Simple Nature - Light and Matter

trous. With very strong damping, the swing comes essentially torest long before the second push. It has lost all its memory, and thesecond push puts energy into the system rather than taking it out.Although the detailed mathematical results with this kind of impulsivedriving force are different, 12 the general results are the sameas for sinusoidal driving: the less damping there is, the greater thepenalty you pay for driving the system off of resonance.High-Q speakers example 48Most good audio speakers have Q ≈ 1, but the resonance curvefor a higher-Q oscillator always lies above the corresponding curvefor one with a lower Q, so people who want their car stereos tobe able to rattle the windows of the neighboring cars will oftenchoose speakers that have a high Q. Of course they could justuse speakers with stronger driving magnets to increase F m , butthe speakers might be more expensive, and a high-Q speakeralso has less friction, so it wastes less energy as heat.One problem with this is that whereas the resonance curve of alow-Q speaker (its “response curve” or “frequency response” inaudiophile lingo) is fairly flat, a higher-Q speaker tends to emphasizethe frequencies that are close to its natural resonance.In audio, a flat response curve gives more realistic reproductionof sound, so a higher quality factor, Q, really corresponds to alower-quality speaker.Another problem with high-Q speakers is discussed in example51 on page 185 .Changing the pitch of a wind instrument example 49⊲ A saxophone player normally selects which note to play bychoosing a certain fingering, which gives the saxophone a certainresonant frequency. The musician can also, however, changethe pitch significantly by altering the tightness of her lips. Thiscorresponds to driving the horn slightly off of resonance. If thepitch can be altered by about 5% up or down (about one musicalhalf-step) without too much effort, roughly what is the Q of asaxophone?⊲ Five percent is the width on one side of the resonance, so thefull width is about 10%, ∆f /f o ≈ 0.1. The equation ∆ω = ω o /Q isdefined in terms of angular frequency, ω = 2π f , and we’ve beengiven our data in terms of ordinary frequency, f . The factors of 2πm / An x-versus-t graph ofthe steady-state motion of aswing being pushed at twiceits resonant frequency by animpulsive force.12 For example, the graphs calculated for sinusoidal driving have resonancesthat are somewhat below the natural frequency, getting lower with increasingdamping, until for Q ≤ 1 the maximum response occurs at ω = 0. In figure m,however, we can see that impulsive driving at ω = 2ω o produces a steady statewith more energy than at ω = ω o.Section 3.3 Resonance 183