5.5: Exercises
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Exercise 5.5.1
In Section 5.2, we encountered the complex frequencies ω±=−iγ±√ω20−γ2. For fixed ω0 and ω0>γ (under-damping), prove that ω± lie along a circular arc in the complex plane.
Exercise 5.5.2
Derive the general solution for the critically damped harmonic oscillator, Eq. (5.3.16), by following these steps:
- Consider the complex ODE, in the under-damped regime ω0>γ. We saw in Section 5.3 that the general solution has the form z(t)=ψ+exp[(−γ−i√ω20−γ2)t]+ψ−exp[(−γ+i√ω20−γ2)t] for some complex parameters ψ+ and ψ−. Define the positive parameter ε=√ω20−γ2. Re-write z(t) in terms of γ and ε (i.e., eliminating ω0).
- The expression for z(t) is presently parameterized by the independent parameters ψ+, ψ−, ε, and γ. We are free to re-define the parameters, by taking α=ψ++ψ−β=−iε(ψ+−ψ−). Using these equations, express z(t) using a new set of independent complex parameters, one of which is ε. Explicitly identify the other independent parameters, and state whether they are real or complex.
- Expand the exponentials in z(t) in terms of the parameter ε. Then show that in the limit ε→0, z(t) reduces to the critically-damped general solution (5.3.16).
Exercise 5.5.3
Repeat the above derivation for the critically-damped solution, but starting from the over-damped regime γ>ω0.
Exercise 5.5.4
Let z(t) be a complex function of a real input t, which obeys the differential equation dzdt=−i(ω1−iγ)z(t), where ω1 and γ are real. Find the general solution for z(t), and hence show that z(t) satisfies the damped oscillator equation [d2dt2+2γddt+ω20]z(t)=0 for some ω20. Finally, show that this harmonic oscillator is always under-damped.
- Answer
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The general solution is z(t)=Aexp[−i(ω1−iγ)t]. It can be verified by direct substitution that this is a solution to the differential equation. It contains one free parameter, and the differential equation is first-order, so it must be a general solution. Next, d2zdt2+2γdzdt=(−i)2(ω1−iγ)2z(t)−2iγ(ω1−iγ)z(t)=[−ω21+γ2+2iγω1−2iγω1−2γ2)]z(t)=−(ω21+γ2)z(t). Hence, z(t) obeys a damped harmonic oscillator equation with ω20=ω21+γ2. This expression for the natural frequency ensures that ω20>γ2 (assuming the parameters γ and ω1 are both real); hence, the harmonic oscillator is always under-damped.