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... Here my dilemma. The stiffness wave satisfies the
differential wave equation, as demonstrated by John ....
Nope. If you re-read my derivation you'll see it explicitly applies
to ***longitudinal*** waves driven by compression. It isn't even
close to right for transverse waves driven by stiffness.
More generally, we have that the extension of the tiny spring is
extension = (dy/dx) delta_x
and the force produced thereby is (using Hooke's law)
F(x) = -(dy/dx) delta_x / (s delta_x)
where
s = compliance per unit length.