Energy in a Fundamental Standing Wave Vs. The Energy in its Harmonic Waves

[Pathways33]

I am seeking to understand the relationship of energy within a standing wave vs. the energy in its first three harmonics.
Is the energy latent within the fundamental wave the same as the energy in its first three harmonics, or is the energy and exponential increase.
IE How much energy is required to encourage the fundamental to oscillate compared with the energy required to encourage the three harmonics to oscillate?
Any assistance much appreciated.

 

[hikaru1221]

Aren't the first three harmonics all standing waves of different frequencies?
I'm not sure what you mean when saying energy depends on the harmonic. Energy doesn't have much to do with harmonic, since harmonic is defined by the frequency the system vibrates at (though in the energy's formula, frequency may appear).

 

[Pathways33]

Thanks. I'm looking for a link between whether the fundamental wave has embedded within it the harmonic waves also?
As each wave whether a moving or standing wave contains kinetic and potential energy, I'm looking for the energy difference inherent in the wave.
From my deductions it appears that there is less energy embedded in the fundamental wave than embedded in the harmonics of that fundamental.
I'm seeking if there is a mathematically formula to confirm this.
I'm looking for the energy relationship (or variance if any) between the fundamental wave and the harmonics of that wave.
Thanks

 

[hikaru1221]

What do you mean by "fundamental wave"? Do you mean wave at fundamental frequency?
Anyway why don't you show us your work?

 

[Pathways33]

Yes, that's correct.
The fundamental frequency, and its first three harmonics.
I want to know that if I put energy into the fundamental frequency, will that also transfer to the harmonic frequencies.
Conversely, if you place energy into say the third harmonic frequency, how much of that energy (if any)will permeate back into the fundamental frequency.
My thinking is that in an electromagnetic medium the fundamental frequency will transmit energy to its adjacent harmonics.
My postulation is that if you 'pump' energy into the fundamental frequency, it will easily transpose (resonate) into its harmonic modes, but if you place energy into any of the harmonic modes, you will see little effect on its fundamental frequency. I would tend to think that if you pump energy into an harmonic frequency, it itself will become a fundamental frequency and will only transpose energy 'down' to its harmonic frequencies and NOT up to its original fundamental frequency where it originated from.
Is there any study of this?
I am linking it to a superposition theory I am exploring.
Cheers

 

[Phrak]

You seem to have some misunderstanding of a standing wave. That's OK.

Say you have a wave A = sin(kx-ωt). As time advances, holding x constant, the wave moves to the right. Add this to another, B = sin(kx+ωt). This wave moves to the left.

A+B is a standing wave. The sum is not composed of harmonics, but can be expressed as a the product of two waves; one stationary in time, and the other, stationary in space:

A+B = 2 cos(ωt) sin(kx)

 

[hikaru1221]

Yes, that's correct.
The fundamental frequency, and its first three harmonics.
I want to know that if I put energy into the fundamental frequency, will that also transfer to the harmonic frequencies.
Conversely, if you place energy into say the third harmonic frequency, how much of that energy (if any)will permeate back into the fundamental frequency.
My thinking is that in an electromagnetic medium the fundamental frequency will transmit energy to its adjacent harmonics.
My postulation is that if you 'pump' energy into the fundamental frequency, it will easily transpose (resonate) into its harmonic modes, but if you place energy into any of the harmonic modes, you will see little effect on its fundamental frequency. I would tend to think that if you pump energy into an harmonic frequency, it itself will become a fundamental frequency and will only transpose energy 'down' to its harmonic frequencies and NOT up to its original fundamental frequency where it originated from.
Is there any study of this?
I am linking it to a superposition theory I am exploring.
Cheers

I'm not sure what you meant by transferring energy from fundamental frequency to its harmonics, because, you know, energy is stored in the vibrating system, not frequency. Can you give us an example or illustrate your point in some way?

 

[AJ Bentley]

My thinking is that in an electromagnetic medium the fundamental frequency will transmit energy to its adjacent harmonics.

Sorry, no.

Energy does not spontaneously transfer between the modes of vibration of a system (including EM waves).

If you have a perfect guitar string on a perfectly solid guitar and you pluck it precisely in the middle, you will get the fundamental. The fundamental will continue to sound until all the energy dissipates.
If you do the same thing but this time cause the second harmonic to sound (bell tone technique) that too will simply continue to sound.
Only when the string and guitar are not 'perfect' will any transfer take place and even then, very little.

EM waves are even more perfect.