Simple Nature - Light and Matter

quality factor, Q, is defined as Q = ω o /2c, and in the limit of weakdamping, where ω ≈ ω o , this can be interpreted as the number ofcycles required for the mechanical energy to fall off by a factor ofe 2π = 535.49 . . . Using this new quantity, we can rewrite the equationfor the frequency of damped oscillations in the slightly more elegantform ω f = ω o√1 − 1/4Q 2 .Summary of Notationk spring constantm mass of the oscillatorb sets the amount of damping,F = −bvT periodf frequency, 1/Tω (Greek letter omega), angularfrequency, 2πf , oftenreferred to simply as “frequency”ω o frequency the oscillatorwould have without damping,k/mω f frequency of the free vibrationsc sets the time scale for theexponential decay envelopee −ct of the free vibrationsF m strength of the drivingforce, which is assumed tovary sinusoidally with frequencyωA amplitude of the steadystateresponseδ phase angle of the steadystateresponseself-check HWhat if we wanted to make a simpler definition of Q, as the number ofoscillations required for the vibrations to die out completely, rather thanthe number required for the energy to fall off by this obscure factor? ⊲Answer, p. 923A graph example 43The damped motion in figure g has Q ≈ 4.5, giving √ 1 − 1/4Q 2 ≈0.99, as claimed at the end of the preceding subsection.Exponential decay in a trumpet example 44⊲ The vibrations of the air column inside a trumpet have a Q ofabout 10. This means that even after the trumpet player stopsblowing, the note will keep sounding for a short time. If the playersuddenly stops blowing, how will the sound intensity 20 cycleslater compare with the sound intensity while she was still blowing?⊲ The trumpet’s Q is 10, so after 10 cycles the energy will havefallen off by a factor of 535. After another 10 cycles we lose anotherfactor of 535, so the sound intensity is reduced by a factorof 535 × 535 = 2.9 × 10 5 .The decay of a musical sound is part of what gives it its character,and a good musical instrument should have the right Q, but theQ that is considered desirable is different for different instruments.A guitar is meant to keep on sounding for a long time after a stringhas been plucked, and might have a Q of 1000 or 10000. One of thereasons why a cheap synthesizer sounds so bad is that the soundsuddenly cuts off after a key is released.3.3.3 Driven motionThe driven case is extremely important in science, technology,and engineering. We have an external driving force F = F m sin ωt,where the constant F m indicates the maximum strength of the forcein either direction. The equation of motion is now[1] ma + bv + kx = F m sin ωt[equation of motion for a driven oscillator] .After the driving force has been applied for a while, we expect thatthe amplitude of the oscillations will approach some constant value.This motion is known as the steady state, and it’s the most interestingthing to find out; as we’ll see later, the most general type ofmotion is only a minor variation on the steady-state motion. For176 Chapter 3 Conservation of Momentum