Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

11.5: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 11.5.1

Find the time-domain Green’s function of the critically-damped harmonic oscillator (γ=ω0).

Exercise 11.5.2

Consider an overdamped harmonic oscillator (γ>ω0) subjected to a random driving force f(t), which fluctuates between random values, which can be either positive or negative, at each time t. The random force satisfies f(t)=0andf(t)f(t)=Aδ(tt), where denotes an average taken over many realizations of the random force and A is some constant. Using the causal Green’s function, find the correlation function x(t1)x(t2) and the mean squared deviation [x(t+Δt)x(t)]2.

Answer

For the over-damped oscillator, the Green’s function is G(t,t)=Θ(tt)eγ(tt)Γsinh[Γ(tt)],whereΓ=γ2ω20. Hence, the response to the force f is x(t)=1mΓtdteγ(tt)sinh[Γ(tt)]f(t). From this, we get the following expression for the desired correlation function: x(t1)x(t2)=1m2Γ2t1dtt2dteγ(t1t)eγ(t2t)×sinh[Γ(t1t)]sinh[Γ(t2t)]f(t)f(t). Note that the can be shifted inside the integrals, because it represents taking the mean over independent sample trajectories. Now, without loss of generality, let us take t1t2. Since f(t)f(t)=Aδ(tt) which vanishes for tt, the double integral only receives contributions from values of t not exceeding t2 (which is the upper limit of the range for t). Thus, we revise t1dt into t2dt. The delta function then reduces the double integral into a single integral, which can be solved and simplified with a bit of tedious algebra: x(t1)x(t2)=Am2Γ2eγ(t1+t2)t2dte2γtsinh[Γ(tt1)]sinh[Γ(tt2)]=A8m2Γ2eγ(t1+t2)[eΓt1e(2γ+Γ)t2γ+Γ+eΓt1e(2γΓ)t2γΓeΓt1e(Γ+2γ)t2+eΓt1e(Γ+2γ)t2γ]=A8m2Γγ[e(γΓ)(t1t2)γΓe(γ+Γ)(t1t2)γ+Γ]. Hence, [x(t+Δt)x(t)]2=2[x(t)2x(t+Δt)x(t)]=A4m2Γγ[1e(γΓ)ΔtγΓ1e(γ+Γ)Δtγ+Γ].


This page titled 11.5: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?