= Rayleigh secular equation
{c}
{title2=$(2-s)^2=4\sqrt{1-s}\sqrt{1-\eta s}$}
For a <Rayleigh wave> at a free surface of a homogeneous isotropic elastic half-space, define $s=c^2/c_T^2$ and $\eta=c_T^2/c_L^2$. Cancelling the two surface <tractions> between the constituents with <Rayleigh wave decay exponents> gives the displayed equation with $0<s<1$. Equivalently $(b_T^2+k^2)^2=4k^2b_Lb_T$. Squaring gives a factor $s$ times $P(s)=s^3-8s^2+(24-16\eta)s-16(1-\eta)$. For stable <Lamé parameters>, $0<\eta<3/4$, so $P(0)<0$, $P(1)=1$, and $P''(s)=6s-16<0$ on $(0,1)$. Its concavity ensures exactly one crossing in this interval. Both sides of the unsquared equation are positive, so this root gives the physical speed. The zero root is a degenerate static cancellation, not a propagating <Rayleigh wave>.
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