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Guided SH modes between rigid and free planes

For a layer

0 < z < h

with rigid lower boundary and traction-free upper boundary, an SH mode

u_{y} = Y (z) e^{i (κ x - ω t)}

has

Y (z) = sin (q_{n} z), q_{n} = \frac{( 2 n + 1 ) π}{2 h},

and dispersion

ω^{2} = c_{s}^{2} (κ^{2} + q_{n}^{2}) .

Its cutoff is

ω_{n} = c_{s} q_{n}

, its phase speed is

ω / κ

, and its group speed is

c_{s}^{2} κ / ω

.

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