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Solving equation \ref{eqn:efreq} yields the wave frequency which dominates energy transport,  \begin{equation}  \label{eqn:omstarnl}  \omega_* \sim \omega_c {\rm max}  \bigg[ \omega_c, \omega_c \bigg(  \frac{ A^2 \rho r^5 \omega_c^4}{\lambda^{3/2} N \dot{E}_0} \bigg]^{-1/(a+3)} \, \bigg)^{-1/(a+3)} \bigg]\,  . \end{equation}  We expect frequency spectra with slopes somewhere near $3 \lesssim a \lesssim 7$. Therefore the exponent in equation \ref{eqn:omstarnl} is quite small, and in most cases, $\omega_*$ does not increase to values much larger than $\omega_c$.