= Periodic seismic multilayer stop band
{title2=$\cos(Kd)=\cos\delta_1\cos\delta_2-\tfrac12(q_1/q_2+q_2/q_1)\sin\delta_1\sin\delta_2$}
The two-layer cell <seismic layer transfer matrix> has reciprocal <eigenvalues> because its <determinant> is one. Writing them as $e^{\pm iKd}$ gives the displayed half-trace relation. At a real <frequency>, an absolute half-trace exceeding one forces an exponentially growing/decaying pair instead of propagating cell modes. A long finite stack then suppresses transmission in this stop band, without dissipating <energy>: the missing transmitted flux is reflected. Quarter-wave layers of unequal <seismic impedance> give half-trace $-\tfrac12(q_1/q_2+q_2/q_1)<-1$.
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