Near-Resonance with Small Damping

We analyze a phenomenon of near-resonance in an oscillator with small damping and make connections to blackbody radiation and the acoustics of string instruments. We find that near-resonance with small damping is characterized by an efficiency index E ≈ 1 as the quotient of the damping energy and the forcing energy, as a result of a phase shift of a quarter of a period beween forcing and velocity. Near-resonance is used in tuning the three string of a piano tone with an offset of about 0.5 Hz to generate longer sustain and a singing quality to the piano. 1 Resonance in Forced Damped Oscillators The problem of resonance is of fundamental importance in the physics of absorption and emission of light/radiation and in the acoustics of string instruments. The analysis of blackbody radiation [3] shows the basic role of a phenomeon of near-resonance in a resonator with small damping. The basic phenomenon can be illustrated for a damped harmonic oscillator modeled by ü(t) + νu(t) + γu̇(t) = f(t), −∞ < t < ∞, (1) where u̇ = du dt , ü = d2u dt2 , ν is a given moderate to large frequency, γ > 0 is a damping parameter and f(t) is a periodic forcing. We seek periodic solutions and measure the relation between the forcing and the damping by the efficiencyE = F R with F = ∫ f (t) dt, R = ∫ γu̇(t) dt , (2) ∗Computer Science and Communication, KTH, SE-10044 Stockholm, Sweden. with integration over a time period. If the forcing f(t) is periodic with the resonance frequency ν, referred to as perfect resonance, then u̇(t) = 1 γ f(t), which gives E = γ with u̇ in phase with f(t). We shall distinguish two basic different cases with the forcing f(t) balanced by the harmonic oscillator term ü(t) + νu(t) and the damping term γu̇(t) in two different ways: 1. γ ≈ 1 with γu̇ ≈ f(t) and |ü(t) + νu(t)| << |f(t)|, 2. γν < 1 with |γu̇(t)| << |f(t)| and ü(t) + νu(t) ≈ f(t), with the case 2. representing near-resonance with small damping, as the case of most interest. We shall define near-resonance at a given frequency ν by flat spectrum centered at ν of width 1. A spectrum of width γ << 1 would then correspond to sharp resonance (with γ not very small this is sometimes referred to as broad resonance). In case 1. the damping is large and the force f(t) is balanced by the damping γu̇(t) with u̇ in phase with f(t). In this case trivially E ≈ 1. In case 2. with near resonance and small damping, f(t) is balanced by the oscillator with u̇ out-of-phase with f(t), and we shall see that also in this case E ≈ 1. The case of near-resonance is to be compared with the case of perfect resonance with f(t) again balanced by γu̇, with now u̇(t) in phase and E = γ. If γ is small there is thus a fundamental difference betwen the case of nearresonance with E ≈ 1 and the case of perfect resonance with E = γ << 1. In applications to blackbody radiation we may view F as input and R as output, but it is also possible to turn this around view R as the input and F as the output, with E = F R then representing an efficiency index. In the case of small damping we then have E ≈ 1 in the case of near-resonance and E << 1 in the case of perfect resonance. In the case of near-resonance the force f(t) is balanced mainly by the excited harmonic oscillator with a small contribution from the damping term, which gives E ≈ 1. In the case of perfect resonance the oscillator does not contribute to the force balance, which requires a large damping term leading to small efficiency. The above discussion concerns time-periodic (equilibrium) states attained after a transient start-up phase, with the forcing now F in-phase with the velocity u̇, in contrast to out-of-phase in equilibrium. The discussion in this note connects to apects of wave vs particle modeling of light and sound [4, 1, 2, 7, 8, 9, 11]. 2 Fourier Analysis of Near-Resonance Although (3) is a maybe the most studied model of all of physics, it appears that the phenomenon of near-resonance has received little attention. We use Fourier transformation in t of (3), writing