power. Figure \[fig:Modes\_Plot\](c) shows mode number $i$ against cut-on frequency $\nu_i$, indicating the number of propagating modes as a function of frequency. Above the four principal modes, the number of modes increases quadratically with frequency.

In numerical work, it is convenient to normalise calculated powers to the power carried by a single principal mode, $P_{qua}$, which defines an effective number of propagating modes: $$N_{eff} = \frac{P_{bal}}{P_{qua}} = \frac{\displaystyle \sum_i \int_{\nu_i}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu}{\int_{0}^\infty B(\nu,T_C) - B(\nu,T_B) \, \mathrm{d}\nu}.$$ $P_{qua}$ is the net power that would be carried by a single elastic mode. In the limit of narrow legs operating at low temperatures, the effective number of modes approaches $N_{eff}=4$. The experimentally measured power $P$ may be substituted for $P_{bal}$, as is done in Section \[sec:Results\_Discussion\], to calculate the actual effective number of modes propagating in test structures, $N_{eff}=P/4P_{qua}$, for which the ballistic case is an upper limit.

In experimental work, it is convenient to normalise the measured power $P$ flowing from the TES island to the heat bath to the straight-leg ballistic limit, $$\epsilon = \frac{P}{4N_{eff}P_{qua}}=\frac{P}{4P_{bal}},
\label{eq:Epsilon}$$ where the factor of 4 arises because each TES has 4 legs. In the case of a phononic thermal filter, $\epsilon$ 