Find the reponse function for the damped harmonic oscillator (mass m, spring constant k, damping α) whose equation of motion is given by
m\ddot{x} + α\dot{x} + kx = f (33.74)
and show that eqn 33.73 holds.
\boxed{\widetilde{χ}^{\prime } (0) = \mathcal{P} \int_{-\infty }^{\infty }{\frac{dω^{\prime } }{\pi } }\frac{\widetilde{χ} ^{\prime \prime }(ω^{\prime) } }{ω^{\prime}} }. (33.73)
Writing the resonant frequency ω^{2} _{0} = k/m , and writing the damping γ = α/m, we have
\ddot{x} + γ\dot{x } + ω^{2}_{0} x = \frac{f}{m}, (33.75)
and Fourier transforming this gives immediately that
\widetilde{χ} (ω)= \frac{\widetilde{x} (ω) }{ \widetilde{f} (ω)} =\frac{1}{m}\left[\frac{1}{ω^{2}_{0} − ω^{2} −iωγ} \right]. (33.76)
Hence, the imaginary part of the response function is
\widetilde{χ}^{\prime \prime } (ω)=\frac{1}{m}\left[\frac{ωγ}{(ω^{2} − ω^{2}_{0} )^{2} +(ωγ)^{2} } \right], (33.77)
and the static susceptibility is
\widetilde{χ} ^{\prime } (0) = \frac{1}{ mω^{2}_{0} } = \frac{1}{k} . (33.78)
The real and imaginary parts of \widetilde{χ}(ω) are plotted in Fig. 33.2(a). The imaginary part shows a peak near ω_{0}. Equation 33.77 shows that \widetilde{χ} ^{\prime \prime } (ω)/ω =(γ/m)[(ω^{2} − ω^{2}_{0} )+(ωγ)^{2}] and straightforward integration shows that \int_{-\infty }^{\infty }{} ( \widetilde{χ} ^{\prime \prime } (ω)/ω)dω = π/(mω^{2}_{0} )=π\widetilde{χ} ^{\prime } (0) and hence that eqn 33.73 holds. This is illustrated in Fig. 33.2(b)