= Strict mode count for a rigid-backed elastic layer
{title2=$N=\lceil A/\pi-1/2\rceil$}
For $A,\epsilon>0$, consider $\tan\theta=-\epsilon\theta/\sqrt{A^2-\theta^2}$ on $0<\theta<A$. On each negative branch of the <tangent>, the left side increases strictly, while the right side decreases strictly. A unique intersection exists precisely when the branch starts at $p_n=(n-1/2)\pi<A$, since the difference increases from negative infinity to a positive value or positive infinity. Consequently the number of trapped <normal modes> is $\lceil A/\pi-1/2\rceil$. The <floor function> expression $\lfloor A/\pi+1/2\rfloor$ agrees except at $A=p_n$, where it counts an inadmissible cutoff state with zero exterior <exponential decay> rate.
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