Question about rms value of a sine wave

[qwas]

Sorry if this sounds like a dumb question, but why is the effective value of a sine wave 0.707, as opposed to 0.637 which is the value generated by finding the definite integral over the domain [0,∏] divided length of the domain?

 

[jbunniii]

RMS means "root mean SQUARED", so we have to square the sine wave before integrating. Regarding why we use this measurement, it is essentially a generalization of Euclidean distance: $\sqrt{\int |x(t)|^2 dt}$ is a limiting form of $\sqrt{|x(t_1)|^2 + |x(t_2)|^2 + \ldots + |x(t_n)|^2}$, which is the distance between the point $(x(t_1), x(t_2), \ldots, x(t_n))$ and the origin. There are many other reasons to prefer the RMS as well: it plays nicely with how we measure the energy in a random quantity (variable or process), namely the standard deviation. Also, the RMS of a function/signal is preserved when we transform to the frequency domain via the Fourier transform.

Mathematically, "RMS" is also a common way to measure the norm ("size") of a function: we call it the $L^2$ norm. Working in the $L^2$ space is very nice because it is a Hilbert space, unlike the other $L^p$ spaces, and because the Fourier transform is an isometry on the $L^2$ space. Don't worry if these terms are unfamiliar - you may see them eventually if you study advanced mathematics or physics, but otherwise you can probably live a perfectly happy life if you never hear about them again.

 

[jbunniii]

By the way, the calculation you performed is also a common way of measuring the size of a function/signal. In mathematics we call it the $L^1$ norm: $\int |x(t)| dt$. It is a limiting form of $|x(t_1)| + |x(t_2)| + \ldots + |x(t_n)|$, which is another way of measuring the distance between a point and the origin, assuming you are constrained to travel along an orthogonal "grid" to get to the point.

 

[qwas]

Alright, thanks for the help!

 

[NascentOxygen]

Sorry if this sounds like a dumb question, but why is the effective value of a sine wave 0.707, as opposed to 0.637 which is the value generated by finding the definite integral over the domain [0,∏] divided length of the domain?

The 0.707 figure is relevant where we are concerned with heating, or the heat produced by that waveform. So a sinewave of amplitude A_v produces heat in a resistance equivalent to that produced by DC of amplitude 0.707A_v, since instantaneous power = i²(t).R

The 0.637 figure also has its uses, but to situations where we are concerned with average. For example, an electromagnet is roughly linear, so if you applied a rectified sinewave of amplitude A_v to the windings of an electromagnet, the field strength produced will have an average value equal to that produced by applying DC of magnitude 0.637A_v to the windings.