22.6: Frequency Multiples

Last updated
Save as PDF

Page ID: 30506

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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 \(\frac{1}{3} \alpha x^{3}\) in the potential (so an additional force \(\alpha x^{2}\) in the equation of motion) can generate a resonance near \(\gamma=\frac{1}{2} \omega_{0}\). We’ve only considered a quartic addition to the potential, \(\frac{1}{4} \beta x^{4}, \text { a force } \beta x^{3}\), we can show that gives a resonance near \(\gamma=\frac{1}{3} \omega_{0}\), and presumably this is the small bump near the beginning of the curves above for large driving strength.

\(\text { We have } \ddot{x}+2 \lambda \dot{x}+\omega_{0}^{2} x=(f / m) \cos \gamma t-\beta x^{3}\)

\(\text { We'll write } x=x^{(0)}+x^{(1)}+\ldots\)

\(\text { Let's define } x^{(0)} \text { by }\)

\[\ddot{x}^{(0)}+2 \lambda \dot{x}^{(0)}+\omega_{0}^{2} x^{(0)}=(f / m) \cos \gamma t\]

\(\text { So } x^{(0)}=b \cos (\gamma t+\delta) . \text { Then }\)

\[\begin{aligned}
\ddot{x}^{(1)}+2 \lambda \dot{x}^{(1)}+\omega_{0}^{2} x^{(1)} &=-\beta\left(x^{(0)}\right)^{3} \\
&=-\beta b^{3} \cos ^{3}(\gamma t+\delta) \\
&=-\beta b^{3}\left[\frac{3}{4} \cos (\gamma t+\delta)+\frac{1}{4} \cos (3 \gamma t+\delta)\right]
\end{aligned}\]

Then, for \(\gamma=\frac{1}{3} \omega_{0}\), the second term, \(-\beta b^{3} \frac{1}{4} \cos (3 \gamma t+\delta)=-\beta b^{3} \frac{1}{4} \cos \left(\omega_{0} t+\delta\right)\), will have a resonant response, although it is proportional to the (small) amplitude cubed. Similar arguments work for other fractional frequencies.