Ask Uncle Colin: The RMS Line

Dear Uncle Colin,

My textbook claims that the RMS voltage of a sine wave is $\sqrt{\frac{1}{2}}$ because between $x=0$ and $x=\pi$, the signed area between the curves $y=\sin (x)$ and $y=\sqrt{2}$ is 0 – but I checked and it isn’t. What’s going on?

Really A Disappointing Instructional Overview

Hi, RADIO, and thanks for your message!

You’re quite right – we can use the symmetry of the graph to restrict ourselves to $0\le x\le \frac{\pi }{2}$, and then we just need to look at the integral ${\int }_0^{\frac{\pi }{2}}(\sin (x)-\sqrt{\frac{1}{2}})\mathrm{d}x$.

That’s ${[-\cos (x)-x\sqrt{\frac{1}{2}}]}_0^{\frac{\pi }{2}}$, or $1-\frac{\pi }{2}\sqrt{\frac{1}{2}}$. That’s clearly not 0 (it’s about -0.11).

So, two questions: one, where should the equal-area line lie; and two, what should the RMS explanation have said?

We’ve done most of the work for the first one already: we just need to replace $\sqrt{\frac{1}{2}}$ with $k$ and solve for it.

The integral is ${[-\cos (x)-kx]}_0^{\frac{\pi }{2}}$, or $1-k\frac{\pi }{2}=0$ – which means $k=\frac{2}{\pi }$. But that’s not the RMS.

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The RMS of a function $f(x)$ over an interval $a<x<b$ is defined as $y_{RMS}=\sqrt{\frac{1}{b-a}{\int }_a^b[f(x){]}^2\mathrm{d}x}$.

I find it slightly nicer to square and cross-multiply to give $(b-a)y_{RMS}^2={\int }_a^b[f(x){]}^2\mathrm{d}x$, which at least looks like you’ve got the same kind of thing on both sides.

In any case, we need to work out ${\int }_0^{\frac{\pi }{2}}{\sin }^2(x)\mathrm{d}x$. Using the identity ${\sin }^2(x)\equiv \frac{1-\cos (2x)}{2}$, this integrates to ${[\frac{x}{2}-\frac{\sin (2x)}{4}]}_0^{\frac{\pi }{2}}$, which is $\frac{\pi }{4}$.

So we have $\left(\frac{\pi }{2}\right)y_{RMS}^2={\left(\frac{\pi }{4}\right)}^2$, which simplifies down to $y_{RMS}=\frac{1}{\sqrt{2}}$, which is a relief.

It turns out that it’s not that the curves have the same area beneath them – it’s that the volumes of revolution they define (about the $x$-axis) have the same volume.

Hope that helps!

- Uncle Colin