Simple Nature - Light and Matter

i / Example 45: a viola withouta mute (left), and with a mute(right). The mute doesn’t touchthe strings themselves.k / The definition of ∆ω, thefull width at half maximum.Steady state, with dampingThe extension of the analysis to the damped case involves somelengthy algebra, which I’ve outlined on page 912 in appendix 2. Theresults are shown in figure j. It’s not surprising that the steady stateresponse is weaker when there is more damping, since the steadystate is reached when the power extracted by damping matches thepower input by the driving force. The maximum amplitude, at thepeak of the resonance curve, is approximately proportional to Q.self-check IFrom the final result of the analysis on page 912, substitute ω = ω o ,and satisfy yourself that the result is proportional to Q. Why is A res ∝ Qonly an approximation? ⊲ Answer, p. 923What is surprising is that the amplitude is strongly affected bydamping close to resonance, but only weakly affected far from it. Inother words, the shape of the resonance curve is broader with moredamping, and even if we were to scale up a high-damping curve sothat its maximum was the same as that of a low-damping curve, itwould still have a different shape. The standard way of describingthe shape numerically is to give the quantity ∆ω, called the fullwidth at half-maximum, or FWHM, which is defined in figure k.Note that the y axis is energy, which is proportional to the squareof the amplitude. Our previous observations amount to a statementthat ∆ω is greater when the damping is stronger, i.e., when the Qis lower. It’s not hard to show from the equations on page 912 thatfor large Q, the FWHM is given approximately by∆ω ≈ ω o /Q .Another thing we notice in figure j is that for small values of Qthe frequency ω res of the maximum A is less than ω o . 11 At even11 The relationship is ω max A/ω o = √ 1 − 1/2Q 2 , which is similar in form tothe equation for the frequency of the free vibration, ω f /ω o = √ 1 − 1/4Q 2 . Asubtle point here is that although the maximum of A and the maximum of A 2must occur at the same frequency, the maximum energy does not occur, as wemight expect, at the same frequency as the maximum of A 2 . This is becausethe interaction energy is proportional to A 2 regardless of frequency, but thekinetic energy is proportional to A 2 ω 2 . The maximum energy actually occursare precisely ω o.180 Chapter 3 Conservation of Momentum