# ცვლილებები

,  9 წლის წინ
სადაც ''t'' არის დრო, ხოლო ''ω'' არის [[კუთხური სიხშირე]] (''ω''&nbsp;=&nbsp;2π/''T'', სადაც ''T'' არის ტალღის პერიოდი).

ვინაიდან $I_{\mathrm{p}}$ არის დადებითი მუდმივამუდმივი სიდიდე:

:$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {\sin^2(\omega t)}\, dt}}.$

ტრიგონომეტრიული იგივობების გამოყენებით მივიღებთ:
Using a [[List of trigonometric identities|trigonometric identity]] to eliminate squaring of trig function:

:$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {{1 - \cos(2\omega t) \over 2}}\, dt}}$
:$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2} -{ \sin(2\omega t) \over 4\omega}} \right ]_{T_1}^{T_2} }$

მაგრამ ვინაიდან ინტეგრების ინტერვალი მოიცავს რხევის ციკლების მთელ რაოდენობას $\sin$ შემცველი წევრები გაბათილდება და გვექნება:
but since the interval is a whole number of complete cycles (per definition of RMS), the $\sin$ terms will cancel, leaving:

:$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2}} \right ]_{T_1}^{T_2} } = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {{{T_2-T_1} \over 2}} } = {I_\mathrm{p} \over {\sqrt 2}}.$

A similar analysis leads to the analogous equation for sinusoidal voltage:

:$V_{\mathrm{RMS}} = {V_\mathrm{p} \over {\sqrt 2}}.$

Where $I_{\mathrm{P}}$ represents the peak current and $V_{\mathrm{P}}$ represents the peak voltage. It bears repeating that these two solutions are for a sinusoidal wave only.

Because of their usefulness in carrying out power calculations, listed [[voltage]]s for power outlets, e.g. 120 V (USA) or 230 V (Europe), are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies ''V''<sub>''p''</sub>&nbsp;=&nbsp;''V''<sub>RMS</sub>&nbsp;×&nbsp;√2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120&nbsp;×&nbsp;√2, or about 170 volts. The peak-to-peak voltage, being twice this, is about 340 volts. A similar calculation indicates that the peak-to-peak mains voltage in Europe is about 650 volts.

It is also possible to calculate the RMS power of a signal. By analogy with RMS voltage and RMS current, RMS power is the square root of the mean of the square of the power over some specified time period. This quantity, which would be expressed in units of watts (RMS), has no physical significance. However, the term "RMS power" is sometimes used in the audio industry as a synonym for "mean power" or "average power". For a discussion of audio power measurements and their shortcomings, see [[Audio power]].

===Root mean square velocity===
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