods in the space where there is one for the blue curve. That means periods occur twice as often for the red curve, or we can say that they are one-half as long.

From the general view, as $\omega$ changes from 2 to 1, the period should shrink, but actually the period grows. Since $\omega$ can only measure the period, but not the real period, we use the notation $T$ to represent the period. $T=\frac{2\pi}{\omega}$, this means the smaller $\omega$ is, the bigger the period is.

Change of The Phase Shift

The black curve, which is almost identical, but just shifted a little bit from the original function, is $g_3(x)=2\sin(\frac{1}{2}x)\,$.

Recall that the red curve is $f(x)=2\sin(\frac{1}{2}x+\frac{1}{6}\pi)\,$. Although the phase shift for the red curve is $\frac{1}{6}\pi\,$, the actual shift is $\frac{1}{3}\pi$.

The actual shift in the graph is different from $\varphi\,$, because the the period and $\omega\,$ affect the shift. As the original function can be written as $f(x)=2\sin(\frac{1}{2}(x+\frac{1}{3}\pi))$, the real shift of the original function away from 0 is $\frac{1}{3}\pi$.

Notice that, if plus a number for $\varphi\,$, the graph will shift to left; if minus a number, the graph will shift to right.

### Graphs of Other Trignometric Functions

This graph can be created by moving function $y=\sin x$ to the left $\frac{\pi}{2}$ units.

## Non-sinusoidal Waves

Non-sinusoidal waves are waves that are not pure sine waves. They are usually derived from simple math functions. While a pure sine consists of a single frequency, non-sinusoidal waveforms can be described as containing multiple sine waves of different frequencies. These "component" sine waves may, or may not, be multiples of a fundamental or "lowest" frequency. The frequency and amplitude of each component can be found using a mathematical technique known as Fourier analysis.

Non-sinusoidal waveforms include square waves, rectangular waves, ramp waves, triangle waves, spiked waves, trapezoidal waves and sawtooth waves...

Non-sinusoidal waveforms include square waves, rectangular waves, ramp waves, triangle waves, spiked waves, trapezoidal waves and sawtooth waves. Here we use the three most common waves as examples for non-sinusoindal waves. $\lfloor t \rfloor$ means the floor of $t$, and is the greatest integer smaller or equal to $t$.

• Sawtooth wave

The sawtooth wave has a shape of saw, and this wave can be represented as the piecewise linear function as $x(t) = t - \lfloor t \rfloor$. Sawtooth wave is found often in time bases for display scanning. It is used as the starting point for subtractive synthesis, as a sawtooth wave of a constant period contains odd and even harmonics that fall off at −6 dB/octave.

• Square wave

This wave can be represented as a piece linear function as $x(t)=A(-1)^{\lfloor \frac{2(x-x_0)}{T} \rfloor}$. This waveform is commonly used to represent digital information. A square wave of a constant period contains odd harmonics that fall off at −6 dB/octave.

• Triangle wave

The wave can be represented as a piecewise linear function $x(t)=\frac{2}{a} \left (t-a \left \lfloor\frac{t}{a}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{t}{a}-\frac{1}{2} \right \rfloor$. Triangle wave contains odd harmonics that fall off at −12 dB/octave.

## Fourier Series

The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound. A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency f and integer multiples of that frequency.

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.

The decomposition process itself is called a Fourier Transform.

The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

Both sinusoidal waves and non-sinusoidal waves can be written in Fourier Series format.

The computation of the Fourier series is based on the integral identities:

$F(\omega) = \mathcal{F}(f)(\omega) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty f(t) e^{-i\omega t}\,dt$.

Expanding the integrand by means of Euler's formula results in:

$F(\omega)=\frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty f(t)(\cos\,{\omega t} - i\,\sin{ \,\omega t})\,dt$,

which may be written as the sum of two integrals:

$F(\omega)=\frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty f(t)\cos\,{\omega t} \,dt - \frac{i}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty f(t)\sin\,{\omega t}\,dt$.

For more about Euler's formula, see Complex Numbers.

Using the method for a generalized Fourier series, the usual Fourier series involving sines and cosines is obtained by taking $f_1(x)=\cos x$ and $f_2(x)=\sin x$. Since these functions form a complete orthogonal system over $[-\pi,\pi]$, the Fourier series of a function $f(x)$ is given by

$f(x)=\frac{1}{2} a_0 + \sum_{n=1}^ \infty a_n \cos(nx)+ \sum_{n=1}^ \infty b_n \sin(nx),$

where

$a_0=\frac{1}{\pi} \int_{-\pi}^\pi f(x) dx$
$a_n=\frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) dx$
$b_n=\frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) dx$

For example, all the non-sinuoidal waves can be written in term of Fourier series, since they are periodic functions.

1. The sawtooth wave can be written as $\frac{1}{2}-\frac{1}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(\frac{n \pi x}{L})$.
2. The triangle wave can be written as $\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(\frac{n \pi x}{L})$, where $n$ could only be odd integers.
3. The square wave can be written as $\frac{8}{\pi^2} \sum_{n=1}^{\infty} \dfrac{(-1)^{\tfrac{n-1}{2}}}{n^2} \sin(\frac{n \pi x}{L})$, where $n$ could only be odd integers.

From the images above, we can see the process of the Fourier transform. Plug different $n$ into the fourier series, we can get different graphs. $n$ is bigger, the graph is closer to the true graph. As the images show, if $n=1$, which represented in orange line, the graph is a sine wave; if $n=20$, which represented in blue line, the graph is almost the true wave form.

# Why It's Interesting

There are lots of waves in nature, and some are made by human beings, like electronic waves.

## Waves In Nature - Mechanical Waves

Physical waves, or mechanical waves, form through the vibration of a medium, or a string, the Earth's crust, or particles of gases and fluids. Waves have mathematical properties that can be analyzed to understand the motion of the wave. Most physical waves are sinusoidal waves.

There are two types of mechanical waves.

• A transverse wave is such that the displacements of the medium are perpendicular (transverse) to the direction of travel of the wave along the medium. Vibrating a string in periodic motion, so the waves move along it, is a transverse wave, as are waves in the ocean.
• A longitudinal wave is such that the displacements of the medium are back and forth along the same direction as the wave itself. Sound waves, where the air particles are pushed along in the direction of travel, is an example of a longitudinal wave.
• Seismic waves, or in general earthquakes, are the waves of energy caused by the sudden breaking of rock within the earth or an explosion. They are the energy that travels through the earth and is recorded on seismographs. Seismic waves have both transverse waves and longitudinal waves. In general, there are two types of seismic waves: body waves and surface waves. Body waves can travel through the earth's inner layers, but surface waves can only move along the surface of the planet like ripples on water. Earthquakes radiate seismic energy as both body and surface waves.
• Body waves travel through the interior of the earth. They arrive before the surface waves emitted by an earthquake. These waves are of a higher frequency than surface waves.
• Primary wave or P wave. This is the fastest kind of seismic wave, and, consequently, the first to 'arrive' at a seismic station. The P wave can move through solid rock and fluids, like water or the liquid layers of the earth. It pushes and pulls the rock it moves through just like sound waves push and pull the air. Have you ever heard a big clap of thunder and heard the windows rattle at the same time? The windows rattle because the sound waves were pushing and pulling on the window glass much like P waves push and pull on rock. Sometimes animals can hear the P waves of an earthquake. Dogs, for instance, commonly begin barking hysterically just before an earthquake 'hits' (or more specifically, before the surface waves arrive). Usually people can only feel the bump and rattle of these waves.
Primary Wave
• Secondary wave or S wave, which is the second wave you feel in an earthquake. An S wave is slower than a P wave and can only move through solid rock, not through any liquid medium. It is this property of S waves that led seismologists to conclude that the Earth's outer core is a liquid. S waves move rock particles up and down, or side-to-side--perpindicular to the direction that the wave is traveling in (the direction of wave propagation).
Seconaary Wave
• Surface waves Travel only through the crust. They are of a lower frequency than body waves, and are easily distinguished on a seismogram as a result. Though they arrive after body waves, it is surface waves that are almost enitrely responsible for the damage and destruction associated with earthquakes. This damage and the strength of the surface waves are reduced in deeper earthquakes.
• Love wave, named after A.E.H. Love, a British mathematician who worked out the mathematical model for this kind of wave in 1911. It's the fastest surface wave and moves the ground from side-to-side. Confined to the surface of the crust, Love waves produce entirely horizontal motion.
Love Wave
• Rayleigh wave, named for John William Strutt, Lord Rayleigh, who mathematically predicted the existence of this kind of wave in 1885. A Rayleigh wave rolls along the ground just like a wave rolls across a lake or an ocean. Because it rolls, it moves the ground up and down, and side-to-side in the same direction that the wave is moving. Most of the shaking felt from an earthquake is due to the Rayleigh wave, which can be much larger than the other waves.
Rayleigh Wave

Even though the waves discussed here will refer to travel in a medium, the mathematics introduced can be used to analyze properties of non-mechanical waves. Electromagnetic radiation, for example, is able to travel through empty space, but still has the same mathematical properties as other waves.

### What Causes A Wave

1. Waves can be viewed as a disturbance in the medium around an equilibrium state, which is generally at rest. The energy of this disturbance is what causes the wave motion. A pool of water is at equilibrium when there are no waves, but as soon as a stone is thrown in it, the equilibrium of the particles is disturbed and the wave motion begins.
2. The disturbance of the wave travels, or propogates, with a definite speed, called the wave speed.
3. Waves transport energy, but not matter. The medium itself doesn't travel; the individual particles undergo back-and-forth or up-and-down motion around the equilibrium position.

## Waves In Electronics - Sine Wave Generation

A function generator is a piece of electronic test equipment or software used to generate electrical waveforms. These waveforms can be either repetitive or single-shot, in which case some kind of triggering source is required (internal or external).

Function Generators are used in development, testing and repair of electronic equipment, e.g. as a signal source to test amplifiers, or to introduce an error signal into a control loop. Producing and manipulating the sine wave function is common used in Physic area. Sine wave circuits is a significant design challenge because they requre a constantly controlled linear oscillator. Sine wave circuitry is required in a number of diverse areas including audio testing, calibration equipment, transducer drives, power conditioning and automatic test equipment (ATE).

Sine wave generator can produce a sine wave of certain amplitude and frequency without an input signal. From the energy view, it is a circuit, which transforms the direct current to alternating current.

### The Components of oscillator

• The amplifier
• The positive feedback
• RC network, LC network, or Quartz notch filter
• RC network, or RC filter, or RC circuit, is A resistor–capacitor network. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit. We use the RC network to produce a low frequency signal that around several hundred kHz.
• LC network, or LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, and a capacitor. Using for create a high frequency signal, that is around several hundred kHz.
• Quartz notch filters, or Quartz crystal filters, are more likely to use to produce a stable high frequency signal over a wide temperature range, since quartz has a very low coefficient of thermal expansion. Quartz filters have much highter quality than RC and LC filters.

# About the Creator of this Image

My name is Xah Lee. (李杀) Am Chinese by blood, but lived most of my adult life in California, USA. I do computer programing for a living, since 1995.

Education: i attended Foothill college and DeAnza college (California, USA) during ~1991 to 1994. These are 2-years community colleges. I took all math courses they offered. Pretty much highest being different equations and linear algebra. That's pretty much my formal education.

Profession: I'm a programer by profession. The companies i worked for are notably Wolfram Research in 1995 for 6 months as a intern, and WebOrder/Netopia during 1999 to 2002. (WebOrder was bought by Netopia in ~1999, and Netopia is bought by Motorola in 2007). My expertise is unix admin, web application development, using unix, Apache, perl. Besides web app dev, my expertise also lies in programing geometry visualization.

My credential are mostly my website xahlee.org, developed since 1996 and is on going. Notably, visited by 9 thousands visitors per day, linked from hundreds of math edu institutions and programing websites, and cited in a few math text books and math journals. I taught once geometry visualization programing using Mathematica to grad math students at National Center for Theoretical Sciences, Taiwan, in 2003. Thanks to professor Richard Palais for the invitation.

# References

[2] Valdez-Sanchez, Luis. Table of Trigonometric Identities. S.O.S. MATH. Dec 13, 1996. Retrieved from http://www.sosmath.com/trig/Trig5/trig5/trig5.html

[6] Welz, Gary Leo. (n.d.). Wave Mathematics - Trigonometric Functions. From Visionlearning. Retrieved from http://www.visionlearning.com/library/module_viewer.php?mid=131

[7] Khamsi, M.A. (n.d.). Fourier Sine and Cosine Series. From S.O.S. MATH. Retrieved from http://www.sosmath.com/fourier/fourier2/fourier2.html

[8] Nave, C.R. Fourier Analysis and Synthesis. From HyperPhysics. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/audio/fourier.html

[13] Weisstein, Eric W. (n.d.). Fourier Series. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FourierSeries.html

[14] Michigan Technological University. (n.d.). What Is Seismology and What Are Seismic Waves? Retrieved from http://www.geo.mtu.edu/UPSeis/waves.html

• Add more things to illustrate the sine curve in other dimensions.
• More detail about Fourier Series.