For the past two days my head has been processing at maximum load!! Thanks goes to an anonymous poster. Mr. Anonymous has brought my attention to an older post, titled "Inconsistent Circuits Formula". His comments triggered my thoughts, and as some do notice, our thoughts aren't clear all the time. At the time I posted my old entry, I had a limited vision - Today, I come with new understanding: An understanding of the nature of the phenomena of resonance.
I will not post my newly-reached understanding in this post. I have a clear vision of the phenomena of resonance, but I want to support it with some mathematical justification, and more importantly, numerical analysis data.
Mr. Anonymous has argued about my bold statement "We cannot prove two physical (even mathematical) quantities to be equal". This post should indirectly justify my position. I am still considering to post a more direct argument can be found here. Just keep in mind, my position is strictly regarding inclusion of equalities for continuous quantity spectrum.
Finally, this mathematical challenge question was tailored by me in order to clarify a point I want to make, so try to solve it, and tell me your results.
The Mathematical Challenge:
The Final Answer:
ZERO (assuming that no noise exists in the signal)
ANY VALUE is the general solution
The interval [0, 2*PI/a] represents one period of the sinusoidal signal cos(ax). When integrating a sinusoidal signal over one period the answer is ZERO. In short, the answer is ZERO regardless of the value of a.
A source of confusion might arise due to the use of the value ZERO for 'a'. The easiest approach is to devise an answer which is independent of 'a', as has been explained above. Note that the sinusoidal signal has a constant value of ONE over the finite region. The interval of the integration extends to ( 2*PI/a ), which also extends to the infinity. For this reason, the transformation of cos(ax) to ONE is NOT valid, because the sinusoidal signal is equal to 1 ONLY in the finite region, but the integration extends to the infinite region which implies that this transformation leads to incorrect results.
This solution applies only if we consider that there is NO NOISE in the sinusoidal signal. If we consider the possibility of existence of noise, then the integration will take the value of the integral of noise over the whole interval.
Update 1: The answer to the challenge has been posted.
Update 2: The paper about resonance has been posted here.