Eugene Salamin <gene_salamin(a)yahoo.com> wrote:
> Actually, the conjecture for sums of multiple sine waves is false.
Oops, you're right. When it comes to the sum of sine waves, I'm more
of a radio and electronics guy than a mathematician. As was perhaps
evident from my using the term "RMS" rather than "quadratic mean."
> 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.
Yes, that's what I was probably thinking of. The first time I learned
that power was proportional to the square of the amplitude, that
struck me as very strange. Two identical radio transmitters emit
four times the power of either of them alone?
Eventually, I realized that a similar principle applies in lots of
contexts. For instance:
* In quantum mechanics, the probability of finding a particle at a
given point is proportional to the square of the absolute magnitude
of the complex amplitude there. (Or, equivalently, the product of
the amplitude with its complex conjugate.)
* The power in a sound wave is proportional to the square of the RMS
overpressure.
* The power in a water wave is proportional to the square of the RMS
height of the wave.
* The power of a gravitational wave is proportional to the square of
the RMS strain (i.e. of the degree to which space is stretched or
squished).
* The energy in an electric or magnetic field is proportional to the
square of the field strength.
* Bosons tend to be in the same state as their neighbors because we're
adding amplitudes, then measuring the squares of those amplitudes.
* The work it takes to shovel snow is proportional to the square of
the snow depth. (X times the depth of snow means X times as much
snow per unit area, and also means you have to lift it X times as
high to get it above the surrounding snow.)
* The work it takes to dig a hole is proportional to the square of the
depth of the hole.
My intended solution to the RMS paradox was that as you add more and
more sine waves to make a square wave, the corners continue to have
spikes that rise above the flat part. Adding more terms will make
those spikes narrower, but never make them shorter. At the limit,
they're infinitely narrow but still no less tall than they ever were.
How about the radio paradox I alluded to above? How can two identical
transmitters radiating the same frequency and phase radiate four times
the power as either one alone? One answer is that the antennas are
generally in different places, so the waves are out of phase in some
places and in phase in others. But what if the antennas are an
infinitesimal distance apart, e.g. two identical parallel dipoles
separated by much less than one wavelength? Then the waves would
be in phase everywhere.
Brent Meeker <meekerdb(a)verizon.net> 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).
Actually, my mistaken claim does work there, since the peak amplitude
of the sum of those two waves is zero, and if you divide that by the
square root of 2, you get zero, which is indeed the RMS.
