is the inertia an $l=0$ modes would have at a frequency $\nu_{nl}$. Multiplying the frequency shifts by $Q_{nl}$ removes the $l$ dependence of the frequency shifts at fixed frequency, as can be seen in Fig. \[figure\[shift vs freq\]\]. This makes the dependence a function of frequency alone. Collapsing the $l$ dependence in this manner allows the frequency shifts of a wide range of $l$ to be combined, thereby reducing any uncertainties associated with the shifts and allowing tighter constraints to be placed on the frequency dependence.

Dependence of solar cycle frequency shifts on mode frequency {#section[freq dependence of shifts]}
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The frequency dependence of the frequency shifts, which can be seen in the right-hand panel of Fig. \[figure\[shift vs freq\]\], is a telltale indicator that the observed 11-year signal must be the result of changes in acoustic properties in the few hundred kilometres just beneath the visible surface of the Sun, a region to which the higher-frequency modes are much more sensitive than their lower-frequency counterparts because of differences in the upper boundaries of the cavities in which the modes are trapped [@1990Natur.345..779L; @1991sia..book..401C]. We now go into more detail.

As mentioned earlier, for a given $l$ the upper turning 