Near Resonance with Small Damping

In my article Computational Blackbody Radiation in the Sky Dragon book I derived the Rayleigh-Jeans and Planck Laws of Radiation from a wave equation model with small damping, as a consequence of a phenomenon of near-resonance in a resonator with small damping.

This phenomenon appears to also be fundamental in the acoustics of string instruments with strings representing the damping and the body or soundboard of the instrument the resonator.

The phenomenon can be studied in the most basic of all models of physics, the harmonic oscillator, subject to small damping, as I do in the new article Near-Resonance with Small Damping. The article is a good complement to the Sky Dragon article.

The key point is that in near-resonance the forcing is balanced only to a small part by the damping force, the main part being balanced by the resonator, which reflects that forcing and velocity are out-of-phase. In this case the resonator acts as an amplifier of the damping (the soundboard amplifies the sound of a string).

Near-resonace is fundamentally different from perfect resonance with forcing and velocity in-phase and the damping force balancing the forcing without amplification from the resonator.

(Near-resonance is of course also different from the case of no resonance).

The importance of near-reonance is well-known to a piano tuners who tunes the two or three strings of a tone (except the single deep bass string) at slightly different pitches to create a longer sustain and singing quality of the piano.

The analysis in the article exhibits the interaction of the vibrating string and the vibrating resonating body, with the string pumping energy into the body during a start-up phase and

then changing role to sustain output from the body, all the time with the string vibrating in-phase with the body with out-of-phase output from the body in equilibrium.

The model suggests that there is a principal similarity between the radiation spectrum of a radiating body and the acoustic spectrum of a multi-string instrument from repeated arpeggios over the strings (with a capo d'astro in high position so that fundamental low frequencies are not involved).

For radiation the spectrum scales with the frequency squared as the result of a damping related to accelleration, while the corresponding spectrum in acoustics is flat in frequency because the damping in this case relates to velocity.

Near-resonane amplification conforms with the experience that the resonating body of a string instrument functions over a wide spectrum of string frequencies.

Near-resonance connects to broad resonance with a frequency band of larger width than that of sharp resonance scaling with the damping.