22.6: Frequency Multiples
- Page ID
- 30506
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The above analysis is for frequencies not very far from \(\omega_{0}\). But nonlinear terms can cause resonance to occur at frequencies which are rational multiples of \(\omega_{0}\). Landau shows that a small \(\dfrac{1}{3} \alpha x^{3}\) in the potential (so an additional force \(\alpha x^{2}\) in the equation of motion) can generate a resonance near \(\gamma=\dfrac{1}{2} \omega_{0}\). We’ve only considered a quartic addition to the potential, \(\dfrac{1}{4} \beta x^{4}, \text { a force } \beta x^{3}\), we can show that gives a resonance near \(\gamma=\dfrac{1}{3} \omega_{0}\), and presumably this is the small bump near the beginning of the curves above for large driving strength.
\(\text { We have } \ddot{x}+2 \lambda \dot{x}+\omega_{0}^{2} x=(f / m) \cos \gamma t-\beta x^{3}\)
\(\text { We'll write } x=x^{(0)}+x^{(1)}+\ldots\)
\(\text { Let's define } x^{(0)} \text { by }\)
\[\ddot{x}^{(0)}+2 \lambda \dot{x}^{(0)}+\omega_{0}^{2} x^{(0)}=(f / m) \cos \gamma t\]
\(\text { So } x^{(0)}=b \cos (\gamma t+\delta) . \text { Then }\)
\[\begin{aligned}
\ddot{x}^{(1)}+2 \lambda \dot{x}^{(1)}+\omega_{0}^{2} x^{(1)} &=-\beta\left(x^{(0)}\right)^{3} \\
&=-\beta b^{3} \cos ^{3}(\gamma t+\delta) \\
&=-\beta b^{3}\left[\dfrac{3}{4} \cos (\gamma t+\delta)+\dfrac{1}{4} \cos (3 \gamma t+\delta)\right]
\end{aligned}\]
Then, for \(\gamma=\dfrac{1}{3} \omega_{0}\), the second term, \(-\beta b^{3} \dfrac{1}{4} \cos (3 \gamma t+\delta)=-\beta b^{3} \dfrac{1}{4} \cos \left(\omega_{0} t+\delta\right)\), will have a resonant response, although it is proportional to the (small) amplitude cubed. Similar arguments work for other fractional frequencies.