= Compatibility condition for pure nonlinear transverse waves
{title2=$\partial_qP_{11}(1,q,0)=0$}
For a plane material-coordinate <elastic wave> with $u=(0,w(X_1,t),0)$, set $q=w_{,X_1}$. The longitudinal equation is $0=\partial_{X_1}P_{11}(1,q,0)$. Arbitrary nonuniform pure transverse profiles thus require the displayed constitutive compatibility, which does not follow from <isotropy> alone. If it holds and $\tau'(q)>0$, the shear equation closes as $\rho_0w_{tt}=\partial_{X_1}\tau(q)$ with speed $c(q)^2=\tau'(q)/\rho_0$.
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