Sine wave
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave, it is named after the function sine. It occurs in pure and applied mathematics, as well as physics, signal processing and many other fields, its most basic form as a function of time is: y = A sin = A sin where: A, the peak deviation of the function from zero. F, ordinary frequency, the number of oscillations that occur each second of time. Ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second φ, specifies where in its cycle the oscillation is at t = 0. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds. A negative value represents a delay, a positive value represents an advance; the sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform; this property makes it acoustically unique.
In general, the function may have: a spatial variable x that represents the position on the dimension on which the wave propagates, a characteristic parameter k called wave number, which represents the proportionality between the angular frequency ω and the linear speed ν. The wavenumber is related to the angular frequency by:. K = ω v = 2 π f v = 2 π λ where λ is the wavelength, f is the frequency, v is the linear speed; this equation gives a sine wave for a single dimension. 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, 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; this wave pattern occurs in nature, including wind waves, sound waves, light waves. A cosine wave is said to be sinusoidal, because cos = sin , a sine wave with a phase-shift of π/2 radians.
Because of this head start, it is 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. To the human ear, a sound, made of more than one sine wave will have perceptible harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, the reason why the same musical note played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier 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 used in signal processing and the statistical analysis of time series.
Since sine waves propagate without changing form in distributed linear systems, they are used to analyze wave propagation. Sine waves traveling in two directions in space can be represented as u = A sin When two waves having the same amplitude and frequency, traveling in opposite directions, superpose each other a standing wave pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed end
Root mean square
In statistics and its applications, the root mean square is defined as the square root of the mean square. The RMS is known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle. For alternating electric current, RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load. In estimation theory, the root mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data; the RMS value of a set of values is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current is the "value of the direct current that dissipates power in a resistor." In the case of a set of n values, the RMS is x. The corresponding formula for a continuous function f defined over the interval T 1 ≤ t ≤ T 2 is f r m s = 1 T 2 − T 1 ∫ T 1 T 2 2 d t, the RMS for a function over all time is f r m s = lim T → ∞ 1 T ∫ 0 T 2 d t.
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sequence of spaced samples. Additionally, the RMS value of various waveforms can be determined without calculus, as shown by Cartwright. In the case of the RMS statistic of a random process, the expected value is used instead of the mean. If the waveform is a pure sine wave, the relationships between amplitudes and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is: Peak-to-peak = 2 2 × R M S ≈ 2.8 × R M S. For other waveforms the relationships are not the same. For example, for either a triangular or sawtooth wave Peak-to-peak = 2 3 × R M S ≈ 3.5 × R M S. Waveforms made by summing known simple waveforms have an RMS, the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal.
R M S Total = R M S 1 2 + R M S 2 2 + ⋯ + R M S n 2 A special case of RMS of waveform combinations is: R M S AC+DC = R M S DC 2 + R M S AC 2 where R M S DC {\displaystyl
Wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is 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 standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.
A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.
In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.
For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω
Logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; the logarithm of x to base b is denoted as logb . More exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is: log b = y if b y = x. For example, log2 64 = 6, as 26 = 64; the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e as its base; the binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations.
They were adopted by navigators, scientists and others to perform computations more using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: log b = log b x + log b y, provided that b, x and y are all positive and b ≠ 1; the present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel is a unit used to express ratio as logarithms for signal power and amplitude. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, in measurements of the complexity of algorithms and of geometric objects called fractals, they help describing frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, can aid in forensic accounting.
In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant. Addition and exponentiation are three fundamental arithmetic operations. Addition, the simplest of these, can be undone by subtraction: adding, say, 2 to 3 gives 5; the process of adding 2 can be undone by subtracting 2: 5 − 2 = 3. Multiplication, the next-simplest operation, can be undone by division: doubling a number x, i.e. multiplying x by 2, the result is 2x. To get back x, it is necessary to divide by 2. For example 2 ⋅ 3 = 6 and the process of multiplying by 2 is undone by dividing by 2: 6 / 2 = 3; the idea and purpose of logarithms is to undo a fundamental arithmetic operation, namely raising a number to a certain power, an operation known as exponentiation. For example, raising 2 to the third power yields 8, because 8 is the product of three factors of 2: 2 3 = 2 × 2 × 2 = 8 The logarithm of 8 is 3, reflecting the fact that 2 was raised to the third power to get 8.
This subsection contains a short overview of the exponentiation operation, fundamental to understanding logarithms. Raising b to the n-th power, where n is a natural number, is done by multiplying n factors equal to b; the n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors Exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b − 1 is the reciprocal of b. Raising b to the power 1/2 is the square root of b. More raising b to a rational power p/q, where p and q are integers, is given by b p / q = b p q, the q-th root of bp. Any irrational number y can be approximated to arbitrary precision by rational numbers; this can be used to compute the y-th power of b: for example 2 ≈ 1.414... and
Doppler spectroscopy
Doppler spectroscopy is an indirect method for finding extrasolar planets and brown dwarfs from radial-velocity measurements via observation of Doppler shifts in the spectrum of the planet's parent star. 669 extrasolar planets were discovered using Doppler spectroscopy, as of April 2016. Otto Struve proposed in 1952 the use of powerful spectrographs to detect distant planets, he described how a large planet, as large as Jupiter, for example, would cause its parent star to wobble as the two objects orbit around their center of mass. He predicted that the small Doppler shifts to the light emitted by the star, caused by its continuously varying radial velocity, would be detectable by the most sensitive spectrographs as tiny red shifts and blue shifts in the star's emission. However, the technology of the time produced radial-velocity measurements with errors of 1,000 m/s or more, making them useless for the detection of orbiting planets; the expected changes in radial velocity are small – Jupiter causes the Sun to change velocity by about 12.4 m/s over a period of 12 years, the Earth's effect is only 0.1 m/s over a period of 1 year – so long-term observations by instruments with a high resolution are required.
Advances in spectrometer technology and observational techniques in the 1980s and 1990s produced instruments capable of detecting the first of many new extrasolar planets. The ELODIE spectrograph, installed at the Haute-Provence Observatory in Southern France in 1993, could measure radial-velocity shifts as low as 7 m/s, low enough for an extraterrestrial observer to detect Jupiter's influence on the Sun. Using this instrument, astronomers Michel Mayor and Didier Queloz identified 51 Pegasi b, a "Hot Jupiter" in the constellation Pegasus. Although planets had been detected orbiting pulsars, 51 Pegasi b was the first planet found orbiting a main-sequence star, the first detected using Doppler spectroscopy. In November 1995, the scientists published their findings in the journal Nature. Since that date, over 700 exoplanet candidates have been identified, most have been detected by Doppler search programs based at the Keck and Anglo-Australian Observatories, teams based at the Geneva Extrasolar Planet Search.
Beginning in the early 2000s, a second generation of planet-hunting spectrographs permitted far more precise measurements. The HARPS spectrograph, installed at the La Silla Observatory in Chile in 2003, can identify radial-velocity shifts as small as 0.3 m/s, enough to locate many rocky, Earth-like planets. A third generation of spectrographs is expected to come online in 2017. With measurement errors estimated below 0.1 m/s, these new instruments would allow an extraterrestrial observer to detect Earth. A series of observations is made of the spectrum of light emitted by a star. Periodic variations in the star's spectrum may be detected, with the wavelength of characteristic spectral lines in the spectrum increasing and decreasing over a period of time. Statistical filters are applied to the data set to cancel out spectrum effects from other sources. Using mathematical best-fit techniques, astronomers can isolate the tell-tale periodic sine wave that indicates a planet in orbit. If an extrasolar planet is detected, a minimum mass for the planet can be determined from the changes in the star's radial velocity.
To find a more precise measure of the mass requires knowledge of the inclination of the planet's orbit. A graph of measured radial velocity versus time will give a characteristic curve, the amplitude of the curve will allow the minimum mass of the planet to be calculated using the binary mass function; the Bayesian Kepler periodogram is a mathematical algorithm, used to detect single or multiple extrasolar planets from successive radial-velocity measurements of the star they are orbiting. It involves a Bayesian statistical analysis of the radial-velocity data, using a prior probability distribution over the space determined by one or more sets of Keplerian orbital parameters; this analysis may be implemented using the Markov chain Monte Carlo method. The method has been applied to the HD 208487 system, resulting in an apparent detection of a second planet with a period of 1000 days. However, this may be an artifact of stellar activity; the method is applied to the HD 11964 system, where it found an apparent planet with a period of 1 year.
However, this planet was not found in re-reduced data, suggesting that this detection was an artifact of the Earth's orbital motion around the Sun. Although radial-velocity of the star only gives a planet's minimum mass, if the planet's spectral lines can be distinguished from the star's spectral lines the radial-velocity of the planet itself can be found and this gives the inclination of the planet's orbit and therefore the planet's actual mass can be determined; the first non-transiting planet to have its mass found this way was Tau Boötis b in 2012 when carbon monoxide was detected in the infra-red part of the spectrum. The graph to the right illustrates the sine curve using Doppler spectroscopy to observe the radial velocity of an imaginary star, being orbited by a planet in a circular orbit. Observations of a real star would produce a similar graph, although eccentricity in the orbit will distort the curve and complicate the calculations below; this theoretical star's velocity shows a periodic variance of ±1 m/s, suggesting an orbiting mass, creating a gravitational pull on this star.
Using Kepler's third law of planetary
Exoplanet
An exoplanet or extrasolar planet is a planet outside the Solar System. The first evidence of an exoplanet was not recognized as such; the first scientific detection of an exoplanet was in 1988. The first confirmed detection occurred in 1992; as of 1 April 2019, there are 4,023 confirmed planets in 3,005 systems, with 656 systems having more than one planet. There are many methods of detecting exoplanets. Transit photometry and Doppler spectroscopy have found the most, but these methods suffer from a clear observational bias favoring the detection of planets near the star. In several cases, multiple planets have been observed around a star. About 1 in 5 Sun-like stars have an "Earth-sized" planet in the habitable zone. Assuming there are 200 billion stars in the Milky Way, it can be hypothesized that there are 11 billion habitable Earth-sized planets in the Milky Way, rising to 40 billion if planets orbiting the numerous red dwarfs are included; the least massive planet known is Draugr, about twice the mass of the Moon.
The most massive planet listed on the NASA Exoplanet Archive is HR 2562 b, about 30 times the mass of Jupiter, although according to some definitions of a planet, it is too massive to be a planet and may be a brown dwarf instead. There are planets that are so near to their star that they take only a few hours to orbit and there are others so far away that they take thousands of years to orbit; some are so far out. All of the planets detected so far are within the Milky Way. Nonetheless, evidence suggests that extragalactic planets, exoplanets farther away in galaxies beyond the local Milky Way galaxy, may exist; the nearest exoplanet is Proxima Centauri b, located 4.2 light-years from Earth and orbiting Proxima Centauri, the closest star to the Sun. The discovery of exoplanets has intensified interest in the search for extraterrestrial life. There is special interest in planets that orbit in a star's habitable zone, where it is possible for liquid water, a prerequisite for life on Earth, to exist on the surface.
The study of planetary habitability considers a wide range of other factors in determining the suitability of a planet for hosting life. Besides exoplanets, there are rogue planets, which do not orbit any star; these tend to be considered as a separate category if they are gas giants, in which case they are counted as sub-brown dwarfs, like WISE 0855−0714. The rogue planets in the Milky Way number in the billions; the convention for designating exoplanets is an extension of the system used for designating multiple-star systems as adopted by the International Astronomical Union. For exoplanets orbiting a single star, the designation is formed by taking the name or, more designation of its parent star and adding a lower case letter; the first planet discovered in a system is given the designation "b" and planets are given subsequent letters. If several planets in the same system are discovered at the same time, the closest one to the star gets the next letter, followed by the other planets in order of orbital size.
A provisional IAU-sanctioned standard exists to accommodate the designation of circumbinary planets. A limited number of exoplanets have IAU-sanctioned proper names. Other naming systems exist. For centuries scientists and science fiction writers suspected that extrasolar planets existed, but there was no way of detecting them or of knowing their frequency or how similar they might be to the planets of the Solar System. Various detection claims made in the nineteenth century were rejected by astronomers; the first evidence of an exoplanet was not recognized as such. The first suspected scientific detection of an exoplanet occurred in 1988. Shortly afterwards, the first confirmed detection came in 1992, with the discovery of several terrestrial-mass planets orbiting the pulsar PSR B1257+12; the first confirmation of an exoplanet orbiting a main-sequence star was made in 1995, when a giant planet was found in a four-day orbit around the nearby star 51 Pegasi. Some exoplanets have been imaged directly by telescopes, but the vast majority have been detected through indirect methods, such as the transit method and the radial-velocity method.
In February 2018, researchers using the Chandra X-ray Observatory, combined with a planet detection technique called microlensing, found evidence of planets in a distant galaxy, stating "Some of these exoplanets are as small as the moon, while others are as massive as Jupiter. Unlike Earth, most of the exoplanets are not bound to stars, so they're wandering through space or loosely orbiting between stars. We can estimate. In the sixteenth century the Italian philosopher Giordano Bruno, an early supporter of the Copernican theory that Earth and other planets orbit the Sun, put forward the view that the fixed stars are similar to the Sun and are accompanied by planets. In the eighteenth century the same possibility was mentioned by Isaac Newton in the "General Scholium" that concludes his Principia. Making a comparison to the Sun's planets, he wrote "And if the fixed stars are the centres of similar systems, they will all be constructed according to a similar design and subject to the dominion of One."In 1952, more than 40 years before the first hot Jupiter was discovere
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics and fluid dynamics; the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the wave equation is a partial differential equation that may constrain some scalar function u = u of a time variable t and one or more spatial variables x1, x2, … xn. The quantity u may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions; the equation is ∂ 2 u ∂ t 2 = c 2. Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as u ¨ = c 2 ∇ 2 u where ◻ ¨ denotes double time derivative, ∇ is the nabla operator, ∇2 = ∇ · ∇ is the Laplacian operator: u ¨ = ∂ 2 u ∂ t 2 ∇ = ∇ 2 = ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 + ⋯ + ∂ 2 ∂ x n 2 A solution of this equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed c.
This analysis is possible. This property is called the superposition principle in physics; the wave equation alone does not specify a physical solution. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments; the wave equation is the simplest example of a hyperbolic differential equation. It, its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity and many other scientific and technical disciplines; the wave equation in one space dimension can be written as follows: ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2. This equation is described as having only one space dimension x, because the only other independent variable is the time t; the dependent variable u may represent a second space dimension, if, for example, the displacement u takes place in y-direction, as in the case of a string, located in the x–y plane.
The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string, vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. Another physical setting for deriva
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This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".
This article incorporates