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62-501: Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals ( sound ), radio waves , and light . For example, if a heart beats at a frequency of 120 times per minute (2 hertz), the period—the time interval between beats—is half a second (60 seconds divided by 120). For cyclical phenomena such as oscillations , waves , or for examples of simple harmonic motion ,

124-633: A sequence of real numbers , oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set ). Medium (optics) In optics , an optical medium is material through which light and other electromagnetic waves propagate. It is a form of transmission medium . The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The optical medium has an intrinsic impedance , given by where E x {\displaystyle E_{x}} and H y {\displaystyle H_{y}} are

186-440: A specific range of frequencies . The audible frequency range for humans is typically given as being between about 20 Hz and 20,000 Hz (20 kHz), though the high frequency limit usually reduces with age. Other species have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz. In many media, such as air, the speed of sound is approximately independent of frequency, so

248-411: A central value (often a point of equilibrium ) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current . Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example

310-411: A fractional error of Δ f f = 1 2 f T m {\textstyle {\frac {\Delta f}{f}}={\frac {1}{2fT_{\text{m}}}}} where T m {\displaystyle T_{\text{m}}} is the timing interval and f {\displaystyle f} is the measured frequency. This error decreases with frequency, so it

372-540: A harmonic oscillator near equilibrium. An example of this is the Lennard-Jones potential , where the potential is given by: U ( r ) = U 0 [ ( r 0 r ) 12 − ( r 0 r ) 6 ] {\displaystyle U(r)=U_{0}\left[\left({\frac {r_{0}}{r}}\right)^{12}-\left({\frac {r_{0}}{r}}\right)^{6}\right]} The equilibrium points of

434-416: A known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a diode . This creates a heterodyne or "beat" signal at the difference between the two frequencies. If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. This process only measures the difference between the unknown frequency and

496-461: A mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control and control theory (e.g. in sliding mode control ), where the aim is convergence to stable state . In these cases it is called chattering or flapping, as in valve chatter, and route flapping . The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension . Such

558-418: A new restoring force in the opposite sense. If a constant force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period . The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by

620-549: A quadratic equation. ( 3 k − m ω 2 ) ( k − m ω 2 ) = 0 ω 1 = k m , ω 2 = 3 k m {\displaystyle {\begin{aligned}&\left(3k-m\omega ^{2}\right)\left(k-m\omega ^{2}\right)=0\\&\omega _{1}={\sqrt {\frac {k}{m}}},\;\;\omega _{2}={\sqrt {\frac {3k}{m}}}\end{aligned}}} Depending on

682-402: A repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the period. For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz . {\displaystyle f={\frac {71}{15\,{\text{s}}}}\approx 4.73\,{\text{Hz}}.} If

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744-706: A similar solution, but now there is a different equation for every direction. x ( t ) = A x cos ⁡ ( ω t − δ x ) , y ( t ) = A y cos ⁡ ( ω t − δ y ) , ⋮ {\displaystyle {\begin{aligned}x(t)&=A_{x}\cos(\omega t-\delta _{x}),\\y(t)&=A_{y}\cos(\omega t-\delta _{y}),\\&\;\,\vdots \end{aligned}}} With anisotropic oscillators, different directions have different constants of restoring forces. The solution

806-483: A sinusoidal position function: x ( t ) = A cos ⁡ ( ω t − δ ) {\displaystyle x(t)=A\cos(\omega t-\delta )} where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between

868-455: A system approaches continuity ; examples include a string or the surface of a body of water . Such systems have (in the classical limit ) an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically propagate. The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of

930-411: A system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position, establishing

992-494: Is called a radio wave . Likewise, an electromagnetic wave with a frequency higher than 8 × 10 Hz will also be invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higher-frequency waves are called X-rays , and higher still are gamma rays . All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation . They all travel through vacuum at

1054-458: Is expressed with the unit reciprocal second (s) or, in the case of radioactivity, with the unit becquerel . It is defined as a rate , f = N /Δ t , involving the number of entities counted or the number of events happened ( N ) during a given time duration (Δ t ); it is a physical quantity of type temporal rate . Oscillation Oscillation is the repetitive or periodic variation, typically in time , of some measure about

1116-411: Is generally a problem at low frequencies where the number of counts N is small. An old method of measuring the frequency of rotating or vibrating objects is to use a stroboscope . This is an intense repetitively flashing light ( strobe light ) whose frequency can be adjusted with a calibrated timing circuit. The strobe light is pointed at the rotating object and the frequency adjusted up and down. When

1178-1749: Is given by resolving the motion into normal modes . The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses. { m 1 x ¨ 1 = − ( k 1 + k 2 ) x 1 + k 2 x 2 m 2 x ¨ 2 = k 2 x 1 − ( k 2 + k 3 ) x 2 {\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}\\m_{2}{\ddot {x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}\end{cases}}} The equations are then generalized into matrix form. F = M x ¨ = k x , {\displaystyle F=M{\ddot {x}}=kx,} where M = [ m 1 0 0 m 2 ] {\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}} , x = [ x 1 x 2 ] {\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}} , and k = [ k 1 + k 2 − k 2 − k 2 k 2 + k 3 ] {\displaystyle k={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}} The values of k and m can be substituted into

1240-400: Is red light, 800 THz ( 8 × 10 Hz ) is violet light, and between these (in the range 400–800 THz) are all the other colors of the visible spectrum . An electromagnetic wave with a frequency less than 4 × 10 Hz will be invisible to the human eye; such waves are called infrared (IR) radiation. At even lower frequency, the wave is called a microwave , and at still lower frequencies it

1302-652: Is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is quasiperiodic . This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat. All real-world oscillator systems are thermodynamically irreversible . This means there are dissipative processes such as friction or electrical resistance which continually convert some of

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1364-472: Is the frequency and λ {\displaystyle \lambda } is the wavelength of the electromagnetic waves. This equation also may be put in the form where ω {\displaystyle \omega } is the angular frequency of the wave and k {\displaystyle k} is the wavenumber of the wave. In electrical engineering , the symbol β {\displaystyle \beta } , called

1426-400: Is the speed of light in vacuum, and this expression becomes f = c λ . {\displaystyle f={\frac {c}{\lambda }}.} When monochromatic waves travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can be done in the following ways: Calculating the frequency of

1488-469: Is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system. Some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some fluid flow. For example, the phenomenon of flutter in aerodynamics occurs when an arbitrarily small displacement of an aircraft wing (from its equilibrium) results in an increase in

1550-805: Is then found, and used to be the effective potential constant: γ eff = d 2 U d r 2 | r = r 0 = U 0 [ 12 ( 13 ) r 0 12 r − 14 − 6 ( 7 ) r 0 6 r − 8 ] = 114 U 0 r 2 {\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}\right]\\[1ex]&={\frac {114U_{0}}{r^{2}}}\end{aligned}}} The system will undergo oscillations near

1612-648: Is used to emphasise that the frequency is characterised by the number of occurrences of a repeating event per unit time. The SI unit of frequency is the hertz (Hz), named after the German physicist Heinrich Hertz by the International Electrotechnical Commission in 1930. It was adopted by the CGPM (Conférence générale des poids et mesures) in 1960, officially replacing the previous name, cycle per second (cps). The SI unit for

1674-419: The angle of attack of the wing on the air flow and a consequential increase in lift coefficient , leading to a still greater displacement. At sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. Resonance occurs in a damped driven oscillator when ω = ω 0 , that is, when the driving frequency is equal to the natural frequency of

1736-474: The electric field and magnetic field , respectively. In a region with no electrical conductivity , the expression simplifies to: For example, in free space the intrinsic impedance is called the characteristic impedance of vacuum , denoted Z 0 , and Waves propagate through a medium with velocity c w = ν λ {\displaystyle c_{w}=\nu \lambda } , where ν {\displaystyle \nu }

1798-405: The simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion . In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely

1860-491: The alternating current in household electrical outlets is 60 Hz (between the tones B ♭ and B; that is, a minor third above the European frequency). The frequency of the ' hum ' in an audio recording can show in which of these general regions the recording was made. Aperiodic frequency is the rate of incidence or occurrence of non- cyclic phenomena, including random processes such as radioactive decay . It

1922-404: The beating of the human heart (for circulation), business cycles in economics , predator–prey population cycles in ecology , geothermal geysers in geology , vibration of strings in guitar and other string instruments , periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy . The term vibration is precisely used to describe

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1984-491: The coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum , where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring. Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to

2046-401: The energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator. Damped oscillators are created when a resistive force is introduced, which is dependent on

2108-419: The equilibrium point. The force that creates these oscillations is derived from the effective potential constant above: F = − γ eff ( r − r 0 ) = m eff r ¨ {\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}} This differential equation can be re-written in

2170-766: The existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium. In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is: F = − k x {\displaystyle F=-kx} By using Newton's second law , the differential equation can be derived: x ¨ = − k m x = − ω 2 x , {\displaystyle {\ddot {x}}=-{\frac {k}{m}}x=-\omega ^{2}x,} where ω = k / m {\textstyle \omega ={\sqrt {k/m}}} The solution to this differential equation produces

2232-802: The first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b . This example assumes a linear dependence on velocity. m x ¨ + b x ˙ + k x = 0 {\displaystyle m{\ddot {x}}+b{\dot {x}}+kx=0} This equation can be rewritten as before: x ¨ + 2 β x ˙ + ω 0 2 x = 0 , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=0,} where 2 β = b m {\textstyle 2\beta ={\frac {b}{m}}} . This produces

2294-1022: The form of a simple harmonic oscillator: r ¨ + γ eff m eff ( r − r 0 ) = 0 {\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0} Thus, the frequency of small oscillations is: ω 0 = γ eff m eff = 114 U 0 r 2 m eff {\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}} Or, in general form ω 0 = d 2 U d r 2 | r = r 0 {\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}} This approximation can be better understood by looking at

2356-421: The frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. Then the frequency can be read from the calibrated readout on the stroboscope. A downside of this method is that an object rotating at an integer multiple of

2418-556: The function are then found: d U d r = 0 = U 0 [ − 12 r 0 12 r − 13 + 6 r 0 6 r − 7 ] ⇒ r ≈ r 0 {\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}\right]\\\Rightarrow r&\approx r_{0}\end{aligned}}} The second derivative

2480-573: The general solution. ( k − M ω 2 ) a = 0 [ 2 k − m ω 2 − k − k 2 k − m ω 2 ] = 0 {\displaystyle {\begin{aligned}\left(k-M\omega ^{2}\right)a&=0\\{\begin{bmatrix}2k-m\omega ^{2}&-k\\-k&2k-m\omega ^{2}\end{bmatrix}}&=0\end{aligned}}} The determinant of this matrix yields

2542-608: The general solution: x ( t ) = e − β t ( C 1 e ω 1 t + C 2 e − ω 1 t ) , {\displaystyle x(t)=e^{-\beta t}\left(C_{1}e^{\omega _{1}t}+C_{2}e^{-\omega _{1}t}\right),} where ω 1 = β 2 − ω 0 2 {\textstyle \omega _{1}={\sqrt {\beta ^{2}-\omega _{0}^{2}}}} . The exponential term outside of

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2604-628: The matrices. m 1 = m 2 = m , k 1 = k 2 = k 3 = k , M = [ m 0 0 m ] , k = [ 2 k − k − k 2 k ] {\displaystyle {\begin{aligned}m_{1}=m_{2}=m,\;\;k_{1}=k_{2}=k_{3}=k,\\M={\begin{bmatrix}m&0\\0&m\end{bmatrix}},\;\;k={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}\end{aligned}}} These matrices can now be plugged into

2666-544: The number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T m {\textstyle \Delta f={\frac {1}{2T_{\text{m}}}}} , or

2728-465: The occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency . Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as Arnold Tongues , can lead to highly complex phenomena as for instance chaotic dynamics. In physics, a system with a set of conservative forces and an equilibrium point can be approximated as

2790-586: The oscillation is said to be driven . The simplest example of this is a spring-mass system with a sinusoidal driving force. x ¨ + 2 β x ˙ + ω 0 2 x = f ( t ) , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=f(t),} where f ( t ) = f 0 cos ⁡ ( ω t + δ ) . {\displaystyle f(t)=f_{0}\cos(\omega t+\delta ).} This gives

2852-445: The others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description

2914-404: The parenthesis is the decay function and β is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where β < ω 0 ; over-damped, where β > ω 0 ; and critically damped, where β = ω 0 . In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case

2976-632: The period, as for all measurements of time, is the second . A traditional unit of frequency used with rotating mechanical devices, where it is termed rotational frequency , is revolution per minute , abbreviated r/min or rpm. 60 rpm is equivalent to one hertz. As a matter of convenience, longer and slower waves, such as ocean surface waves , are more typically described by wave period rather than frequency. Short and fast waves, like audio and radio, are usually described by their frequency. Some commonly used conversions are listed below: For periodic waves in nondispersive media (that is, media in which

3038-500: The positive and negative amplitude forever without friction. In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions. F → = − k r → {\displaystyle {\vec {F}}=-k{\vec {r}}} This produces

3100-423: The potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between potential energy and force. d U d t = − F ( r ) {\displaystyle {\frac {dU}{dt}}=-F(r)} By thinking of

3162-432: The potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between r min {\displaystyle r_{\text{min}}} and r max {\displaystyle r_{\text{max}}} . This approximation is also useful for thinking of Kepler orbits . As the number of degrees of freedom becomes arbitrarily large,

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3224-409: The reference frequency. To convert higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies ( optical heterodyne detection ). Visible light is an electromagnetic wave , consisting of oscillating electric and magnetic fields traveling through space. The frequency of the wave determines its color: 400 THz ( 4 × 10 Hz)

3286-560: The rotation rate of a shaft, mechanical vibrations, or sound waves , can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. As of 2018, frequency counters can cover the range up to about 100 GHz. This represents the limit of direct counting methods; frequencies above this must be measured by indirect methods. Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly utilizing heterodyning ( frequency conversion ). A reference signal of

3348-414: The same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies. c = f λ , {\displaystyle \displaystyle c=f\lambda ,} where c is the speed of light ( c in vacuum or less in other media), f is the frequency and λ is the wavelength. In dispersive media , such as glass, the speed depends somewhat on frequency, so

3410-1065: The solution: x ( t ) = A cos ⁡ ( ω t − δ ) + A t r cos ⁡ ( ω 1 t − δ t r ) , {\displaystyle x(t)=A\cos(\omega t-\delta )+A_{tr}\cos(\omega _{1}t-\delta _{tr}),} where A = f 0 2 ( ω 0 2 − ω 2 ) 2 + 4 β 2 ω 2 {\displaystyle A={\sqrt {\frac {f_{0}^{2}}{(\omega _{0}^{2}-\omega ^{2})^{2}+4\beta ^{2}\omega ^{2}}}}} and δ = tan − 1 ⁡ ( 2 β ω ω 0 2 − ω 2 ) {\displaystyle \delta =\tan ^{-1}\left({\frac {2\beta \omega }{\omega _{0}^{2}-\omega ^{2}}}\right)} The second term of x ( t )

3472-410: The starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system. More special cases are

3534-461: The strobing frequency will also appear stationary. Higher frequencies are usually measured with a frequency counter . This is an electronic instrument which measures the frequency of an applied repetitive electronic signal and displays the result in hertz on a digital display . It uses digital logic to count the number of cycles during a time interval established by a precision quartz time base. Cyclic processes that are not electrical, such as

3596-439: The system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations. The harmonic oscillator and the systems it models have a single degree of freedom . More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of

3658-498: The term frequency is defined as the number of cycles or repetitions per unit of time. The conventional symbol for frequency is f or ν (the Greek letter nu ) is also used. The period T is the time taken to complete one cycle of an oscillation or rotation. The frequency and the period are related by the equation f = 1 T . {\displaystyle f={\frac {1}{T}}.} The term temporal frequency

3720-473: The wave speed is independent of frequency), frequency has an inverse relationship to the wavelength , λ ( lambda ). Even in dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ . {\displaystyle f={\frac {v}{\lambda }}.} In the special case of electromagnetic waves in vacuum , then v = c , where c

3782-417: The wavelength is not quite inversely proportional to frequency. Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances. In general, frequency components of a sound determine its "color", its timbre . When speaking about the frequency (in singular) of a sound, it means the property that most determines its pitch . The frequencies an ear can hear are limited to

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3844-462: The wavelength of the sound waves (distance between repetitions) is approximately inversely proportional to frequency. In Europe , Africa , Australia , southern South America , most of Asia , and Russia , the frequency of the alternating current in household electrical outlets is 50 Hz (close to the tone G), whereas in North America and northern South America, the frequency of

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