tailieunhanh - Basic Theoretical Physics: A Concise Overview P4

Basic Theoretical Physics: A Concise Overview P4. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 20 4 Mechanics of the Damped and Driven Harmonic Oscillator Thus far we have only dealt with the free harmonic oscillator. 1 We now consider a forced harmonic oscillator restricting ourselves at first to the simple periodic driving force f t f a- However due to the superposition principle this is no real restriction since almost every driving force f t can be written by Fourier integration as the sum or integral of such terms f t y duAf uA e1 A t with f uA 2n -1 y dtf t e1WA 1 . tt tt For x t in the above-mentioned case after relaxation of a transient process the following stationary solution results x t xa e1UAt with complex amplitude -fA XA ZJ2 w2 2i . M - A 2iV If one now plots for the case of weak friction . for rx0 1 the fraction XA a as a function of the driving frequency vA the following typical amplitude resonance curve is obtained XA i t Ta wA 1 w0 _ wA 2 TA This curve has a very sharp maximum . a spike of height at the resonance frequency vA w0 for slight deviations positive or negative from resonance . for w A Wo t the amplitude almost immediately becomes smaller by a factor of compared to the maximum. For small frequencies . for wA C wo the amplitude xa is in phase with the driving force for high frequencies . for wA w0 they are of opposing phase out of phase by 180 or at resonance the motion xa is exactly 90 or behind the driving force a. The transition from in phase to opposite phase behavior is very rapid 4 Mechanics of the Damped and Driven Harmonic Oscillator 21 occurring in the narrow interval given by the values -A V - . T The dimensionless ratio t vq 1 is called the quality factor of the resonance. It can be of the order of or or even higher. 2 In the ballistic case there is again no restriction. For a sequence of ultrashort and ultra-strong pulses h t t with t between t t0 and t t . for the formal case t f to dt g t S t t with the Dirac 6-function 6 x a very high and very narrow bell-shaped function of .