Data collection is the process of gathering and measuring information on targeted variables in an established system, which enables one to answer relevant questions and evaluate outcomes. Data collection is a component of research in all fields of study including physical and social sciences and business. While methods vary by discipline, the emphasis on ensuring accurate and honest collection remains the same; the goal for all data collection is to capture quality evidence that allows analysis to lead to the formulation of convincing and credible answers to the questions that have been posed. Regardless of the field of study or preference for defining data, accurate data collection is essential to maintaining the integrity of research. Both the selection of appropriate data collection instruments and delineated instructions for their correct use reduce the likelihood of errors occurring. A formal data collection process is necessary as it ensures that the data gathered are both defined and accurate.
This way, subsequent decisions based on arguments embodied in the findings are made using valid data. The process provides both a baseline from which to measure and in certain cases an indication of what to improve. Consequences from improperly collected data include: Inability to answer research questions accurately. Distorted findings result in wasted resources and can mislead other researchers into pursuing fruitless avenues of investigation. Bureau of Statistics, Guyana by Arun Sooknarine
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon and the Sun, the rotation of the Earth. Tide tables can be used for any given locale to find the predicted times and amplitude; the predictions are influenced by many factors including the alignment of the Sun and Moon, the phase and amplitude of the tide, the amphidromic systems of the oceans, the shape of the coastline and near-shore bathymetry. They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. Many shorelines experience low tides each day. Other locations have a diurnal tide -- one low tide each day. A "mixed tide" – two uneven magnitude tides a day – is a third regular category. Tides vary on timescales ranging from hours to years due to a number of factors, which determine the lunitidal interval. To make accurate records, tide gauges at fixed stations measure water level over time. Gauges ignore; these data are compared to the reference level called mean sea level.
While tides are the largest source of short-term sea-level fluctuations, sea levels are subject to forces such as wind and barometric pressure changes, resulting in storm surges in shallow seas and near coasts. Tidal phenomena are not limited to the oceans, but can occur in other systems whenever a gravitational field that varies in time and space is present. For example, the shape of the solid part of the Earth is affected by Earth tide, though this is not as seen as the water tidal movements. Tide changes proceed via the following stages: Sea level rises over several hours, covering the intertidal zone; the water rises to its highest level. Sea level falls over several hours; the water stops reaching low tide. Oscillating currents produced by tides are known as tidal streams; the moment that the tidal current ceases is called slack tide. The tide reverses direction and is said to be turning. Slack water occurs near high water and low water, but there are locations where the moments of slack tide differ from those of high and low water.
Tides are semi-diurnal, or diurnal. The two high waters on a given day are not the same height; the two low waters each day are the higher low water and the lower low water. The daily inequality is not consistent and is small when the Moon is over the Equator. From the highest level to the lowest: Highest astronomical tide – The highest tide which can be predicted to occur. Note that meteorological conditions may add extra height to the HAT. Mean high water springs – The average of the two high tides on the days of spring tides. Mean high water neaps – The average of the two high tides on the days of neap tides. Mean sea level – This is the average sea level; the MSL is constant for any location over a long period. Mean low water neaps – The average of the two low tides on the days of neap tides. Mean low water springs – The average of the two low tides on the days of spring tides. Lowest astronomical tide and Chart Datum – The lowest tide which can be predicted to occur. Modern charts use this as the chart datum.
Note that under certain meteorological conditions the water may fall lower than this meaning that there is less water than shown on charts. Tidal constituents are the net result of multiple influences impacting tidal changes over certain periods of time. Primary constituents include the Earth's rotation, the position of the Moon and Sun relative to the Earth, the Moon's altitude above the Earth's Equator, bathymetry. Variations with periods of less than half a day are called harmonic constituents. Conversely, cycles of days, months, or years are referred to as long period constituents. Tidal forces affect the entire earth. In contrast, the atmosphere is much more fluid and compressible so its surface moves by kilometers, in the sense of the contour level of a particular low pressure in the outer atmosphere. In most locations, the largest constituent is the "principal lunar semi-diurnal" known as the M2 tidal constituent, its period is about 12 hours and 25.2 minutes half a tidal lunar day, the average time separating one lunar zenith from the next, thus is the time required for the Earth to rotate once relative to the Moon.
Simple tide clocks track this constituent. The lunar day is longer than the Earth day because the Moon orbits in the same direction the Earth spins; this is analogous to the minute hand on a watch crossing the hour hand at 12:00 and again at about 1:05½. The Moon orbits the Earth in the same direction as the Earth rotates on its axis, so it takes more than a day—about 24 hours and 50 minutes—for the Moon to return to the same location in the sky. During this time, it has passed overhead once and underfoot once, so in many places the period of strongest tidal forcing is the above-mentioned, about 12 hours and 25 minutes; the moment of highest tide is not when the Moon is nearest to zenith or nadir, but the period of the forcing still determines the time between high tides. Because the gravitational field created by the Moon weakens
In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables or canonically conjugate variables such as position x and momentum p, can be known. Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more the position of some particle is determined, the less its momentum can be known, vice versa; the formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard that year and by Hermann Weyl in 1928: where ħ is the reduced Planck constant, h/. The uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level as a physical "explanation" of quantum uncertainty.
It has since become clearer, that the uncertainty principle is inherent in the properties of all wave-like systems, that it arises in quantum mechanics due to the matter wave nature of all quantum objects. Thus, the uncertainty principle states a fundamental property of quantum systems and is not a statement about the observational success of current technology, it must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics observe aspects of it. Certain experiments, may deliberately test a particular form of the uncertainty principle as part of their main research program; these include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems. Applications dependent on the uncertainty principle for their operation include low-noise technology such as that required in gravitational wave interferometers.
The uncertainty principle is not apparent on the macroscopic scales of everyday experience. So it is helpful to demonstrate how it applies to more understood physical situations. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle; the wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in a way that generalizes more easily. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another. A nonzero function and its Fourier transform cannot both be localized. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, a delocalized sine wave.
In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, momentum is its Fourier conjugate, assured by the de Broglie relation p = ħk, where k is the wavenumber. In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value. For example, if a measurement of an observable A is performed the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable. According to the de Broglie hypothesis, every object in the universe is a wave, i.e. a situation which gives rise to this phenomenon. The position of the particle is described by a wave function Ψ.
The time-independent wave function of a single-moded plane wave of wavenumber k0 or momentum p0 is ψ ∝ e i k 0 x = e i p 0 x / ℏ. The Born rule states that this should be interpreted as a probability density amplitude function in the sense that the probability of finding the particle between a and b is P = ∫ a b | ψ | 2 d x. In the case of the single-moded plane wave, | ψ | 2 is a uniform distribution. In other words, the particle position is uncertain in the sense that it could be esse
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not given in the general case, certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean. Consider ƒ an integrable function on the interval. For such an ƒ the Fourier coefficients f ^ are defined by the formula f ^ = 1 2 π ∫ 0 2 π f e − i n t d t, n ∈ Z, it is common to describe its Fourier series by f ∼ ∑ n f ^ e i n t. The notation ~ here means. To investigate this more the partial sums must be defined: S N = ∑ n = − N N f ^ e i n t; the question here is: Do the functions S N converge to ƒ and in which sense? Are there conditions on ƒ ensuring this or that type of convergence? This is the main problem discussed in this article.
Before continuing, the Dirichlet kernel must be introduced. Taking the formula for f ^, inserting it into the formula for S N and doing some algebra gives that S N = f ∗ D N where ∗ stands for the periodic convolution and D N is the Dirichlet kernel, which has an explicit formula, D n = sin sin ; the Dirichlet kernel is not a positive kernel, in fact, its norm diverges, namely ∫ | D n | d t → ∞ a fact that plays a crucial role in the discussion. The norm of Dn in L1 coincides with the norm of the convolution operator with Dn, acting on the space C of periodic continuous functions, or with the norm of the linear functional ƒ → on C. Hence, this family of linear functionals on C is unbounded, when n → ∞. In applications, it is useful to know the size of the Fourier coefficient. If f is an continuous function, | f ^ | ≤ K | n | for K a constant that only depends on f. If f is a bounded variation function, | f ^ | ≤ v a r 2 π | n |. If f ∈ C p | f ^ | ≤ ‖ f ‖ L 1 | n | p. If f ∈ C p and f has modulus of continuity ω p, | f ^ | ≤ ω
In physics and related fields, a wave is a disturbance of a field in which a physical attribute oscillates at each point or propagates from each point to neighboring points, or seems to move through space. The waves most studied in physics are mechanical and electromagnetic. A mechanical wave is a local deformation in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves in air are variations of the local pressure that propagate by collisions between gas molecules. Other examples of mechanical waves are seismic waves, gravity waves and shock waves. An electromagnetic wave consists of a combination of variable electric and magnetic fields, that propagates through space according to Maxwell's equations. Electromagnetic waves can travel through vacuum. Other types of waves include gravitational waves, which are disturbances in a gravitational field that propagate according to general relativity.
Mechanical and electromagnetic waves may seem to travel through space. In mathematics and electronics waves are studied as signals. On the other hand, some waves do not appear to move at all, like hydraulic jumps. Some, like the probability waves of quantum mechanics, may be static in both space. A plane seems to travel in a definite direction, has constant value over any plane perpendicular to that direction. Mathematically, the simplest waves are the sinusoidal ones. Complicated waves can be described as the sum of many sinusoidal plane waves. A plane wave can be transverse, if its effect at each point is described by a vector, perpendicular to the direction of propagation or energy transfer. While mechanical waves can be both transverse and longitudinal, electromagnetic waves are transverse in free space. Consider a traveling transverse wave on a string. Consider the string to have a single spatial dimension. Consider this wave as traveling in the x direction in space. For example, let the positive x direction be to the right, the negative x direction be to the left.
With constant amplitude u with constant velocity v, where v is independent of wavelength independent of amplitude. With constant waveform, or shapeThis wave can be described by the two-dimensional functions u = F u = G or, more by d'Alembert's formula: u = F + G. representing two component waveforms F and G traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation 1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2. General solutions are based upon Duhamel's principle; 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, 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. In the case of a periodic function F with period λ, that is, F = F, 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.
In a similar fashion, this periodicity of F implies a periodicity in time as well: F = F 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. The amplitude of a wave may be constant, 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 w
Theory of tides
The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans under the gravitational loading of another astronomical body or bodies. In 1609 Johannes Kepler suggested that the gravitation of the Moon causes the tides, basing his argument upon ancient observations and correlations; the influence of the Moon on tides was mentioned in Ptolemy's Tetrabiblos as having derived from ancient observation. In 1616, Galileo Galilei wrote Discourse in a letter to Cardinal Orsini. In this discourse, he tried to explain the occurrence of the tides as the result of the Earth's rotation and revolution around the Sun. Galileo believed that the oceans moved like water in a large basin: as the basin moves, so does the water. Therefore, as the Earth revolves, the force of the Earth's rotation causes the oceans to "alternately accelerate and retardate", his view on the oscillation and "alternately accelerated and retardated" motion of the Earth's rotation is a "dynamic process" that deviated from the previous dogma, which proposed "a process of expansion and contraction of seawater."
However, Galileo's theory was erroneous. In subsequent centuries, further analysis led to the current tidal physics. Galileo rejected Kepler's explanation of the tides. Newton, in the Principia, provided a correct explanation for the tidal force, which can be used to explain tides on a planet covered by a uniform ocean, but which takes no account of the distribution of the continents or ocean bathymetry; the dynamic theory of tides predicts the actual real behavior of ocean tides. While Newton explained the tides by describing the tide-generating forces and Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides, developed by Pierre-Simon Laplace in 1775, describes the ocean's real reaction to tidal forces. Laplace's theory of ocean tides took into account friction and natural periods of ocean basins, it predicted the large amphidromic systems in the world's ocean basins and explains the oceanic tides that are observed. The equilibrium theory, based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, other important effects, could not explain the real ocean tides.
Since measurements have confirmed the theory, many things have possible explanations now, like how the tides interact with deep sea ridges and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface. The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters. Satellite observations confirm the accuracy of the dynamic theory, the tides worldwide are now measured to within a few centimeters. Measurements from the CHAMP satellite match the models based on the TOPEX data. Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels. In 1776, Pierre-Simon Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity.
Laplace obtained these equations by simplifying the fluid dynamic equations. But they can be derived from energy integrals via Lagrange's equation. For a fluid sheet of average thickness D, the vertical tidal elevation ζ, as well as the horizontal velocity components u and v satisfy Laplace's tidal equations: ∂ ζ ∂ t + 1 a cos = 0, ∂ u ∂ t − v + 1 a cos ∂ ∂ λ = 0 and ∂ v ∂ t + u +
In music, harmony considers the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. This means occurring frequencies, pitches, or chords; the study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is said to refer to the "vertical" aspect of music, as distinguished from melodic line, or the "horizontal" aspect. Counterpoint, which refers to the relationship between melodic lines, polyphony, which refers to the simultaneous sounding of separate independent voices, are thus sometimes distinguished from harmony. In popular and jazz harmony, chords are named by their root plus various terms and characters indicating their qualities. In many types of music, notably baroque, romantic and jazz, chords are augmented with "tensions". A tension is an additional chord member that creates a dissonant interval in relation to the bass. In the classical common practice period a dissonant chord "resolves" to a consonant chord.
Harmonization sounds pleasant to the ear when there is a balance between the consonant and dissonant sounds. In simple words, that occurs; the term harmony derives from the Greek ἁρμονία harmonia, meaning "joint, concord", from the verb ἁρμόζω harmozō, " fit together, join". In the past, harmony referred to the whole field of music, while music referred to the arts in general. In Ancient Greece, the term defined the combination of contrasted elements: a lower note, it is unclear whether the simultaneous sounding of notes was part of ancient Greek musical practice. In the Middle Ages the term was used to describe two pitches sounding in combination, in the Renaissance the concept was expanded to denote three pitches sounding together. Aristoxenus wrote a work entitled Harmonika Stoicheia, thought the first work in European history written on the subject of harmony, it was not until the publication of Rameau's Traité de l'harmonie in 1722 that any text discussing musical practice made use of the term in the title, although that work is not the earliest record of theoretical discussion of the topic.
The underlying principle behind these texts is that harmony sanctions harmoniousness by conforming to certain pre-established compositional principles. Current dictionary definitions, while attempting to give concise descriptions highlight the ambiguity of the term in modern use. Ambiguities tend to arise from either aesthetic considerations or from the point of view of musical texture (distinguishing between harmonic and "contrapuntal". In the words of Arnold Whittall: While the entire history of music theory appears to depend on just such a distinction between harmony and counterpoint, it is no less evident that developments in the nature of musical composition down the centuries have presumed the interdependence—at times amounting to integration, at other times a source of sustained tension—between the vertical and horizontal dimensions of musical space; the view that modern tonal harmony in Western music began in about 1600 is commonplace in music theory. This is accounted for by the replacement of horizontal composition, common in the music of the Renaissance, with a new emphasis on the vertical element of composed music.
Modern theorists, tend to see this as an unsatisfactory generalisation. According to Carl Dahlhaus: It was not that counterpoint was supplanted by harmony but that an older type both of counterpoint and of vertical technique was succeeded by a newer type, and harmony comprises not only the structure of chords but their movement. Like music as a whole, harmony is a process. Descriptions and definitions of harmony and harmonic practice may show bias towards European musical traditions. For example, South Asian art music is cited as placing little emphasis on what is perceived in western practice as conventional harmony. Pitch simultaneity in particular is a major consideration. Many other considerations of pitch are relevant to the music, its theory and its structure, such as the complex system of Rāgas, which combines both melodic and modal considerations and codifications within it. So, intricate pitch combinations that sound do occur in Indian classical music—but they are studied as teleological harmonic or contrapuntal progressions—as with notated Western music.
This contrasting emphasis manifests itself in the different methods of performance adopted: in Indian Music improvisation takes a major role in the structural framework of a piece, whereas in Western Music improvisation has been uncommon since the end of the 19th century. Where it does occur in Western music, the improvisation either embellishes pre-notated music or draws from musical models established in notated compositions, therefore uses familiar harmonic schemes. Emphasis on the precomposed in European art music and th