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[[File:Harmonic partials on strings.svg|thumb|250px|[[Vibration]] and [[standing wave]]s in a string, The fundamental and the first 6 [[overtone]]s]] | |||
The '''fundamental frequency''', often referred to simply as the '''fundamental''', is defined as the lowest [[frequency]] of a [[periodic signal|periodic]] [[waveform]]. In terms of a superposition of [[Sine wave|sinusoid]]s (e.g. [[Fourier series]]), the fundamental frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as '''''f''<sub>0</sub>''' (or '''FF'''), indicating the lowest frequency [[Zero-based numbering|counting from zero]].<ref>{{cite web|url=http://www.phon.ucl.ac.uk/home/johnm/sid/sidf.htm |title=sidfn |publisher=Phon.ucl.ac.uk |date= |accessdate=2012-11-27}}</ref><ref>{{cite web|url=http://www.acoustics.hut.fi/publications/files/theses/lemmetty_mst/chap3.html |title=Phonetics and Theory of Speech Production |publisher=Acoustics.hut.fi |date= |accessdate=2012-11-27}}</ref><ref>{{cite web|url=http://fourier.eng.hmc.edu/e101/lectures/Fundamental_Frequency.pdf |title=Fundamental Frequency of Continuous Signals |publisher=Fourier.eng.hmc.edu |accessdate=2012-11-27}}</ref> In other contexts, it is more common to abbreviate it as '''''f''<sub>1</sub>''', the first [[harmonic]].<ref>{{cite web|url=https://nchsdduncanapphysics.wikispaces.com/file/view/Standing+Waves+in+a+Tube+II.pdf |title=Standing Wave in a Tube II - Finding the Fundamental Frequency |publisher=Nchsdduncanapphysics.wikispaces.com |accessdate=2012-11-27}}</ref><ref>{{cite web|url=http://physics.kennesaw.edu/P11_standwaves3.pdf |title=Physics: Standing Waves |publisher=Physics.kennesaw.edu |accessdate=2012-11-27}}</ref><ref>{{cite web|url=http://www.colorado.edu/physics/phys1240/phys1240_fa05/notes/lect24_Tu11_22_4up.pdf |title=Phys 1240: Sound and Music |publisher=Colorado.edu |accessdate=2012-11-27}}</ref><ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html |title=Standing Waves on a String |publisher=Hyperphysics.phy-astr.gsu.edu |date= |accessdate=2012-11-27}}</ref><ref>[http://openlearn.open.ac.uk/mod/oucontent/view.php?id=397877§ion=5.13.2]{{dead link|date=November 2012}}</ref> (The second harmonic is then f<sub>2</sub> = 2⋅f<sub>1</sub>, etc. In this context, the zeroth harmonic would be 0 Hz.) | |||
All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. A single period is thus the smallest repeating unit of a signal, and one period describes the signal completely. We can show a waveform is periodic by finding some period ''T'' for which the following equation is true: | |||
:<math> x(t) = x(t + T)\text{ for all }t \in \mathbb{R} </math> | |||
Where ''x''(''t'') is the function of the waveform. | |||
This means that for multiples of some period T the value of the signal is always the same. The least possible value of T for which this is true is called the fundamental period and the fundamental frequency (''f''<sub>0</sub>) is: | |||
:<math> f_0 = \frac{1}{T}</math> | |||
Where ''f''<sub>0</sub> is the fundamental frequency and ''T'' is the fundamental period. | |||
[[File:F0leftclosed.gif|thumb|F0leftclosed]] | |||
[[File:F0rightclosed.gif|thumb|F0rightclosed]] | |||
For a tube of length ''L'' with one end closed and the other end open the wavelength of the fundamental harmonic is 4''L'', as indicated by the top two animations on the right. Hence, | |||
:<math>\lambda_0 = 4L.</math> | |||
Therefore, using the relation | |||
:<math> \lambda_0 = \frac{v}{f_0}</math> , | |||
where ''v'' is the speed of the wave, we can find the fundamental frequency in terms of the speed of the wave and the length of the tube: | |||
:<math> f_0 = \frac{v}{4L}.</math> | |||
[[File:F0bothclosed.gif|thumb|F0bothclosed]][[File:F0bothopen.gif|thumb|F0bothopen]]If the ends of the same tube are now both closed or both opened as in the bottom two animations on the right, the wavelength of the fundamental harmonic becomes 2''L''. By the same method as above, the fundamental frequency is found to be | |||
:<math> f_0 = \frac{v}{2L}.</math> | |||
At 20 °C (68 °F) the [[speed of sound]] in air is 343 m/s (1129 ft/s). This speed is [[Speed of sound#Practical formula for dry air|temperature dependent]] and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature (1.1 ft/s for every increase of 1 °F). | |||
The velocity of a sound wave at different temperatures:- | |||
*v = 343.2 m/s at 20 °C | |||
*v = 331.3 m/s at 0 °C | |||
== Mechanical systems == | |||
Consider a spring, fixed at one end and having a mass attached to the other; this would be a single degree of freedom (SDoF) oscillator. Once set into motion it will oscillate at its natural frequency. For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness; (providing the system is undamped). The radian frequency, ''ω''<sub>n</sub>, can be found using the following equation: | |||
:<math> \omega_\mathrm{n}^2 = \frac{k}{m} \, </math> | |||
Where:<br/> | |||
''k'' = [[stiffness]] of the spring<br/> | |||
''m'' = mass <br/> | |||
''ω''<sub>n</sub> = radian frequency (radians per second) | |||
From the radian frequency, the natural frequency, ''f''<sub>n</sub>, can be found by simply dividing ''ω''<sub>n</sub> by 2''π''. Without first finding the radian frequency, the natural frequency can be found directly using: | |||
:<math>f_\mathrm{n} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \,</math> | |||
Where:<br/> | |||
''f''<sub>n</sub> = natural frequency in hertz (cycles/second)<br/> | |||
''k'' = stiffness of the spring (Newtons/meter or N/m)<br/> | |||
''m'' = mass(kg) <br/> | |||
while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency. | |||
==See also== | |||
*[[Missing fundamental]] | |||
*[[Natural frequency]] | |||
*[[Oscillation]] | |||
*[[Hertz]] | |||
*[[Electronic tuner]] | |||
*[[Scale of harmonics]] | |||
*[[Pitch detection algorithm]] | |||
==References== | |||
{{Reflist}} | |||
{{Acoustics}} | |||
{{Timbre}} | |||
{{DEFAULTSORT:Fundamental Frequency}} | |||
[[Category:Musical tuning]] | |||
[[Category:Acoustics]] | |||
[[Category:Fourier analysis]] |
Revision as of 22:49, 25 January 2014
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids (e.g. Fourier series), the fundamental frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as f0 (or FF), indicating the lowest frequency counting from zero.[1][2][3] In other contexts, it is more common to abbreviate it as f1, the first harmonic.[4][5][6][7][8] (The second harmonic is then f2 = 2⋅f1, etc. In this context, the zeroth harmonic would be 0 Hz.)
All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. A single period is thus the smallest repeating unit of a signal, and one period describes the signal completely. We can show a waveform is periodic by finding some period T for which the following equation is true:
Where x(t) is the function of the waveform.
This means that for multiples of some period T the value of the signal is always the same. The least possible value of T for which this is true is called the fundamental period and the fundamental frequency (f0) is:
Where f0 is the fundamental frequency and T is the fundamental period.
For a tube of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the top two animations on the right. Hence,
Therefore, using the relation
where v is the speed of the wave, we can find the fundamental frequency in terms of the speed of the wave and the length of the tube:
If the ends of the same tube are now both closed or both opened as in the bottom two animations on the right, the wavelength of the fundamental harmonic becomes 2L. By the same method as above, the fundamental frequency is found to be
At 20 °C (68 °F) the speed of sound in air is 343 m/s (1129 ft/s). This speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature (1.1 ft/s for every increase of 1 °F).
The velocity of a sound wave at different temperatures:-
- v = 343.2 m/s at 20 °C
- v = 331.3 m/s at 0 °C
Mechanical systems
Consider a spring, fixed at one end and having a mass attached to the other; this would be a single degree of freedom (SDoF) oscillator. Once set into motion it will oscillate at its natural frequency. For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness; (providing the system is undamped). The radian frequency, ωn, can be found using the following equation:
Where:
k = stiffness of the spring
m = mass
ωn = radian frequency (radians per second)
From the radian frequency, the natural frequency, fn, can be found by simply dividing ωn by 2π. Without first finding the radian frequency, the natural frequency can be found directly using:
Where:
fn = natural frequency in hertz (cycles/second)
k = stiffness of the spring (Newtons/meter or N/m)
m = mass(kg)
while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency.
See also
- Missing fundamental
- Natural frequency
- Oscillation
- Hertz
- Electronic tuner
- Scale of harmonics
- Pitch detection algorithm
References
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