Periodic seismic multilayer stop band ($\cos (Kd)=\cos {\delta }_1\cos {\delta }_2-\frac{1}{2}(q_1/q_2+q_2/q_1)\sin {\delta }_1\sin {\delta }_2$)

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 $-\frac{1}{2}(q_1/q_2+q_2/q_1)<-1$.