# Difference between revisions of "RMS amplitude"

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=\tfrac{A}{\sqrt{2}} (5) | =\tfrac{A}{\sqrt{2}} (5) | ||

− | </math><math>fRMS=\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2} [f(t)]^2}dt</math><math>f_RMS = \sqrt{\tfrac{1}{T}\int_{0}^{T}[f(2)]^2dt }</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math> | + | </math><math>fRMS=\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2} [f(t)]^2}dt</math><math>f_RMS = \sqrt{\tfrac{1}{T}\int_{0}^{T}[f(2)]^2dt }</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math> |

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=== Definition === | === Definition === | ||

In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup> | In statistics, RMS is typical value of a number (n) of values of a quantity (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>…) equal to the square root of the sum of the squares of the values divided by n. <sup>[1]</sup> |

## Revision as of 09:09, 21 October 2019

**Failed to parse (unknown function "\pift"): {\displaystyle y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)}**
**Failed to parse (unknown function "\omegat"): {\displaystyle Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}}**
**Failed to parse (unknown function "\omegat"): {\displaystyle = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}}**
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}}**
**Failed to parse (syntax error): {\displaystyle x_RMS=\sqrt{\tfrac{1}{n}\sum_{i=1}^nx_i^2} (1) The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is f_{RMS}=\sqrt{\tfrac{T_2}{T_1}\textstyle \int_{T_1}^{T_2} \displaystyle [f(t)]^2}dt (2) and the RMS for a function over all time is f_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [f(t)]^2}dt (3) For a sine wave y(t)=A sin(2πft+φ)=Asin(ωt+φ) (4) where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero) Y_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [Asin(\omegat)]^2dt} =A\sqrt{\tfrac{1}{T}\int\limits_{0}^{T} \tfrac{1-cos(2\omegat)}{2}dt} =A\sqrt{\tfrac{1}{T}[\tfrac{T}{2}-\tfrac{sin(2\omegat)}{4\omega}]_0^T} =\tfrac{A}{\sqrt{2}} (5) }**

### Definition

In statistics, RMS is typical value of a number (n) of values of a quantity (x_{1}, x_{2}, x_{3}…) equal to the square root of the sum of the squares of the values divided by n. ^{[1]}

In geophysics, RMS amplitude is the square root of the average of the squares of a series of measurements. The auto correlation value (without normalizing) for zero lag is the mean square value. For a sine wave, the RMS value is () times the peak amplitude.^{[2]}

In a set of n values {}, the RMS is

The RMS of the corresponding formula for a continuous waveform *f(t)* defined over the interval [T_{1}, T_{2}] is

### Mathematical Expression

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-time waveform. ^{[3]} It’s also known as the quadratic mean of amplitude and is a particular case of the generalized mean with exponent 2.

**References**

[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009.

[2] Encyclopedic Dictionary of Applied Geophysics (Forth Edition). SEG.2002

[3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.