Solution
= Solution
Seek an SH displacement
$$
u=(0,Y(z),0)e^{i(\kappa x-\omega t)}.
$$
The S-wave equation gives
$$
Y''+q^2Y=0,\qquad
q^2=\frac{\omega^2}{c_s^2}-\kappa^2.
$$
Rigidity at $z=0$ gives $Y(0)=0$, while zero tangential traction at the free surface gives
$\mu Y'(h)=0$. Thus
$$
Y(z)=C\sin(qz),\qquad \cos(qh)=0,
$$
so
$$
q=q_n=\frac{(2n+1)\pi}{2h},\qquad
\boxed{\omega^2=c_s^2(\kappa^2+q_n^2)}.
$$
These are the <guided SH modes between rigid and free planes>.