= Eigenvalue equation for heat conduction in two joined slabs
{title2=$k_1\alpha_1\cos u\sin v+k_2\alpha_2\sin u\cos v=0$}
For two length-$L$ slabs with zero exterior <temperatures> and perfect <thermal contact>, put $\alpha_i=\sqrt{\rho_ic_i/k_i}$, $u=\omega\alpha_1L$, and $v=\omega\alpha_2L$. A <separated solution> with time factor $e^{-\omega^2t}$ and $\omega\ne0$ exists precisely when the displayed interface <determinant> vanishes. Dividing by $\cos u\cos v$ gives a tangent <characteristic equation> only away from its poles. Common zeros of both cosines are legitimate <normal modes> and must be retained. The zero-frequency case gives only the trivial <solution of a differential equation> with these positive material parameters and exterior <boundary conditions>.
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