The graphs of the sine and cosine functions are sinusoids of different phases.

The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = A \cdot \sin(\omega t + \phi)

where:

• A, the amplitude, is the peak deviation of the function from its center position.
• ω, the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per second
• φ, the phase, specifies where in its cycle the oscillation begins at t = 0.
  • When the phase 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.

	Sine wave 5 seconds of a 220 Hz sine wave
Problems listening to this file? See media help.

The oscillation of an undamped spring-mass system around the equilibrium is a sine wave.

The sine wave is important in physics because it retains its waveshape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

General form[link]

In general, the function may also have:

• a spatial dimension, x (aka position), with wavenumber k
• a non-zero center amplitude, D

which looks like this:

Failed to parse (Missing texvc executable; please see math/README to configure.): y(x,t) = A\cdot \sin(\omega t - kx + \phi ) + D.\,

The wavenumber is related to the angular frequency by:.

Failed to parse (Missing texvc executable; please see math/README to configure.): k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }

where λ is the wavelength, f is the frequency, and c is the speed of propagation.

This equation gives a sine wave for a single dimension, thus the generalized equation given above gives the amplitude of the wave at a position x at time t along a single line. This could, for example, be considered 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 wavenumber 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.

Occurrences[link]

Illustrating the cosine wave's fundamental relationship to the circle.

This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves.

A cosine wave is said to be "sinusoidal", because Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(x) = \sin(x + \pi/2),

which is also a sine wave with a phase-shift of π/2. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.

The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork.

To the human ear, a sound that is made up of more than one sine wave will either sound "noisy" or will have detectable harmonics; this may be described as a different timbre.

Fourier series[link]

Sine, square, triangle, and sawtooth waveforms

In 1822, Joseph Fourier, a French mathematician, discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform including square waves. Fourier used it as an analytical tool in the study of waves and heat flow. It is frequently used in signal processing and the statistical analysis of time series.

Traveling and standing waves[link]

Since sine waves propagate without changing form in distributed linear systems, they are often used to analyze wave propagation. Sine waves traveling in two directions can be represented as

Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = A \sin(\omega t - kx)

 and Failed to parse (Missing texvc executable; please see math/README to configure.): y(t)= A \sin(\omega t + kx)

.

When two waves having the same amplitude and frequency, and traveling in opposite directions, superpose each other, then a standing wave pattern is created.

See also[link]

• Wave (physics)
• Crest (physics)
• Fourier transform
• Harmonic series (mathematics)
• Harmonic series (music)
• Helmholtz equation
• Instantaneous phase
• Pure tone
• Sawtooth wave
• Sinusoidal model
• Simple harmonic motion
• Square wave
• Triangle wave
• Wave equation

References[link]

• Hazewinkel, Michiel, ed. (2001), "Sinusoid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=s085640 ^{[dead link]}

Sine
Basic features
Parity	odd
Domain	(-∞,∞)
Codomain	[-1,1]
Period	2π
Specific values
At zero	0
Maxima	((2k+½)π,1)
Minima	((2k-½)π,-1)
Specific features
Root	kπ
Critical point	kπ-π/2
Inflection point	kπ
Fixed point	0
Variable k is an integer. Domain and codomain are given only for real (not complex) numbers.

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}}
For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

The sine function graphed on the Cartesian plane. In this graph, the angle x is given in radians (π = 180°).

The sine and cosine functions are related in multiple ways. The derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(x)

is Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(x)

. Also they are out of phase by 90°: Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(\pi/2 - x)

= Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(x)

. And for a given angle, cos and sin give the respective x, y coordinates on a unit circle.

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.

Sine is usually listed first amongst the trigonometric functions.

Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.

The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^[1] The word "sine" comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.^[2]

Right-angled triangle definition[link]

For any similar triangle the ratio of the length of the sides remains the same. For example, if the hypotenuse is twice as long, so are the other sides. Therefore respective trigonometric functions, depending only on the size of the angle, express those ratios: between the hypotenuse and the "opposite" side to an angle A in question (see illustration) in the case of sine function; or between the hypotenuse and the "adjacent" side (cosine) or between the "opposite" and the "adjacent" side (tangent), etc.

To define the trigonometric functions for an acute angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

• The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
• The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
• The adjacent side is the side that is in contact with (adjacent to) both the angle we are interested in (angle A) and the right angle, in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate the inside angles of every triangle total 180° (π radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be greater than 0° and less than 90°. The following definition applies to such angles.

The angle A (having measure α) is the angle between the hypotenuse and the adjacent line.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.

Note that this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.

Relation to slope[link]

The trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line.

• When the length of the line segment is 1, sine takes an angle and tells the rise
• Sine takes an angle and tells the rise per unit length of the line segment.
• Rise is equal to sin θ multiplied by the length of the line segment

In contrast, cosine is used for the telling the run from the angle; and tangent is used for telling the slope from the angle. Arctan is used for telling the angle from the slope.

The line segment is the equivalent of the hypotenuse in the right-triangle, and when it has a length of 1 it is also equivalent to the radius of the unit circle.

Relation to the unit circle[link]

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The point's distance from the origin is always 1.

Unlike the definitions with the right or left triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

The unit circle.

Illustration of a unit circle. The radius has a length of 1. The variable t is an angle measure.

Point P(x,y) on the circle of unit radius at an obtuse angle θ > π/2

Animation showing the graphing process of y = sin x (where x is the angle in radians) using a unit circle

Identities[link]

Exact identities (using radians):

These apply for all values of Failed to parse (Missing texvc executable; please see math/README to configure.): \theta .

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin \theta & = \cos \left(\frac{\pi}{2} - \theta \right) \\ & = \frac{1}{\csc \theta} \end{align}

Reciprocal[link]

The reciprocal of sine is cosecant, i.e. the reciprocal of sin(A) is csc(A), or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

Failed to parse (Missing texvc executable; please see math/README to configure.): \csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.

Inverse[link]

The usual principal values of the arcsin(x) function graphed on the cartesian plane. Arcsin is the inverse of sin.

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin⁻¹). As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value.

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \sin^{-1} \left( \frac {a} {h} \right).

k is some integer:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin(y) = x \ \Leftrightarrow\ & y = \arcsin(x) + 2k\pi , \text{ or }\\ & y = \pi - \arcsin(x) + 2k\pi \end{align}

Arcsin satisfies:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(\arcsin x) = x\!

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin(\sin \theta) = \theta\quad\text{for }-\pi/2 \leq \theta \leq \pi/2.

Calculus[link]

For the sine function:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \sin x \,

The derivative is:

Failed to parse (Missing texvc executable; please see math/README to configure.): f'(x) = \cos x \,

The antiderivative is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int f(x)\,dx = -\cos x + C

C denotes the constant of integration.

Other trigonometric functions[link]

The four quadrants of a Cartesian coordinate system.

It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).

Sine in terms of the other common trigonometric functions:

Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.

The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos^2\theta + \sin^2\theta = 1\!

where sin²x means (sin(x))².

Properties relating to the quadrants[link]

Over the four quadrants of the sine function is as follows.

Points between the quadrants. k is an integer.

The quadrants of the unit circle and of sin x, using the Cartesian coordinate system.

For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad): Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(\alpha + 360^\circ) = \sin(\alpha) , or use Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(\alpha + 180^\circ) = -\sin(\alpha) . Or use Failed to parse (Missing texvc executable; please see math/README to configure.): cos(x)= \frac{e^{ix}+e^{-ix}}{2}

 and  Failed to parse (Missing texvc executable; please see math/README to configure.): sin(x)=\frac{e^{ix}-e^{-ix}}{2i}  

For complement of sine, we have Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(180^\circ-\alpha) = \sin(\alpha)

Series definition[link]

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine.

Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin^{(4n+k)}(0)=\begin{cases} 0 & \text{when } k=0 \\ 1 & \text{when } k=1 \\ 0 & \text{when } k=2 \\ -1 & \text{when } k=3 \end{cases}

This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians) :^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt] & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt] \end{align}

If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin x_\mathrm{deg} & = \sin y_\mathrm{rad} \\ & = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots . \end{align}

The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin 0 = 0 & \text{ and } \sin{2x} = 2 \sin x \cos x \\ \cos^2 x + \sin^2 x = 1 & \text{ and } \cos{2x} = \cos^2 x - \sin^2 x \\ \end{align}

The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin x \approx x \text{ when } x \approx 0.

The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.

In general, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.

A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.

Continued fraction[link]

The sine function can also be represented as a generalized continued fraction:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin x = \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 + \cfrac{2\cdot3 x^2}{4\cdot5-x^2 + \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.

The continued fraction representation expresses the real number values, both rational and irrational, of the sine function.

Fixed point[link]

This section requires expansion.

The fixed point iteration x_n+1 = sin x_n with initial value x₀ = 2 converges to 0.

Zero is a fixed point of the sine function because sin(0) = 0.

Law of sines[link]

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.

This is equivalent to the equality of the first three expressions below:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,

where R is the triangle's circumradius.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Values[link]

sin(x)

Some common angles (θ) shown on the unit circle. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos θ, sin θ).

A memory aid (note it does not include 15° and 75°):

90 degree increments:

Other values not listed above:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{60}=\sin 3^\circ=\tfrac{1}{16} \left[2(1-\sqrt3)\sqrt{5+\sqrt5}+\sqrt2(\sqrt5-1)(\sqrt3+1)\right]\,

 A019812

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{30}=\sin 6^\circ=\tfrac{1}{8} \left[\sqrt{6(5-\sqrt5)}-\sqrt5-1\right]\,

 A019815

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{20}=\sin 9^\circ=\tfrac{1}{8} \left[\sqrt2(\sqrt5+1)-2\sqrt{5-\sqrt5}\right]\,

 A019818

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{15}=\sin 12^\circ=\tfrac{1}{8} \left[\sqrt{2(5+\sqrt5)}-\sqrt3(\sqrt5-1)\right]\,

 A019821

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{10}=\sin 18^\circ=\tfrac{1}{4}\left(\sqrt5-1\right)=\tfrac{1}{2}\varphi^{-1}\,

 A019827

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{7\pi}{60}=\sin 21^\circ=\tfrac{1}{16}\left[2(\sqrt3+1)\sqrt{5-\sqrt5}-\sqrt2(\sqrt3-1)(1+\sqrt5)\right]\,

 A019830

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{8}=\sin 22.5^\circ=\tfrac{1}{2}(\sqrt{2-\sqrt{2}}),

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{2\pi}{15}=\sin 24^\circ=\tfrac{1}{8}\left[\sqrt3(\sqrt5+1)-\sqrt2\sqrt{5-\sqrt5}\right]\,

 A019833

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{3\pi}{20}=\sin 27^\circ=\tfrac{1}{8}\left[2\sqrt{5+\sqrt5}-\sqrt2\;(\sqrt5-1)\right]\,

 A019836

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{11\pi}{60}=\sin 33^\circ=\tfrac{1}{16}\left[2(\sqrt3-1)\sqrt{5+\sqrt5}+\sqrt2(1+\sqrt3)(\sqrt5-1)\right]\,

 A019842

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\pi}{5}=\sin 36^\circ=\tfrac14[\sqrt{2(5-\sqrt5)}]\,

 A019845

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{13\pi}{60}=\sin 39^\circ=\tfrac1{16}[2(1-\sqrt3)\sqrt{5-\sqrt5}+\sqrt2(\sqrt3+1)(\sqrt5+1)]\,

 A019848

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{7\pi}{30}=\sin 42^\circ=\frac{\sqrt6\sqrt{5+\sqrt5}-\sqrt5+1}{8}\,

 A019851

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine periodic function with period 2π:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\theta = \sin\left(\theta + 2\pi k \right),\,

for any angle θ and any integer k.

The primitive period (the smallest positive period) of sine is a full circle, i.e. 2π radians or 360 degrees.

Relationship to complex numbers[link]

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Sine is used to determine the imaginary part of a complex number given in polar coordinates (r,φ):

Failed to parse (Missing texvc executable; please see math/README to configure.): z = r(\cos \varphi + i\sin \varphi )\,

the imaginary part is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Im}(z) = r \sin \varphi

r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is a complex number.

Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take a complex number as an argument.

Sine with a complex argument[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin z\,

Domain coloring of sin(z) over (-π,π) on x and y axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude.

sin(z) as a vector field

The definition of the sine function for complex arguments z:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin z & = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \\ & = \frac{e^{i z} - e^{-i z}}{2i}\, \\ & = \frac{\sinh \left( i z\right) }{i} \end{align}

where i ² = −1. This is an entire function. Also, for purely real x,

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin x = \operatorname{Im}(e^{i x}) \,

For purely imaginary numbers:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin iy = i \sinh y \,

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin (x + iy) &= \sin x \cos iy + \cos x \sin iy \\ &= \sin x \cosh y + i \cos x \sinh y \end{align}

Partial fraction and product expansions of complex sine[link]

Using the partial fraction expansion technique in Complex Analysis, one can find that the infinite series

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sum_{n = -\infty}^{\infty}\frac{(-1)^n}{z-n} = \frac{1}{z} + \sum_{n = 1}^{\infty}\frac{2z}{n^2-z^2} \end{align}

both converge and are equal to Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi}{\sin \pi z} .

Similarly we can find

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \frac{\pi^2}{\sin^2 \pi z} = \sum_{-\infty}^\infty \frac{1}{(z-n)^2}. \end{align}

Using product expansion technique, one can derive

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \sin \pi z = \pi z \prod_{n = 1}^\infty ( 1- \frac{z^2}{n^2}) \end{align}

Usage of complex sine[link]

sin z is found in the functional equation for the Gamma function,

Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s}

.

Which in turn is found in the functional equation for the Riemann zeta-function,

Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s)

As a holomorphic function, sin z is a 2D solution of Laplace's equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta u(x_1, x_2) = 0

Complex graphs[link]

Sine function in the complex plane

real component	imaginary component	magnitude

Sin^-1 in the complex plane

real component	imaginary component	magnitude

History[link]

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD).

The function sine (and cosine) can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^[1]

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician Albert Girard; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.^[4] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).^[5] Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.^[6]

Etymology[link]

Look up sine in Wiktionary, the free dictionary.

Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب , abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب , which means "bosom", when the Arabic text was translated in the 12th century into Latin by Gerard of Cremona. The translator used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold") ^[7][8] The English form sine was introduced in the 1590s.

Software implementations[link]

The sin function, along with other trigonometric functions, is widely available across programming languages and platforms. Some CPU architectures have a built-in instruction for sin, including the Intel x86 FPU. In programming languages, sin is usually either a built in function or found within the language's standard math library. There is no standard algorithm for calculating sin. IEEE 754-2008, the most widely used standard for floating-point computation, says nothing on the topic of calculating trigonometric functions such as sin.^[9] Algorithms for calculating sin may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to very different results for different algorithms in special circumstances, such as for very large inputs, e.g. sin(10²²).

A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sin values, for example a value per degree. This allowed results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method typically offers no advantage.

See also[link]

Wikimedia Commons has media related to: Sine function

• Sine wave
• Trigonometric functions
• Sinusoidal model
• Discrete sine transform
• Law of sines
• Sine and cosine transforms
• Sine quadrant
• Aryabhata's sine table
• Madhava's sine table
• Madhava series
• Bhaskara I's sine approximation formula
• Hyperbolic function
• List of trigonometric identities
• Proofs of trigonometric identities
• Euler's formula
• Polar sine — a generalization to vertex angles
• Optical sine theorem
• Generalized trigonometry
• Sine–Gordon equation

Notes[link]

Surface waves in water

In physics, a wave is a disturbance or oscillation that travels through spacetime, accompanied by a transfer of energy. Wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.

There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

The second type of wave, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations in electrical and magnetic fields generated by charged particles, and can therefore travel through a vacuum. These types of waves vary in wavelength, and include radio waves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

A wave can be transverse or longitudinal depending on the direction of its oscillation. Transverse waves occur when a disturbance creates oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations are parallel to the direction of propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse.

General features[link]

A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line.

The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall 1980, p. 8). However, this notion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic (e.g., light) waves in a vacuum, where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are water waves on the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.

It may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved. For example, in the case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh waves, dispersion; and so on.

Other properties, however, although usually described in terms of origin, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently of their physical origin.^[1] For example, based on the mechanical origin of acoustic waves, a moving disturbance in space–time can exist if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. This is impossible because it would violate general relativity. On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion. Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.

Similarly, wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802, in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specific contexts (for example, scattering of sound by sound), is still a researched area in the study of sound.

Mathematical description of one-dimensional waves[link]

Wave equation[link]

Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

Wavelength λ, can be measured between any two corresponding points on a waveform

• in the Failed to parse (Missing texvc executable; please see math/README to configure.): x

direction in space. E.g., let the positive Failed to parse (Missing texvc executable; please see math/README to configure.): x
direction be to the right, and the negative Failed to parse (Missing texvc executable; please see math/README to configure.): x
direction be to the left.

• with constant amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): u

• with constant velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v

, where Failed to parse (Missing texvc executable; please see math/README to configure.): v

is

• • independent of wavelength (no dispersion)
  • independent of amplitude (linear media, not nonlinear).^[2]
• with constant waveform, or shape

This wave can then be described by the two-dimensional functions

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = F(x - v \ t)

 (waveform Failed to parse (Missing texvc executable; please see math/README to configure.): F
traveling to the right)

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = G(x + v \ t)

 (waveform Failed to parse (Missing texvc executable; please see math/README to configure.): G
traveling to the left)

or, more generally, by d'Alembert's formula:^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x,t)=F(x-vt)+G(x+vt). \,

representing two component waveforms Failed to parse (Missing texvc executable; please see math/README to configure.): F

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
traveling through the medium in opposite directions. This wave can also be represented by the partial differential equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,

General solutions are based upon Duhamel's principle.^[4]

Wave forms[link]

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x-direction at velocity v (and G will propagate at the same speed in the negative x-direction).^[5]

In the case of a periodic function F with period λ, that is, F(x + λ − vt) = F(x − vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(x − v(t + T)) = F(x − vt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.^[6]

Amplitude and modulation[link]

Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.

The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form:^[7][8][9]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = A(x, \ t)\sin (kx - \omega t + \phi) \ ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): A(x,\ t)

is the amplitude envelope of the wave, Failed to parse (Missing texvc executable; please see math/README to configure.): k
is the wavenumber and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
is the phase. If the group velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v_g
(see below) is wavelength-independent, this equation can be simplified as:[10]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = A(x - v_g \ t)\sin (kx - \omega t + \phi) \ ,

showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.^[10][11]

Phase velocity and group velocity[link]

Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity.

There are two velocities that are associated with waves, the phase velocity and the group velocity. To understand them, one must consider several types of waveform. For simplification, examination is restricted to one dimension.

This shows a wave with the Group velocity and Phase velocity going in different directions.

The most basic wave (a form of plane wave) may be expressed in the form:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (x, \ t) = A e^{i \left( kx - \omega t \right)} \ ,

which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, Failed to parse (Missing texvc executable; please see math/README to configure.): kx-\omega t = \left(\frac{2\pi}{\lambda}\right)(x - vt) , makes clear that this expression describes a vibration of wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{2\pi}{k}

traveling in the x-direction with a constant phase velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v_p = \frac{\omega}{k}\,

.^[12]

The other type of wave to be considered is one with localized structure described by an envelope, which may be expressed mathematically as, for example:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (x, \ t) = \int_{-\infty} ^{\infty}\ dk_1 \ A(k_1)\ e^{i\left(k_1x - \omega t \right)} \ ,

where now A(k₁) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k₁ = k. In exponential form:

Failed to parse (Missing texvc executable; please see math/README to configure.): A = A_o (k_1) e^ {i \alpha (k_1)} \ ,

with A_o the magnitude of A. For example, a common choice for A_o is a Gaussian wave packet:^[13]

Failed to parse (Missing texvc executable; please see math/README to configure.): A_o (k_1) = N\ e^{-\sigma^2 (k_1-k)^2 / 2} \ ,

where σ determines the spread of k₁-values about k, and N is the amplitude of the wave.

The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k₁), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.^[12] However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of stationary phase for evaluation of such integrals.^[14]) The condition for φ to vary slowly is that its rate of change with k₁ be small; this rate of variation is:^[12]

Failed to parse (Missing texvc executable; please see math/README to configure.): \left . \frac{d \varphi }{d k_1} \right | _{k_1 = k } = x - t \left . \frac{d \omega}{dk_1}\right | _{k_1 = k } +\left . \frac{d \alpha}{d k_1}\right | _{k_1 = k } \ ,

where the evaluation is made at k₁ = k because A(k₁) is centered there. This result shows that the position x where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_g = \frac{d \omega}{dk} \ .

The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantum mechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)²/(2m). Consequently, for that wave situation, the group velocity is

Failed to parse (Missing texvc executable; please see math/README to configure.): v_g = \frac {\hbar k}{m} \ ,

showing that the velocity of a localized particle in quantum mechanics is its group velocity.^[12] Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.^[15] In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wave propagates.

Sinusoidal waves[link]

Sinusoidal waves correspond to simple harmonic motion.

Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): u

described by the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t)= A \sin (kx - \omega t + \phi) \ ,

where

• Failed to parse (Missing texvc executable; please see math/README to configure.): A

is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.

• Failed to parse (Missing texvc executable; please see math/README to configure.): x

is the space coordinate

• Failed to parse (Missing texvc executable; please see math/README to configure.): t

is the time coordinate

• Failed to parse (Missing texvc executable; please see math/README to configure.): k

is the wavenumber

• Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

is the angular frequency

• Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the phase.

The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter).

The wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber Failed to parse (Missing texvc executable; please see math/README to configure.): k

, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation

Failed to parse (Missing texvc executable; please see math/README to configure.): k = \frac{2 \pi}{\lambda}. \,

The period Failed to parse (Missing texvc executable; please see math/README to configure.): T

is the time for one complete cycle of an oscillation of a wave. The frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f
is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:

Failed to parse (Missing texvc executable; please see math/README to configure.): f=\frac{1}{T}. \,

In other words, the frequency and period of a wave are reciprocals.

The angular frequency Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

represents the frequency in radians per second. It is related to the frequency or period by

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = 2 \pi f = \frac{2 \pi}{T}. \,

The wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

of a sinusoidal waveform traveling at constant speed Failed to parse (Missing texvc executable; please see math/README to configure.): v
is given by:[16]

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{v}{f},

where Failed to parse (Missing texvc executable; please see math/README to configure.): v

is called the phase speed (magnitude of the phase velocity) of the wave and Failed to parse (Missing texvc executable; please see math/README to configure.): f
is the wave's frequency.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.^[17]

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.^[18] Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.^[19] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.^[20][21]

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.^[22][23] In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.^[24] When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves[link]

Standing waves[link]

Standing wave in stationary medium. The red dots represent the wave nodes

A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.

• 

  One-dimensional standing waves; the fundamental mode and the first 6 overtones.

• 

  A two-dimensional standing wave on a disk; this is the fundamental mode.

• 

  A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

Physical properties[link]

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves exhibit common behaviors under a number of standard situations, e.g.,

Transmission and media[link]

Waves normally move in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:

• A bounded medium if it is finite in extent, otherwise an unbounded medium
• A linear medium if the amplitudes of different waves at any particular point in the medium can be added
• A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space
• An anisotropic medium if one or more of its physical properties differ in one or more directions
• An isotropic medium if its physical properties are the same in all directions

Absorption[link]

Reflection[link]

When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.

Interference[link]

Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.

Refraction[link]

Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.

Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.

Diffraction[link]

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

Polarization[link]

A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of a polarizing filter. The polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel.

Longitudinal waves such as sound waves do not exhibit polarization. For these waves the direction of oscillation is along the direction of travel.

Dispersion[link]

Schematic of light being dispersed by a prism. Click to see animation.

A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colours and that these colours cannot be decomposed any further.^[25]

Mechanical waves[link]

Waves on strings[link]

The speed of a wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ):

Failed to parse (Missing texvc executable; please see math/README to configure.): v=\sqrt{\frac{T}{\mu}}, \,

where the linear density μ is the mass per unit length of the string.

Acoustic waves[link]

Acoustic or sound waves travel at speed given by

Failed to parse (Missing texvc executable; please see math/README to configure.): v=\sqrt{\frac{B}{\rho_0}}, \,

or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).

Water waves[link]

• Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
• Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas;
• Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
• Ocean surface waves, which are perturbations that propagate through water.

Seismic waves[link]

Shock waves[link]

Other[link]

• Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves^[26]

• Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.

Electromagnetic waves[link]

(radio, micro, infrared, visible, uv)

An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet and X-rays.

Quantum mechanical waves[link]

The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.^[27]

de Broglie waves[link]

Louis de Broglie postulated that all particles with momentum have a wavelength

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{h}{p},

where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10⁻¹³ m.

A wave representing such a particle traveling in the k-direction is expressed by the wave function:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (\mathbf{r}, \ t=0) =A\ e^{i\mathbf{k \cdot r}} \ ,

where the wavelength is determined by the wave vector k as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac {2 \pi}{k} \ ,

and the momentum by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf p = \hbar \mathbf{k} \ .

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,^[28] a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.^[29] Gaussian wave packets also are used to analyze water waves.^[30]

For example, a Gaussian wavefunction ψ might take the form:^[31]

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x,\ t=0) = A\ \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) \ ,

at some initial time t = 0, where the central wavelength is related to the central wave vector k₀ as λ₀ = 2π / k₀. It is well known from the theory of Fourier analysis,^[32] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.^[33] Given the Gaussian:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = e^{-x^2 / (2\sigma^2)} \ ,

the Fourier transform is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} \ .

The Gaussian in space therefore is made up of waves:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk \ ;

that is, a number of waves of wavelengths λ such that kλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

Gravitational waves[link]

Researchers believe that gravitational waves also travel through space, although gravitational waves have never been directly detected. Not to be confused with gravity waves, gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

WKB method[link]

In a nonuniform medium, in which the wavenumber k can depend on the location as well as the frequency, the phase term kx is typically replaced by the integral of k(x)dx, according to the WKB method. Such nonuniform traveling waves are common in many physical problems, including the mechanics of the cochlea and waves on hanging ropes.

References[link]

See also[link]

Sources[link]

• Campbell, M. and Greated, C. (1987). The Musician’s Guide to Acoustics. New York: Schirmer Books.
• French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 0-393-09936-9. OCLC 163810889. 
• Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company. ISBN 0-534-00758-9. .
• Hunt, F. V. (1992) (^{[dead link]}). Origins in Acoustics. New York: Acoustical Society of America Press. http://asa.aip.org/publications.html#pub17. .
• Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 0-8018-5870-4. .
• Vassilakis, P.N. (2001). Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.

External links[link]

Wikimedia Commons has media related to: Wave

Look up wave in Wiktionary, the free dictionary.

• Interactive Visual Representation of Waves
• Science Aid: Wave properties—Concise guide aimed at teens
• Simulation of diffraction of water wave passing through a gap
• Simulation of interference of water waves
• Simulation of longitudinal traveling wave
• Simulation of stationary wave on a string
• Simulation of transverse traveling wave
• Sounds Amazing—AS and A-Level learning resource for sound and waves
• chapter from an online textbook
• Simulation of waves on a string
• of longitudinal and transverse mechanical wave
• MIT OpenCourseWare 8.03: Vibrations and Waves Free, independent study course with video lectures, assignments, lecture notes and exams.