The sine wave or sinusoid is a function that occurs often in mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, music Music is an art form whose medium is sound. Common elements of music are pitch , rhythm (and its associated concepts tempo, meter, and articulation), dynamics, and the sonic qualities of timbre and texture. The word derives from Greek μουσική (mousike), "(art) of the Muses", physics Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the world and universe behave, signal processing Signal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals. Depending upon the application, a useful operation could be control, data compression, data transmission, denoising, prediction,, audition Hearing is one of the traditional five senses. It is the ability to perceive sound by detecting vibrations via an organ such as the ear. The inability to hear is called deafness, electrical engineering Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of engineering that deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and, and many other fields. Its most basic form is:
which describes a wavelike function of time (t) with:
- peak deviation from center = A (aka amplitude Amplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation. If the variable undergoes regular oscillations, and a graph of the system)
- angular frequency In physics , angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a ω, (radians The radian is represented by the symbol "rad" or, more rarely, by the superscript c . For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2c" (the second symbol is often mistaken for a degree: "1.2°"). However, the radian is mathematically considered a "pure number" that needs per second)
- phase The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, = θ
- When the phase is non-zero, the entire waveform appears to be shifted in time by the amount θ/ω seconds. A negative value represents a delay, and a positive value represents a "head-start".
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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. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
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General form
In general, the function may also have:
- a spatial dimension, x (aka position), with frequency k (also called wavenumber Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly 2π times that, or the number of radians of phase per unit distance)
- a non-zero center amplitude, D (also called DC Direct current is the undirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also be through semiconductors, insulators, or even through a vacuum as in offset)
which looks like this:
The wavenumber is related to the angular frequency by:.
where λ is the wavelength In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats. It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and, f is the frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency, and c is the speed of propagation The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase speed is given in terms of the.
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 In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector if position x and wavenumber k are interpreted as vectors, and their product as a dot product In mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the orthonormal Euclidean space. It contrasts with the cross product which produces a vector result. 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
Illustrating the sine wave's fundamental relationship to the circle.This wave A wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum, including electromagnetic radiation and probably pattern occurs often in nature, including ocean waves In fluid dynamics, wind waves or, more precisely, wind-generated waves are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They usually result from the wind blowing over a vast enough stretch of fluid surface. Some waves in the oceans can travel thousands of miles before, sound Sound is a travelling wave which is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations waves, and light Light is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye . In physics, the term light sometimes refers to electromagnetic radiation of any wavelength, whether visible or not waves. Also, a rough sinusoidal pattern can be seen in plotting average daily temperatures for each day of the year, although the graph may resemble an inverted cosine In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications wave.
Graphing the voltage of an alternating current In alternating current the movement (or flow) of electric charge periodically reverses direction. An electric charge would for instance move forward, then backward, then forward, then backward, over and over again. In direct current (DC), the movement (or flow) of electric charge is only in one direction gives a sine wave pattern.
A cosine In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications wave is said to be "sinusoidal", because cos(x) = sin(x + π / 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 The ear is the organ that detects sound. The vertebrate ear shows a common biology from fish to humans, with variations in structure according to order and species. It not only acts as a receiver for sound, but plays a major role in the sense of balance and body position. The ear is part of the auditory system can recognize single sine waves as sounding clear because sine waves are representations of a single frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency with no harmonics In acoustics and telecommunication, a harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics have frequencies f, 2f, 3f, 4f, etc. The harmonics have the property that they are all periodic at the fundamental frequency,; some sounds that approximate a pure sine wave are whistling Human whistling is the production of sound by means of carefully controlling a stream of air flowing through a small hole. Whistling can be achieved by creating a small opening with one's lips and then blowing air out of the hole or sucking air into the hole. The air is moderated by the lips, tongue, teeth or fingers to create turbulence, and the, a crystal glass Glass generally refers to hard, brittle, transparent material, such as those used for windows, many bottles, or eyewear. Examples of such solid materials include, but are not limited to, soda-lime glass, borosilicate glass, acrylic glass, sugar glass, Muscovy-glass, or aluminium oxynitride. In the technical sense, glass is an inorganic product of set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal (usually steel). It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a moment to allow some high overtones to.
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 In acoustics and telecommunication, a harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics have frequencies f, 2f, 3f, 4f, etc. The harmonics have the property that they are all periodic at the fundamental frequency,; this may be described as a different timbre In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices or musical instruments. The physical characteristics of sound that mediate the perception of timbre include spectrum and envelope. Timbre is also known in psychoacoustics as tone quality or tone color.
Fourier series
Sine, square A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels, triangle Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave, and sawtooth The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw waveforms Main article: Fourier analysis Today the subject of Fourier analysis encompasses a vast spectrum of mathematics with parts that, at first glance, may appear quite different. In the sciences and engineering the process of decomposing a function into simpler pieces is often called an analysis. The corresponding operation of rebuilding the function from these pieces is known asIn 1822, Joseph Fourier Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their application to problems of heat transfer. The Fourier transform and Fourier's Law are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect, a French mathematician, discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform including square waves A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels. Fourier used it as an analytical tool in the study of waves and heat flow. It is frequently used in signal processing Signal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals. Depending upon the application, a useful operation could be control, data compression, data transmission, denoising, prediction, and the statistical analysis of time series In statistics, signal processing, and many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying context of the data points (Where did they come from?.
See also
- Wave (physics) A wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum, including electromagnetic radiation and probably
- Crest (physics) A crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle
- Fourier transform In mathematics, the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. In such applications as signal processing, the domain of the original function is typically time and is accordingly called the time domain. That of the new function is frequency, and so the Fourier transform is often
- Harmonic series (mathematics) Its name derives from the concept of overtones, or harmonics, in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the term harmonic mean likewise derives from music
- Harmonic series (music) Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with
- Helmholtz equation The Helmholtz equation often arises in the study of physical problems involving partial differential equations in both space and time. The Helmholtz equation, which represents the time-independent form of the original equation, results from applying the technique of separation of variables to reduce the complexity of the analysis
- Instantaneous phase When is constrained to an interval such as or it is called the wrapped phase. Otherwise it is called unwrapped, which is a continuous function of argument assuming is a continuous function of Unless otherwise indicated, the continuous form should be inferred
- Pure tone A sine wave is characterized by its frequency — the number of cycles per second, or its wavelength — the distance the waveform travels through its medium within a period, and the amplitude — the size of each cycle. A pure tone has the unique property that its waveshape and sound are changed only in amplitude and phase by linear acoustic
- Sinusoidal model where C is constant defining a mean level, α is an amplitude for the sine wave, ω is the frequency, Ti is a time variable, φ is the phase, and Ei is the error sequence in approximating the sequence Yi by the model. This sinusoidal model can be fit using nonlinear least squares; to obtain a good fit, nonlinear least squares routines may require
- Simple harmonic motion In physics, simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction
- Wave equation
References
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Categories: Trigonometry | Wave mechanics | Waves
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Q. I have recorded the heat from the mix of Magnesium turns and hydrochloric acid. It took 165 seconds for it to reach it maximum, and the original temperature was at 24.3*C The equation is (what I think) is 13.55sin(period? 2Pi/165 right? + 0 because of no phase shift) + 24.3 How do I find the right period? Thanks, Matt
Asked by Matt W - Thu Mar 15 18:08:18 2007 - - 1 Answers - 0 Comments
A. You shouldn't expect reaction heat data to be sinusoidal. Your heat increases monotonically. That is, you never have less heat with decreasing time. You function is not periodic. Since the rate of heat generation is likely related to how much reactant is left, I would expect an exponential form asymptotic at the total heat of reaction. The general form is: Q = H(1 - e^(-kt)) At time zero (t=0) the heat out is zero. For long time it approaches H. You can estimate H as the final heat value and the time constant (k) from the initial slope. One final note, Q is proportional to the temperature rise.
Answered by Pretzels - Thu Mar 15 21:06:42 2007
