\widetilde{g}_{2\omega}(\eta,\xi_2)=\frac{C_{2\omega}}{4\pi}\frac{\Delta \widetilde{T}_2(\eta,\xi_2)+\frac{1}{2}P_{\omega}\Lambda_{2\omega}|\mu|/C_{2\omega}}{1+i\eta\tau_{2\omega}+i\mu\xi_2\Lambda_{2\omega}},$$ and the temperature profile $$\label{eq:SubstrateTemp}
\Delta \widetilde{T}_2(\eta,\xi_2) = \frac{\int_0^{\omega_m}P_{\omega}v_{2\omega}\frac{1+i\eta \tau_{2\omega}}{(\Lambda_{2\omega}\xi_2)^2}\log\left[1+\left(\frac{\Lambda_{2\omega}\xi_2}{1+i\eta\tau_{2\omega}}\right)^2\right]d\omega}{\int_0^{\omega_m}\frac{C_{2\omega}}{2\pi\tau_{2\omega}}\left[1-\frac{1}{\Lambda_{2\omega}\xi_2}\text{tan}^{-1}\left(\frac{\Lambda_{\omega}\xi_2}{1+i\eta\tau_{2\omega}}\right)\right]d\omega},$$ where $\xi_2$ is the Fourier variable of $x_2$.

Again, to express $\widetilde{g}_{2\omega}$ only in terms of unknown coefficients $P_{\omega}$, we simply plug Eq. (\[eq:SubstrateTemp\]) into Eq. (\[eq:SubstrateDistribution\]).

Interface condition {#sec:InterfaceCondition}
-------------------

The unknown coefficients in the solutions of transducer film and substrate are obtained by applying appropriate interface conditions. Here, we use the elastic transmission interface condition with mode conversion, closely following the work by Minnich *et al*.[@Minnich2011b] Briefly, for a given mode $i$, the heat fluxes outgoing from the interface, $q_{1\omega}^{i-}$ and $q_{2\omega}^{i+}$, must be equal to the reflected and transmitted heat fluxes incident to the interface, $q_{1\omega}^{i+}$ and $q_{2\omega}^{i-}$. By assuming elastic and diffuse scattering, the transmission and reflection process for each phonon frequency is treated independently and the heat flux equality condition must be satisfied for each frequency and polarization.

The interface conditions are $$\begin{aligned}
\label{eq:IC_1}\nonumber
\int_0^1 g_{2\omega}^{i+} v^i_{2\omega}\mu d\mu &=& \sum_{j}T^{ji}_{12}(\omega)\int_0^1 g_{1\omega}^{j+} v^j_{1\omega}\mu d\mu + \sum_{j}R^{ji}_{21}(\omega) \int_0^1 g_{2\omega}^{j-}v^j_{2\omega}\mu d\mu, \\  \\ \nonumber
\label{eq:IC_2}
\int_0^1 g_{1\omega}^{i-} v^i_{1\omega}\mu d\mu &=& \sum_{j}T^{ji}_{21}(\omega)\int_0^1 g_{2\omega}^{j-} v^j_{2\omega}\mu d\mu+\sum_{j}R^{ji}_{12}(\omega)\int_0^1 g_{1\omega}^{j+} v^j_{1\omega}\mu d\mu, \\\end{aligned}$$ where $T^{ji}_{12}(\omega)$ is the transmission coefficient of mode $j$ at frequency $\omega$ from side 1 to side 2 