# Difference between revisions of "RMS amplitude"

${\displaystyle {\sqrt {2}}/2}$${\displaystyle x_{R}MS={\sqrt {{\tfrac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}}$${\displaystyle f_{R}MS={\sqrt {{\tfrac {1}{T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}[f(t)]^{2}dt}}}$${\displaystyle f_{R}MS={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[f(t)]^{2}dt}}}$$\displaystyle y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)$ $\displaystyle Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}$ $\displaystyle = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}$ $\displaystyle = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}$ ${\displaystyle ={\tfrac {A}{\sqrt {2}}}}$${\displaystyle ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}}$$\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)$ ${\displaystyle fRMS={\sqrt {{\frac {1}{T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}[f(t)]^{2}}}dt}$${\displaystyle f_{R}MS={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[f(2)]^{2}dt}}}$${\displaystyle ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{n}x_{i}^{2}}}}$

### Definition

In statistics, RMS is typical value of a number (n) of values of a quantity (x1, x2, x3…) 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

${\displaystyle x_{RMS}={\sqrt {{\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}}$

The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] 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.