Re: [math-fun] RMS paradox
Eugene Salamin <gene_salamin@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@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.
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.
Amplitude is a measure of voltage. Current through a resistor is proportional to voltage, so doubling voltage also doubles current, and thus power through a resistor increases as the square of the voltage (P = V * I, and I = V/R, so P = V^2/R). Your example assumes that the two signals would simply add their amplitudes. But your infinitesimally spaced dipoles are going to interact in funky ways that your transmitters won't like. Yet in the spirit of spherical cows, perhaps we can imagine a situation where the transmitters cleanly overcome the field created by each other, and end up creating double the amplitude by doubling their current draw, thus preserving conservation of energy. On Mon, Nov 13, 2017 at 8:52 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
Eugene Salamin <gene_salamin@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@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.
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Putting two antennae "an infinitesimal distance apart" does not result in two independent antennae, but rather two pieces of wire coupled by a large capacitance. Each amplifier drives the whole circuit consisting of the two wires, their inductance, the coupling capacitance, and the impedance of the other amplifier.
From this you then calculate the distribution of E and B fields. Big, expensive FE simulators do this.
When the antennae are far enough apart that the coupling capacitance is small compared with the antenna impedance and the impedance of free space (377 Ohms), it is a convenient approximation to consider them as independent and then you get the result discussed below--fields add in some directions and not in others. --R On Sun, Nov 26, 2017 at 3:32 AM, Jason Holt <credentiality@gmail.com> wrote:
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.
Amplitude is a measure of voltage. Current through a resistor is proportional to voltage, so doubling voltage also doubles current, and thus power through a resistor increases as the square of the voltage (P = V * I, and I = V/R, so P = V^2/R).
Your example assumes that the two signals would simply add their amplitudes. But your infinitesimally spaced dipoles are going to interact in funky ways that your transmitters won't like. Yet in the spirit of spherical cows, perhaps we can imagine a situation where the transmitters cleanly overcome the field created by each other, and end up creating double the amplitude by doubling their current draw, thus preserving conservation of energy.
On Mon, Nov 13, 2017 at 8:52 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
Eugene Salamin <gene_salamin@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@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.
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participants (3)
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Jason Holt -
Keith F. Lynch -
Richard Howard