The root mean square (RMS) of a sine wave is always the peak value divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes. But every repeating waveform is equal to the sum of sine waves with different frequencies, phases, and amplitudes. This includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain. I learn a lot by coming up with such paradoxes then figuring out the solution.
Keith, The root mean square (RMS) of a sine wave is always the peak value
divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes.
My skepticism kicks in at that step. For a sum of two sine waves, I might believe it, but not for the sum of arbitrarily many.
But every repeating waveform is equal to the sum of
sine waves with different frequencies, phases, and amplitudes. This
includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain.
Jim Propp
A fourier approximation of a square wave of amplitude X has to include frequencies with amplitude greater than X, so it doesn't strike me as paradoxical that the RMS would average out to the correct value... am I missing the puzzle? On Sun, Nov 12, 2017 at 5:04 PM, James Propp <jamespropp@gmail.com> wrote:
Keith,
The root mean square (RMS) of a sine wave is always the peak value
divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes.
My skepticism kicks in at that step. For a sum of two sine waves, I might believe it, but not for the sum of arbitrarily many.
But every repeating waveform is equal to the sum of
sine waves with different frequencies, phases, and amplitudes. This
includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Actually, the conjecture for sums of multiple sine waves is false. What is true is that the mean square MS = RMS^2 of the sum equals the sum of the mean squares of the component sine waves, with the proviso that the components are orthogonal. The peak value has no bearing on the matter. Consider sin(x)+sin(2x). Lazy guy that I am, I used a spreadsheet to find the peak value 1.76. This divided by sqrt(2) is 1.24. But the RMS of the sum is sqrt(1/2 + 1/2) = 1. -- Gene On Sunday, November 12, 2017, 1:56:12 PM PST, Keith F. Lynch <kfl@KeithLynch.net> wrote: The root mean square (RMS) of a sine wave is always the peak value divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes. But every repeating waveform is equal to the sum of sine waves with different frequencies, phases, and amplitudes. This includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain. I learn a lot by coming up with such paradoxes then figuring out the solution.
If I add two sine waves of equal amplitude but with 180deg phase difference, I get zero - clearly not an RMS of the amplitude/sqrt(2). Brent On 11/12/2017 1:55 PM, Keith F. Lynch wrote:
The root mean square (RMS) of a sine wave is always the peak value divided by the square root of 2. The same is true of the sums of multiple sine waves with different frequencies, phases, and amplitudes. But every repeating waveform is equal to the sum of sine waves with different frequencies, phases, and amplitudes. This includes the square wave, i.e. a function which always equals +X or -X, and never takes any other value. But obviously the RMS of that square wave is simply X, not X divided by the square root of 2. Explain.
I learn a lot by coming up with such paradoxes then figuring out the solution.
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On 11/12/2017 6:58 PM, Marc LeBrun wrote:
If I add two sine waves of equal amplitude but with 180deg phase difference, I get zero - clearly not an RMS of the amplitude/sqrt(2). That fooled me too until I remembered "...with different frequencies ..."
So take twice the frequency. The frequencies you sum to get a square wave are not just some random ones. They are all multiples of a fundamental. Brent
participants (6)
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James Davis -
James Propp -
Keith F. Lynch -
Marc LeBrun