 Compatibility condition for pure nonlinear transverse waves

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 ${\tau }^{\prime }(q)>0$, the shear equation closes as ${\rho }_0{w}_{tt}={\partial }_{X_1}\tau (q)$ with speed $c(q{)}^2={\tau }^{\prime }(q)/{\rho }_0$.