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