Simple Nature - Light and Matter

end up canceling out, however:Q = ω o∆ω= 2πf o2π∆f= f of≈ 10In other words, once the musician stops blowing, the horn willcontinue sounding for about 10 cycles before its energy falls offby a factor of 535. (Blues and jazz saxophone players will typicallychoose a mouthpiece that gives a low Q, so that they canproduce the bluesy pitch-slides typical of their style. “Legit,” i.e.,classically oriented players, use a higher-Q setup because theirstyle only calls for enough pitch variation to produce a vibrato,and the higher Q makes it easier to play in tune.)Q of a radio receiver example 50⊲ A radio receiver used in the FM band needs to be tuned in towithin about 0.1 MHz for signals at about 100 MHz. What is itsQ?⊲ As in the last example, we’re given data in terms of f s, not ωs,but the factors of 2π cancel. The resulting Q is about 1000, whichis extremely high compared to the Q values of most mechanicalsystems.TransientsWhat about the motion before the steady state is achieved?When we computed the undriven motion numerically on page 172,the program had to initialize the position and velocity. By changingthese two variables, we could have gotten any of an infinite numberof simulations. 13 The same is true when we have an equation of motionwith a driving term, ma+bv +kx = F m sin ωt (p. 176, equation[1]). The steady-state solutions, however, have no adjustable parametersat all — A and δ are uniquely determined by the parametersof the driving force and the oscillator itself. If the oscillator isn’tinitially in the steady state, then it will not have the steady-statemotion at first. What kind of motion will it have?The answer comes from realizing that if we start with the solutionto the driven equation of motion, and then add to it anysolution to the free equation of motion, the result,x = A sin(ωt + δ) + A ′ e −ct sin(ω f t + δ ′ ) ,13 If you’ve learned about differential equations, you’ll know that any secondorderdifferential equation requires the specification of two boundary conditionsin order to specify solution uniquely.184 Chapter 3 Conservation of Momentum